Preprints & selected publications:

2011 2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996  1994  1992  1980-1990

 

2011

 

A.N. Gorban, G.S.Yablonsky

Extended detailed balance for systems with irreversible reactions, Chemical Engineering Science 66 (2011) 5388–5399.

The principle of detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process. For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. In this case, the principle of detailed balance cannot be applied directly. We represent irreversible reactions as limits of reversible steps and obtain the principle of detailed balance for complex mechanisms with some irreversible elementary processes. We proved two consequences of the detailed balance for these mechanisms: the structural condition and the algebraic condition that form together the extended form of detailed balance. The algebraic condition is the principle of detailed balance for the reversible part. The structural condition is: the convex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span of the stoichiometric vectors of the reversible reaction. Physically, this means that the irreversible reactions cannot be included in oriented pathways.

The systems with the extended form of detailed balance are also the limits of the reversible systems with detailed balance when some of the equilibrium concentrations (or activities) tend to zero. Surprisingly, the structure of the limit reaction mechanism crucially depends on the relative speeds of this tendency to zero.

 

A.N. Gorban, H.P. Sargsyan and H.A. Wahab
Quasichemical Models of Multicomponent Nonlinear Diffusion, Mathematical Modelling of Natural PhenomenaVolume 6 / Issue 05, (2011), 184−262.
Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicomponent diffusion equations should be ordered and special tools are needed to provide the systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant interaction between components. We develop an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can be included into the general thermodynamic framework, and prove the corresponding dissipation inequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell–jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation

 

A. Gorban and S. Petrovskii

Collective dynamics: when one plus one does not make twoMathematical Medicine and Biology (2011) 28, 85−88.

A brief introduction into the interdisciplinary field of collective dynamics is given, followed by an overview of ‘Mathematical Models of Collective Dynamics in Biology and Evolution’ (University of Leicester, 11–13 May 2009). Collective dynamics—understood as the dynamics arising from the interplay between the constituting elementary argents or parts of a more complex system—has been one of the main paradigms of the natural sciences over the last several decades.

 

A.N. Gorban and M. Shahzad

The Michaelis-Menten-Stueckelberg Theorem. Entropy 201113, 966-1019

We study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE). Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the “rarefied gas” limit or the “molecular chaos” hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density (low density of elementary events). This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann H-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.

 

G. S. Yablonsky, A. N. Gorban, D. Constales, V. V. Galvita and G. B. Marin
Reciprocal relations between kinetic curves, EPL, 93 (2011) 20004.

We study coupled irreversible processes. For linear or linearized kinetics with microreversibilityDescription: \dot{x}=Kx , the kinetic operator K is symmetric in the entropic inner product. This form of Onsager's reciprocal relations implies that the shift in time, exp(Kt), is also a symmetric operator. This generates the reciprocity relations between the kinetic curves. For example, for the Master equation, if we start the process from the i-th pure state and measure the probability pj(t) of the j-th state (ji), and, similarly, measure pi(t) for the process, which starts at the j-th pure state, then the ratio of these two probabilities pj(t)/pi(t) is constant in time and coincides with the ratio of the equilibrium probabilities. We study similar and more general reciprocal relations between the kinetic curves. The experimental evidence provided as an example is from the reversible water gas shift reaction over iron oxide catalyst. The experimental data are obtained using Temporal Analysis of Products (TAP) pulse-response studies. These offer excellent confirmation within the experimental error.

 

A.N. Gorban, D. Roose

PrefaceIn: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. V-VI.

A mathematical model is an intellectual device that works. …

 

 

A.N. Gorban

Self-simplification in Darwin’s Systems, In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. 311-344

We prove that a non-linear kinetic system with conservation of supports for distributions has generically limit distributions with final support only. The conservation of support has a biological interpretation: inheritance. We call systems with inheritance “Darwin’s systems”. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. The finite dimension of limit distributions demonstrates effects of natural selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t → ∞. The drift equations for peak motion are obtained. They describe the asymptotic layer near the omega-limit distributions with finite support .

 

D.J. Packwood, J. Levesley, and A.N. Gorban

Time step expansions and the invariant manifold approach to lattice Boltzmann models, In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. 169-206.

The classical method for deriving the macroscopic dynamics of a lattice Boltzmann system is to use a combination of different approximations and expansions. Usually a Chapman-Enskog analysis is performed, either on the continuous Boltzmann system, or its discrete velocity counterpart. Separately a discrete time approximation is introduced to the discrete velocity Boltzmann system, to achieve a practically useful approximation to the continuous system, for use in computation. Thereafter, with some additional arguments, the dynamics of the Chapman-Enskog expansion are linked to the discrete time system to produce the dynamics of the completely discrete scheme. In this paper we put forward a different route to the macroscopic dynamics. We begin with the system discrete in both velocity space and time. We hypothesize that the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes, give rise to a slow invariant manifold. We perform a time step expansion of the discrete time dynamics using the invariance of the manifold. Finally we calculate the dynamics arising from this system. By choosing the fully discrete scheme as a starting point we avoid mixing approximations and arrive at a general form of the microscopic dynamics up to the second order in the time step. We calculate the macroscopic dynamics of two commonly used lattice schemes up to the first order, and hence find the precise form of the deviation from the Navier-Stokes equations in the dissipative term, arising from the discretization of velocity space.

Finally we perform a short wave perturbation on the dynamics of these example systems, to find the necessary conditions for their stability.

 

A.N. Gorban

Kinetic path summation, multi-sheeted extension of master equation, and evaluation of ergodicity coefficientPhysica A 390 (2011) 1009–1025.

We study the master equation with time-dependent coefficients, a linear kinetic equation for the Markov chains or for the monomolecular chemical kinetics. For the solution of this equation a path summation formula is proved. This formula represents the solution as a sum of solutions for simple kinetic schemes (kinetic paths), which are available in explicit analytical form. The relaxation rate is studied and a family of estimates for the relaxation time and the ergodicity coefficient is developed. To calculate the estimates we introduce the multi-sheeted extensions of the initial kinetics. This approach allows us to exploit the internal (‘‘micro’’) structure of the extended kinetics without perturbation of the base kinetics.

 

2010

 

A.N. Gorban, L.I. Pokidysheva,·E,V. Smirnova, T.A. Tyukina.
Law of the Minimum ParadoxesBull Math Biol 73(9) (2011), 2013-2044; Online first 19.11.2010,

The “Law of the Minimum” states that growth is controlled by the scarcest resource (limiting factor). This concept was originally applied to plant or crop growth (Justus von Liebig, 1840) and quantitatively supported by many experiments. Some generalizations based on more complicated “dose-response” curves were proposed. Violations of this law in natural and experimental ecosystems were also reported. We study models of adaptation in ensembles of similar organisms under load of environmental factors and prove that violation of Liebig’s law follows from adaptation effects. If the fitness of an organism in a fixed environment satisfies the Law of the Minimum then adaptation equalizes the pressure of essential factors and, therefore, acts against the Liebig’s law. This is the the Law of the Minimum paradox: if for a randomly chosen pair “organism–environment” the Law of the Minimum typically holds, then in a well-adapted system, we have to expect violations of this law.

For the opposite interaction of factors (a synergistic system of factors which amplify each other), adaptation leads from factor equivalence to limitations by a smaller number of factors.

For analysis of adaptation, we develop a system of models based on Selye’s idea of the universal adaptation resource (adaptation energy). These models predict that under the load of an environmental factor a population separates into two groups (phases): a less correlated, well adapted group and a highly correlated group with a larger variance of attributes, which experiences problems with adaptation. Some empirical data are presented and evidences of interdisciplinary applications to econometrics are discussed.

 

A.N. Gorban, E.V. Smirnova, T.A. Tyukina
Correlations, risk and crisis: From physiology to finance, Physica A, Vol. 389, Issue 16, 2010, 3193-3217. Number 9 in the Top Hottest Articles in the Journal, April to June 2010

We study the dynamics of correlation and variance in systems under the load of environmental factors. A universal effect in ensembles of similar systems under the load of similar factors is described: in crisis, typically, even before obvious symptoms of crisis appear, correlation increases, and, at the same time, variance (and volatility) increases too. This effect is supported by many experiments and observations of groups of humans, mice, trees, grassy plants, and on financial time series.

A general approach to the explanation of the effect through dynamics of individual adaptation of similar non-interactive individuals to a similar system of external factors is developed. Qualitatively, this approach follows Selye’s idea about adaptation energy.

 

A.N. Gorban

Kinetic Path Summation, Multi--Sheeted Extension of Master Equation, and Evaluation of Ergodicity Coefficient, arXiv:1006.4128 [physics.comp-ph].

 We study the Master equation with time--dependent coefficients, a linear kinetic equation for the Markov chains or for the monomolecular chemical kinetics. For the solution of this equation a paths summation formula is proved. This formula represents the solution as a sum of solutions for simple kinetic schemes (kinetic paths), which are available in explicit analytical form. The relaxation rate is studied and a family of estimates for the relaxation time and the ergodicity coefficient is developed. To calculate the estimates we introduce the multi--sheeted extensions} of the initial kinetics. This approach allows us to exploit the internal ("micro")structure of the extended kinetics without perturbation of the base kinetics.

 

A. N. Gorban, A. Zinovyev.

Principal manifolds and graphs in practice: from molecular biology to dynamical systemsInternational Journal of Neural Systems, Vol. 20, No. 3 (2010) 219–232.

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen’s self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative  political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.

 

E. Chiavazzo, I.V. Karlin, A.N. Gorban, K. Boulouchos,

Coupling of the model reduction technique with the lattice Boltzmann methodCombustion and Flame 157 (2010) 1833–1849 Description: http://www.sciencedirect.com/scidirimg/clear.gifdoi:10.1016/j.combustflame.2010.06.009

A new framework of simulation of reactive flows is proposed based on a coupling between accurate reduced reaction mechanism and the lattice Boltzmann representation of the flow phenomena. The model reduction is developed in the setting of slow invariant manifold construction, and the simplest lattice Boltzmann equation is used in order to work out the procedure of coupling of the reduced model with the flow solver. Practical details of constructing slow invariant manifolds of a reaction system under various thermodynamic conditions are reported. The proposed method is validated with the two-dimensional simulation of a premixed counterflow flame in the hydrogen-air mixture.

 

Gorban A.N., Gorban P.A., Judge G.

Entropy: The Markov Ordering ApproachEntropy. 2010; 12(5):1145-1193. GorbanGorbanJudgeEntropy2010.pdf

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant “additivity” properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally “most random” distributions.

 

A. N. Gorban and V. M. Cheresiz,

Slow Relaxations and Bifurcations of the Limit Sets of Dynamical Systems. I. Bifurcations of Limit Sets, Journal of Applied and Industrial Mathematics2010, Vol. 4, No. 1, pp. 54–64.

We consider one-parameter semigroups of homeomorphisms depending continuously on the parameters. We study the phenomenon of slow relaxation that consists in anomalously slow motion to the limit sets. We investigate the connection between slow relaxations and bifurcations of limit sets and other singularities of the dynamics. The statements of some of the problems stem from mathematical chemistry.

 

A. N. Gorban and V. M. Cheresiz,

Slow Relaxations and Bifurcations of the Limit Sets of Dynamical Systems. II. Slow Relaxations of a Family of Semiflows, Journal of Applied and Industrial Mathematics2010, Vol. 4, No. 2, pp. 182–190.

We propose a number of approaches to the notion of the relaxation time of a dynamical system which are motivated by the problems of chemical kinetics, give exact mathematical definitions of slow relaxations, study their possible reasons, among which an important role is played by bifurcations of limit sets.

 

E. Chiavazzo, I.V. Karlin, and A.N. Gorban,

The Role of Thermodynamics in Model Reduction when Using Invariant GridsCommun. Comput. Phys., Vol. 8, No. 4 (2010), pp. 701-734.

In the present work, we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of Invariant Grids (MIG). To this end, reduced models (invariant grids) are obtained by refining initial approximations of slow invariant manifolds, and used for integrating smaller and less stiff systems of equations capable to recover the detailed description with high accuracy. Moreover, we clarify the role played by thermodynamics in model reduction, and carry out a comparison between detailed and reduced solutions for a model hydrogen oxidation reaction.

 

Andrei Zinovyev, Nadya Morozova, Nora Nonne, Emmanuel Barillot, Annick Harel-Bellan, Alexander N Gorban
Dynamical modeling of microRNA action on the protein translation process,
  BMC Systems Biology 2010, 4:13 (24 February 2010)

Description: http://www.biomedcentral.com/bmcimages/browse/highlyaccessed.gif

Background

Protein translation is a multistep process which can be represented as a cascade of biochemical reactions (initiation, ribosome assembly, elongation, etc.), the rate of which can be regulated by small non-coding microRNAs through multiple mechanisms. It remains unclear what mechanisms of microRNA action are the most dominant: moreover, many experimental reports deliver controversal messages on what is the concrete mechanism actually observed in the experiment. Nissan and Parker have recently demonstrated that it might be impossible to distinguish alternative biological hypotheses using the steady state data on the rate of protein synthesis. For their analysis they used two simple kinetic models of protein translation.

Results

In contrary to the study by Nissan and Parker, we show that dynamical data allow to discriminate some of the mechanisms of microRNA action. We demonstrate this using the same models as developed by Nissan and Parker for the sake of comparison but the methods developed (asymptotology of biochemical networks) can be used for other models. We formulate a hypothesis that the effect of microRNA action is measurable and observable only if it affects the dominant system (generalization of the limiting step notion for complex networks) of the protein translation machinery. The dominant system can vary in different experimental conditions that can partially explain the existing controversy of some of the experimental data.

Conclusions

Our analysis of the transient protein translation dynamics shows that it gives enough information to verify or reject a hypothesis about a particular molecular mechanism of microRNA action on protein translation. For multiscale systems only that action of microRNA is distinguishable which affects the parameters of dominant system (critical parameters), or changes the dominant system itself. Dominant systems generalize and further develop the old and very popular idea of limiting step. Algorithms for identifying dominant systems in multiscale kinetic models are straightforward but not trivial and depend only on the ordering of the model parameters but not on their concrete values. Asymptotic approach to kinetic models allows to put in order diverse experimental observations in complex situations when many alternative hypotheses co-exist.

 

A. N. GorbanO. RadulescuA. Y. Zinovyev
Asymptotology of chemical reaction networks, Chemical Engineering Science 65 (2010) 2310–2324 GorbRadZinCES2010Rev.pdf

The concept of the limiting step is extended to the asymptotology of multiscale reaction networks. Complete theory for linear networks with well separated reaction rate constants is developed. We present algorithms for explicit approximations of eigenvalues and eigenvectors of kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms is demonstrated on simple examples. Application of algorithms to nonlinear systems is discussed.

 

2009

A.N. GorbanE.V. SmirnovaT.A. Tyukina
General Laws of Adaptation to Environmental Factors: from Ecological Stress to Financial Crisis. Math. Model. Nat. Phenom. Vol. 4, No. 6, 2009, pp. 1-53

We study ensembles of similar systems under load of environmental factors. The phenomenon of adaptation has similar properties for systems of different nature. Typically, when the load increases above some threshold, then the adapting systems become more different (variance increases), but the correlation increases too. If the stress continues to increase then the second threshold appears: the correlation achieves maximal value, and start to decrease, but the variance continue to increase. In many applications this second threshold is a signal of approaching of fatal outcome. This effect is supported by many experiments and observation of groups of humans, mice, trees, grassy plants, and on financial time series.

A general approach to explanation of the effect through dynamics of adaptation is developed. H. Selye introduced “adaptation energy” for explanation of adaptation phenomena. We formalize this approach in factors – resource models and develop hierarchy of models of adaptation. Different organization of interaction between factors (Liebig’s versus synergistic systems) lead to different adaptation dynamics. This gives an explanation to qualitatively different dynamics of correlation under different types of load and to some deviation from the typical reaction to stress. In addition to the “quasistatic” optimization factor – resource models, dynamical models of adaptation are developed, and a simple model (three variables) for adaptation to one factor load is formulated explicitly.

 

A. N. GorbanA. Y. Zinovyev
Principal Graphs and Manifolds, Chapter 2 in: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods, and Techniques, Emilio Soria Olivas et al. (eds), IGI Global, 
Hershey, PA, USA, 2009, pp. 28-59.

In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found ‘lines and planes of closest fit to system of points’. The famous k-means algorithm solves the approximation problem too, but by finite sets instead of lines and planes. This chapter gives a brief practical introduction into the methods of construction of general principal objects (i.e., objects embedded in the ‘middle’ of the multidimensional data set). As a basis, the unifying framework of mean squared distance approximation of finite datasets is selected. Principal graphs and manifolds are constructed as generalisations of principal components and k-means principal points. For this purpose, the family of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal graphs with controlled complexity is based on the graph grammar approach.

 

A.N. Gorban, L.I. PokidyshevaE.V. SmirnovaT.A. Tyukina 
Law of the Minimum Paradoxes, e-print http://arxiv.org/abs/0907.1965 
The "law of the minimum" states that growth is controlled by the scarcest resource (limiting factor) (Justus von Liebig (1840)). This concept was originally applied to plant or crop growth and quantitatively supported by many experiments. Some generalizations based on more complicated "dose-response" curves were proposed. Violations of this law in natural and experimental ecosystems were also reported. We study models of adaptation in ensembles of similar organisms under load of environmental factors and prove that violation of the Liebig law follows from adaptation effects. If the fitness of an organism in fixed environment satisfies the law of the minimum then adaptation equalizes the pressure of essential factors and therefore acts against the law. This is the the law of the minimum paradox: if for a randomly chosen pair "organism--environment" the law of the minimum typically holds, then, in a well-adapted system, we have to expect violations of this law. For the opposite interaction of factors (a synergistic system of factors which amplify each other) adaptation leads from factor equivalence to limitations by a smaller number of factors. For analysis of adaptation we develop a system of models based on Selye's idea of the universal adaptation resource (adaptation energy). These models predict that under the load of an environmental factor a population separates into two groups (phases): a less correlated, well adapted group and a highly correlated group with a larger variance of attributes, which experiences problems with adaptation. Some empirical data are presented and some evidences of interdisciplinary applications to econometrics are discussed.

 

E. Chiavazzo, I. V. Karlin, A. N. Gorban and K Boulouchos,

Combustion simulation via lattice Boltzmann and reduced chemical kinetics, J. Stat. Mech. (2009) P06013, MIG-LB_StatMech_2009.pdf

We present and validate a methodology for coupling reduced models of detailed combustion mechanisms within the lattice Boltzmann framework. A detailed mechanism (9 species, 21 elementary reactions) for modeling reacting mixtures of air and hydrogen is considered and reduced using the method of invariant grids (MIG). In particular, a 2D quasi-equilibrium grid is constructed, further refined via the MIG method, stored in the form of tables and used to simulate a 1D flame propagating freely through a homogeneous premixed mixture. Comparisons between the detailed and reduced models show that the technique presented enables one to achieve a remarkable speedup in the computations with excellent accuracy.

 

A. N. GorbanE. V. SmirnovaT. A. Tyukina,

Correlations, Risk and Crisis: from Physiology to Finance, e-print: http://arxiv.org/abs/0905.0129. Available at SSRN: http://ssrn.com/abstract=1397677.

We study the dynamics of correlation and variance in systems under the load of environmental factors. A universal effect in ensembles of similar systems under load of similar factors is described: in crisis, typically, even before obvious symptoms of crisis appear, correlation increases, and, at the same time, variance (and volatility) increases too. After the crisis achieves its bottom, it can develop into two directions: recovering (both correlations and variance decrease) or fatal catastrophe (correlations decrease, but variance not). This effect is supported by many experiments and observation of groups of humans, mice, trees, grassy plants, and on financial time series. A general approach to explanation of the effect through dynamics of adaptation is developed. Different organization of interaction between factors (Liebig's versus synergistic systems) lead to different adaptation dynamics. This gives an explanation to qualitatively different dynamics of correlation under different types of load.

 

A. N. GorbanO. RadulescuA. Y. Zinovyev,

Limitation and Asymptotology of Chemical Reaction Networks, e-print: http://arxiv.org/abs/0903.5072

The concept of the limiting step is extended to the asymptotology of multiscale reaction networks. Complete theory for linear networks with well separated reaction rate constants is developed. We present algorithms for explicit approximations of eigenvalues and eigenvectors of kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms is demonstrated on simple examples. Application of algorithms to nonlinear systems is discussed.

 

A. GorbanI. TyukinE. SteurH. Nijmeijer

Positive Invariance Lemmas for Control Problems with Convergence to Lyapunov-unstable Sets, e-print http://arxiv.org/abs/0901.3577

We provide Lyapunov-like characterizations of positive invariance, boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. The systems of this class comprise of a stable part coupled with a one-dimensional unstable or critically stable subsystem. Examples of these systems appear in the problems of nonlinear output regulation, parameter estimation and adaptive control. We demonstrate that, for a large class of systems with unstable equilibria and solutions that might escape to infinity in finite time, it is always possible to determine simple criteria for positive invariance and boundedness of the system's nontrivial solutions. Conversely, it is possible to characterize domains of initial conditions that lead to solutions escaping from the origin. In contrast to other works addressing convergence issues in unstable systems, our results do not rely on the availability of input-output gains or contraction rates that are usually required for the stable compartment.

 

2008

A. N. GorbanA. Y. Zinovyev

Principal Graphs and Manifolds, e-print: http://arxiv.org/abs/0809.0490

In many physical statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found "lines and planes of closest fit to system of points". The famous k-means algorithm solves the approximation problem too, but by finite sets instead of lines and planes. This chapter gives a brief practical introduction into the methods of construction of general principal objects, i.e. objects embedded in the "middle" of the multidimensional data set. As a basis, the unifying framework of mean squared distance approximation of finite datasets is selected. Principal graphs and manifolds are constructed as generalisations of principal components and k-means principal points. For this purpose, the family of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal graphs with controlled complexity is based on the graph grammar approach.

 

Ovidiu RadulescuAlexander N GorbanAndrei Zinovyev, and Alain Lilienbaum

Robust simplifications of multiscale biochemical networks, BMC Systems Biology 2008, 2:86 doi:10.1186/1752-0509-2-86

Description: http://www.biomedcentral.com/bmcimages/browse/highlyaccessed.gifThe most accessed paper in BMC Systems Biology in November 2008

Background

Cellular processes such as metabolism, decision making in development and differentiation, signalling, etc., can be modeled as large networks of biochemical reactions. In order to understand the functioning of these systems, there is a strong need for general model reduction techniques allowing to simplify models without loosing their main properties. In systems biology we also need to compare models or to couple them as parts of larger models. In these situations reduction to a common level of complexity is needed.

Results

We propose a systematic treatment of model reduction of multiscale biochemical networks. First, we consider linear kinetic models, which appear as "pseudo-monomolecular" subsystems of multiscale nonlinear reaction networks. For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in (Gorban and Radulescu 2008). Second, for non-linear systems we develop an algorithm based on dominant solutions of quasi-stationarity equations. For oscillating systems, quasi-stationarity and averaging are combined to eliminate time scales much faster and much slower than the period of the oscillations. In all cases, we obtain robust simplifications and also identify the critical parameters of the model. The methods are demonstrated for simple examples and for a more complex model of NFkB pathway.

Conclusions

Our approach allows critical parameter identification and produces hierarchies of models. Hierarchical modeling is important in "middle-out" approaches when there is need to zoom in and out several levels of complexity. Critical parameter identification is an important issue in systems biology with potential applications to biological control and therapeutics. Our approach also deals naturally with the presence of multiple time scales, which is a general property of systems biology models.

 

 

A.N. Gorban and O. Radulescu,

Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited, Advances in Chemical Engineering 34, 103-173. GorbanRadulescuAdvChemEng2008.pdf Local copy

 

The concept of the limiting step gives the limit simplification: the whole network behaves as a single step. This is the most popular approach for model simplification in chemical kinetics. However, in its elementary form this idea is applicable only to the simplest linear cycles in steady states. For simple cycles the nonstationary behavior is also limited by a single step, but not the same step that limits the stationary rate. In this chapter, we develop a general theory of static and dynamic limitation for all linear multiscale networks. Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ‘‘cycles surgery’’ for reaction graphs. New estimates of eigenvectors for diagonally dominant matrices are used.

 

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (‘‘modes’’) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules

that it is possible to solve the kinetic equation for every module in the explicit form. All such ‘‘solvable’’ networks are described. The obtained multiscale approximations, that we call ‘‘dominant systems’’ are computationally cheap and robust. These dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning.

 

 

A. N. Gorban,

Selection Theorem for Systems with Inheritance, Math. Model. Nat. Phenom., Vol. 2, No. 4, 2007, pp. 1-45. GOtborMMNP2(4)2007.pdf  Local copy. The original publication is available at www.edpsciences.org

 

The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of natural selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t→∞. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of self-synchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinite-dimensional space is discussed, and the notion of “completely thin” sets is introduced.

 

 

R. A. Brownlee, A. N. Gorban, and J. Levesley,
Nonequilibrium entropy limiters in lattice Boltzmann methods, Physica A: Statistical Mechanics and its Applications
Volume 387, Issues 2-3, 15 January 2008, Pages 385-406  BrownGorbLevPhysA2007FinFin.pdf

We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity — nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy “trimming”) and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimation of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers between 2000 and 7500 on a coarse 100×100 grid. All limiter constructions are applicable both for entropic and for non-entropic equilibria.

2007

A.N. Gorban and O. Radulescu

Dynamical robustness of biological networks with hierarchical distribution of time scales, IET Syst. Biol., 2007, 1, (4), pp. 238–246 Gorban2007IEESystemsBiology.pdf

Concepts of distributed robustness and r-robustness proposed by biologists to explain a variety of stability phenomena in molecular biology are analysed. Then, the robustness of the relaxation time using a chemical reaction description of genetic and signalling networks is discussed. First, the following result for linear networks is obtained: for large multiscale systems with hierarchical distribution of time scales, the variance of the inverse relaxation time (as well as the variance of the stationary rate) is much lower than the variance of the separate constants. Moreover, it can tend to 0 faster than 1/n, where n is the number of reactions. Similar phenomena are valid in the nonlinear case as well. As a numerical illustration, a model of signalling network is used for the important transcription factor NFkB.

 

A.N. Gorban and  A.Y. Zinovyev
The Mystery of Two Straight Lines in Bacterial Genome Statistics, Bulletin of Mathematical Biology (2007) DOI 10.1007/s11538-007-9229-6 (Online First) GorbanZinovyev2007BMB1.pdf
In special coordinates (codon position-specific nucleotide frequencies), bacterial genomes form two straight lines in 9-dimensional space: one line for eubacterial genomes, another for archaeal genomes. All the 348 distinct bacterial genomes available in Genbank in April 2007, belong to these lines with high accuracy. The main challenge now is to explain the observed high accuracy. The new phenomenon of complementary symmetry for codon position-specific nucleotide frequencies is observed. The results of analysis of several codon usage models are presented.We demonstrate that the mean-field approximation, which is also known as context-free, or complete independence model, or Segre variety, can serve as a reasonable approximation to the real codon usage. The first two principal components of codon usage correlate strongly with genomic G
+C content and the optimal growth temperature, respectively. The variation of codon usage along the third component is related to the curvature of the mean-field approximation. First three eigenvalues in codon usage PCA explain 59.1%, 7.8% and 4.7% of variation. The eubacterial and archaeal genomes codon usage is clearly distributed along two third order curves with genomic G+C content as a parameter.

 

A.N. Gorban, O. Radulescu
Dynamic and static limitation in reaction networks, revisited,
http://arxiv.org/abs/physics/0703278  [physics.chem-ph] GorRadLimarXiv0703278v2.pdf
The concept of limiting step gives the limit simplification: the whole network behaves as a single step. This is the most popular approach for model simplification in chemical kinetics. However, in its simplest form this idea is applicable only to the simplest linear cycles in steady states. For such the simplest cycles the nonstationary behaviour is also limited by a single step, but not the same step that limits the stationary rate. In this paper, we develop a general theory of static and dynamic limitation for all linear multiscale networks, not only for simple cycles. Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ``cycles surgery" for reaction graphs. New estimates of eigenvectors for diagonally dominant matrices are used.

Multiscale ensembles of reaction networks with well separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (``modes") is presented. In particular, we proved that for systems with well separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose to select such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such ``solvable" networks are described. The obtained multiscale approximations that we call ``dominant systems" are computationally cheap and robust. These dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning.

 

R.A. Brownlee, A.N. Gorban, J. Levesley,
Nonequilibrium entropy limiters in lattice Boltzmann methods,
arXiv:0704.0043v1 [cond-mat.stat-mech] BrowGorLevLimitersArXiv.pdf

We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity - nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy "trimming") and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimate of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100*100 grid. All limiter constructions are applicable for both entropic and non-entropic quasiequilibria.

R. A. Brownlee, A. N. Gorban, and J. Levesley,
Stability and stabilization of the lattice Boltzmann method, Phys. Rev. E 75, 036711 (2007) (17 pages) BGJPhyRev2007.pdf
We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager-Gross-Krook method (LBGK). The LBGK scheme can be recognized as a discrete dynamical system generated by free flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modeling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface—the invariant film (up to second order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free flight. The main instability mechanisms are identified. The simplest recipes for stabilization add no artificial dissipation (up to second order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blowup. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a one-dimensional (1D) shock tube and the unsteady 2D flow around a square cylinder up to Reynolds number Re~20,000.

 

E. Chiavazzo, A.N. Gorban, and I.V. Karlin,

Comparison of Invariant Manifolds for Model Reduction in Chemical Kinetics, Commun. Comput. Phys. Vol. 2, No. 5 (2007), pp. 964-992 CiCP2007vol2_n5_p964.pdf

A modern approach to model reduction in chemical kinetics is often based on the notion of slow invariant manifold. The goal of this paper is to give a comparison of various methods of construction of slow invariant manifolds using a simple Michaelis-Menten catalytic reaction. We explore a recently introduced Method of Invariant Grids (MIG) for iteratively solving the invariance equation. Various initial approximations for the grid are considered such as Quasi Equilibrium Manifold, Spectral Quasi Equilibrium Manifold, Intrinsic Low Dimensional Manifold and Symmetric Entropic Intrinsic Low Dimensional Manifold. Slow invariant manifold was also computed using the Computational Singular Perturbation (CSP) method. A comparison between MIG and CSP is also reported.

 

A.N. Gorban, N.R. Sumner, and A.Y. Zinovyev,
Topological grammars for data approximation, Applied Mathematics Letters Volume 20, Issue 4  (2007),  382-386 GorSummnZinAML2006.pdf
A method of topological grammars is proposed for multidimensional data approximation. For data with complex topology we define a principal cubic complex of low dimension and given complexity that gives the best approximation for the dataset. This complex is a generalization of linear and non-linear principal manifolds and includes them as particular cases. The problem of optimal principal complex construction is transformed into a series of minimization problems for quadratic functionals. These quadratic functionals have a physically transparent interpretation in terms of elastic energy. For the energy computation, the whole complex is represented as a system of nodes and springs. Topologically, the principal complex is a product of one-dimensional continuums (represented by graphs), and the grammars describe how these continuums transform during the process of optimal complex construction. This factorization of the whole process onto one-dimensional transformations using minimization of quadratic energy functionals allows us to construct efficient algorithms.

 

A.N. Gorban,
Order–disorder separation: Geometric revision, Physica A Volume 374, Issue 1 , 15 January 2007, Pages 85-102 GorPhysA2006Order.pdf
After Boltzmann and Gibbs, the notion of disorder in statistical physics relates to ensembles, not to individual states. This disorder is measured by the logarithm of ensemble volume, the entropy. But recent results about measure concentration effects in analysis and geometry allow us to return from the ensemble-based point of view to a state-based one, at least, partially. In this paper, the order–disorder problem is represented as a problem of relation between distance and measure. The effect of strong order–disorder separation for multiparticle systems is described: the phase space could be divided into two subsets, one of them (set of disordered states) has almost zero diameter, the second one has almost zero measure. The symmetry with respect to permutations of particles is responsible for this type of concentration. Dynamics of systems with strong order–disorder separation has high average acceleration squared, which can be interpreted as evolution through a series of collisions (acceleration-dominated dynamics). The time arrow direction from order to disorder follows from the strong order–disorder separation. But, inverse, for systems in space of symmetric configurations with “sticky boundaries” the way back from disorder to order is typical (Natural selection). Recommendations for mining of molecular dynamics results are also presented.

2006

Ovidiu Radulescu, Alexander N. Gorban, Sergei Vakulenko, Andrei Zinovyev
Hierarchies and modules in complex biological systems,
In: Proceedings of European Conference on Complex Systems (paper ECCS06-114), Oxford, UK, September 2006. http://complexsystems.lri.fr/FinalReview/FILES/PDF/p114.pdf or OxfordHiModP114.pdf
We review several mathematical methods allowing to identify modules and hierarchies with several levels of complexity in biological systems. These methods are based either on the properties of the input-output characteristic of the modules or on global properties of the dynamics such as the distribution of timescales or the stratification of attractors with variable dimension. We also discuss the consequences of the hierarchical structure on the robustness of biological processes. Stratified attractors lead to Waddington's type canalization effects. Successive application of the many to one mapping relating parameters of different levels in an hierarchy of models (analogue to the renormalization operation from statistical mechanics) leads to concentration and robustness of those properties that are common to many levels of complexity. Examples such as the response of the transcription factor NF·B to signalling, and the segmentation patterns in the development of Drosophila are used as illustrations of the theoretical ideas.

 

R. A. Brownlee, A. N. Gorban, and J. Levesley,
Stabilization of the lattice Boltzmann method using the Ehrenfests' coarse-graining idea, Phys. Rev. E 74, 037703 (2006) RobBrowGorbLeveslPRE2006.pdf
The lattice Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modeling complex fluid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple technique to stabilize the LBM by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests' steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests' coarse-graining idea.

A.N. Gorban, B.M. Kaganovich, S.P. Filippov, A.V. Keiko, V.A. Shamansky, I.A. Shirkalin,
Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibria,
Springer, Berlin-Heidelberg-New York, 2006.

Model Reduction and Coarse--Graining Approaches for Multiscale Phenomena,
Ed. by Alexander N. Gorban, Nikolaos  Kazantzis, Ioannis G. Kevrekidis, Hans Christian Öttinger, Constantinos Theodoropoulos , Springer, Berlin-Heidelberg-New York, 2006.

A. Gorban, I. Karlin, A. Zinovyev,
Invariant Grids: Method of Complexity Reduction in Reaction Networks, Complexus, V. 2, 110–127. ComPlexUs2006.pdf

Complexity in the description of big chemical reaction networks has both structural (number of species and reactions) and temporal (very different reaction rates) aspects. A consistent way to make model reduction is to construct the invariant manifold which describes the asymptotic system behaviour. In this paper we present a discrete analogue of this object: an invariant grid. The invariant grid is introduced independently from the invariant manifold notion and can serve to represent the dynamic system behaviour as well as to approximate the invariant manifold after refinement. The method is designed for pure dissipative systems and widely uses their thermodynamic properties but allows also generalizations for some classes of open systems. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis-Menten mechanism) and the hydrogen oxidation.

A.N. Gorban,
Basic Types of Coarse-Graining, e-print http://arxiv.org/abs/cond-mat/0602024 (local copy CoaGrWorkSpri7.pdf).

42 pgs, 11 figs. A talk given at the research workshop: "Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena,"

We consider two basic types of coarse-graining: the Ehrenfest's coarse-graining and its extension to a general principle of non-equilibrium thermodynamics, and the coarse-graining based on uncertainty of dynamical models and $\epsilon$-motions (orbits). Non-technical discussion of basic notions and main coarse-graining theorems are presented: the theorem about entropy overproduction for the Ehrenfest's coarse-graining and its generalizations, both for conservative and for dissipative systems, and the theorems about stable properties and the Smale order for $\epsilon$-motions of general dynamical systems including structurally unstable systems. A brief discussion of two other types, coarse-graining by rounding and by small noise, is also presented. Computational kinetic models of macroscopic dynamics are considered. We construct a theoretical basis for these kinetic models using generalizations of the Ehrenfest's coarse-graining.

A.N. Gorban, I.V. Karlin,
Quasi-Equilibrium Closure Hierarchies for the Boltzmann Equation
, Physica A 360 (2006) 325–364 GKQEBoltzPhysA2006.pdf
In this paper, explicit method  of constructing  approximations (the Triangle Entropy Method) is developed for nonequilibrium problems.  This method enables one to treat any complicated nonlinear functionals that fit best the physics  of a problem (such  as, for  example, rates of processes) as new independent variables.
The work of the method was demonstrated on the Boltzmann's - type kinetics. New   macroscopic variables are introduced (moments of the Boltzmann  collision integral, or scattering rates). They are treated  as  independent variables rather than as infinite moment series. This  approach gives the complete  account  of  rates  of scattering  processes. Transport equations for scattering rates are obtained (the  second hydrodynamic chain), similar to the usual moment chain (the  first hydrodynamic chain). Various examples of the closure of the first, of the second, and of the mixed hydrodynamic chains  are considered for the hard spheres model. It is  shown, in particular, that the complete account  of scattering processes  leads to a renormalization of transport coefficients.
The method gives the explicit solution for the closure problem, provides thermodynamic properties of reduced models, and can be applied to any kinetic equation with a thermodynamic Lyapunov function

2005

A. Gorban, A. Zinovyev,
Elastic Principal Graphs and Manifolds and their Practical Applications, Computing 75, 359–379 (2005), (DOI) 10.1007/s00607-005-0122-6 , GorbZin2005Computing.pdf

Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through “the middle” of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. First advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.

 

A.N. Gorban, I.V. Karlin,
Invariance correction to Grad's equations: Where to go beyond approximations? Continuum Mechanics and Thermodynamics, 17(4) (2005), 311–335,  GorKarCMT_05.pdf,  http://arxiv.org/abs/cond-mat/0504221
We review some recent developments of Grad's approach to solving the Boltzmann equation and creating reduced description. The method of invariant manifold is put forward as a unified principle to establish corrections to Grad's equations. A consistent derivation of regularized Grad's equations in the framework the method of invariant manifold is given. A new class of kinetic models to lift the finite-moment description to a kinetic theory in the whole space is established. Relations of Grad's approach to modern mesoscopic integrators such as the entropic lattice Boltzmann method are also discussed.

A.N. GorbanT.G.Popova, A.Yu. Zinovyev,
Codon usage trajectories and 7-cluster structure of 143 complete bacterial genomic sequences Physica A: Statistical and Theoretical Physics,
353C (2005), 365-387. CodonPhysA2005.pdf (Number 11 in TOP25 articles within the journal: Physica A: Statistical Mechanics and its Applications, APR - JUN 2005 Top25.pdf)
Three results are presented. First, we prove the existence of a universal 7-cluster structure in all 143 completely sequenced bacterial genomes available in Genbank in August 2004, and explained its properties. The 7-cluster structure is responsible for the main part of sequence heterogeneity in bacterial genomes. In this sense, our 7 clusters is the basic model of bacterial genome sequence. We demonstrated that there are four basic ``pure" types of this model, observed in nature: ``parallel triangles", ``perpendicular triangles", degenerated case and the flower-like type.
Second, we answered the question: how big are the position-specific information and the contribution connected with correlations between nucleotide. The accuracy of the mean-field (context-free) approximation is estimated for bacterial genomes.
We show that codon usage of bacterial genomes is a multi-linear function of their genomic G+C-content with high accuracy (more precisely, by two similar functions, one for eubacterial genomes and the other one for archaea). Description of these two codon-usage trajectories is the third result.
All 143 cluster animated 3D-scatters are collected in a database and is made available on our web-site: http://www.ihes.fr/~zinovyev/7clusters .

A.N. GorbanT.G.Popova, A.Yu. Zinovyev,
Four basic symmetry types in the universal 7-cluster structure of microbial genomic sequences,
In Silico Biology, 5 (2005), 0039. Internet site CLUSTER STRUCTURE IN GENOME with analysis of all bacterial genomes.
Coding information is the main source of heterogeneity (non-randomness) in the sequences of microbial genomes. The heterogeneity corresponds to a cluster structure in triplet distributions of relatively short genomic fragments (200-400bp). We found a universal 7-cluster structure in microbial genomic sequences and explained its properties. We show that codon usage of bacterial genomes is a multi-linear function of their genomic G+C-content with high accuracy. Based on the analysis of 143 completely sequenced bacterial genomes available in Genbank in August 2004, we show that there are four "pure" types of the 7-cluster structure observed. All 143 cluster animated 3D-scatters are collected in a database which is made available on our web-site (http://www.ihes.fr/~zinovyev/7clusters). The findings can be readily introduced into software for gene prediction, sequence alignment or microbial genomes classification.

A.N. Gorban, I.V. Karlin,
Invariant Manifolds for Physical and Chemical Kinetics, Lect. Notes Phys. 660, Springer, Berlin, Heidelberg, 2005 (498 pages). [Preface-Contents-Introduction(pdf)][Reviews(htm)]

2004

A.N. Gorban, T.G. Popova, A.Yu. Zinovyev,
Four basic symmetry types in the universal 7-cluster structure of 143 complete bacterial genomic sequences E-print: http://arxiv.org/abs/q-bio/0410033
coding information is the main source of heterogeneity (non-randomness) in the sequences of bacterial genomes. This information can be naturally modeled by analysing cluster structures in the "in-phase" triplet distributions of relatively short genomic fragments (200-400bp). We found a universal 7-cluster structure in bacterial genomic sequences and explained its properties. We show that codon usage of bacterial genomes is a multi-linear function of their genomic G+C-content with high accuracy. Based on the analysis of 143 completely sequenced bacterial genomes available in Genbank in August 2004, we show that there are four "pure" types of the 7-cluster structure observed. All 143 cluster animated 3D-scatters are collected in a database and is made available on our web-site: http://www.ihes.fr/~zinovyev/7clusters. The finding can be readily introduced into any software for gene prediction, sequence alignment or bacterial genomes classification

Gorban, I.N.;Zinovyev, A.Y.
Elastic principal manifolds and their practical applications E-print http://arxiv.org/abs/cond-mat/0405648
Principal manifolds defined as lines or surfaces passing through "the middle" of the data distribution serve as useful objects for many practical applications. We propose a new algorithm for fast construction of grid approximations of principal manifolds with given topology. One advantage of the method is a new form of the functional to be minimized, which becomes quadratic at the step of the vertexes positions refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows easily numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its applications with speed performance characteristics.

Gorban, A.N.
Systems with inheritance: dynamics of distributions with conservation of support, natural selection and finite-dimensional asymptotics E-print: http://arxiv.org/abs/cond-mat/0405451
If we find a representation of an infinite-dimensional dynamical system as a nonlinear kinetic system with {\it conservation of supports} of distributions, then (after some additional technical steps) we can state that the asymptotics is finite-dimensional. This conservation of support has a {\it quasi-biological interpretation, inheritance} (if a gene was not presented initially in a isolated population without mutations, then it cannot appear at later time). These quasi-biological models can describe various physical, chemical, and, of course, biological systems. The finite-dimensional asymptotic demonstrates effects of {\it "natural" selection}. The estimations of asymptotic dimension are presented. The support of an individual limit distribution is almost always small. But the union of such supports can be the whole space even for one solution. Possible are such situations: a solution is a finite set of narrow peaks getting in time more and more narrow, moving slower and slower. It is possible that these peaks do not tend to fixed positions, rather they continue moving, and the path covered tends to infinity at $t \to \infty$. The {\it drift equations} for peaks motion are obtained. Various types of stability are studied. In example, models of cell division self-synchronization are studied. The appropriate construction of notion of typicalness in infinite-dimensional spaces is discussed, and the "completely thin" sets are introduced

Gorban, A.N.
Singularities of transition processes in dynamical systems: Qualitative theory of critical delays Electron. J. Diff. Eqns. Monograph 5, 2004, 55 p.Slorelax2004EJDE.pdf Online: http://ejde.math.swt.edu/Monographs/05/abstr.html
This monograph presents a systematic analysis of the singularities in the transition processes for dynamical systems. We study general dynamical systems, with dependence on a parameter, and construct relaxation times that depend on three variables: Initial conditions x, parameters k of the system, and accuracy e of the relaxation. We study the singularities of relaxation times as functions of (x,k) under fixed e, and then classify the bifurcations (explosions) of limit sets. We study the relationship between singularities of relaxation times and bifurcations of limit sets. An analogue of the Smale order for general dynamical systems under perturbations is constructed. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for the Morse-Smale systems

Gorban, A.N.;Gorban, P.A.;Karlin, I.V.
Legendre integrators, post-processing and quasiequilibrium J. Non-Newtonian Fluid Mech. 120 (2004) 149–167GoGoKar2004.pdf Online: http://arxiv.org/abs/cond-mat/0308488
A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical post-processing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The post-processing is the cheapestway to improve the solution obtained by the Legendre integrators. Legendre integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (maximum entropy, MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to the FENE-P model of polymer kinetic theory. The multi-peak model of polymer dynamics is developed.

Gorban, A.N.;Karlin, I.V.
Uniqueness of thermodynamic projector and kinetic basis of molecular individualism Physica A, 336, 2004,  391-432 UniMolIndRepr.pdf Online: http://arxiv.org/abs/cond-mat/0309638
Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given anzatz manifold which is not tangent to the Lyapunov function levels. Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. (The theorem about entropy overproduction.) In example, we apply the thermodynamic projector to derive the equations of reduced kinetics for the Fokker-Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2^m, and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of molecular individualism. This is the third result.

Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
Constructive methods of invariant manifolds for kinetic problems Phys. Rep., 396, 2004, 197-403 PhysRepCorr.pdf Online: http://arxiv.org/abs/cond-mat/0311017
The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.
A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions.
The following examples of applications are presented: nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data ; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, etc.

Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
Invariant grids for reaction kinetics Physica A, 333, 2004 106-154 ChemGrPhA2004.pdf Online: http://arxiv.org/abs/cond-mat/0307076
In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A grid-based version of MIM is developed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics. The most essential new element of this paper is the systematic consideration of a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, {invariant grids}. We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grid's nodes correction. The edges between the nodes appears only in the calculation of the tangent spaces. This fact determines high computational efficiency of the invariant grids method.

 

2003

A. Yu. Zinovyev, A. N. Gorban, T. G. Popova
Self-Organizing Approach for Automated Gene Identification
Open Sys. & Information Dyn., 10, 2003, 321-333 GoZiPo2003final.pdf
Self-training technique for automated gene recognition both in entire genomes and in unassembled ones is proposed. It is based on a simple measure (namely, the vector of frequencies of non-overlapping triplets in sliding window), and needs neither predetermined information, nor preliminary learning. The sliding window length is the only one tuning parameter. It should be chosen close to the average exon length typical to the DNA text under investigation. An essential feature of the technique proposed is preliminary visualization of the set of vectors in the subspace of the first three principal components. It was shown, the distribution of DNA sites has the bullet-like structure with one central cluster (corresponding to non-coding sites) and three or six ank ones (corresponding to protein-coding sites). The bullet-like structure itself revealed in the distribution seems to be very interesting illustration of triplet usage in DNA sequence. The method was examined on several genomes (mitochondrion of P.wickerhamii, bacteria C.crescentus and primitive eukaryot S.cerevisiae). The percentage of truly predicted nucleotides exceeds 90%.
In October 2004 this paper was mentioned as one of the five most viewed paper published in the Journal since 1997 http://www.kluweronline.com/issn/1230-1612 .

A. N. Gorban, A. Yu. Zinovyev, T. G. Popova
Seven clusters in genomic triplet distributions In Silico Biology, 3, 2003, 471-482 (0039), Online: http://arXiv.org/abs/cond-mat/0305681 29 May 2003  Seven03.pdf
Motivation: In several recent papers new algorithms were proposed for detecting coding regions without requiring learning dataset of already known genes. In this paper we studied cluster structure of several genomes in the space of codon usage. This allowed to interpret some of the results obtained in other studies and propose a simpler method, which is, nevertheless, fully functional. Results: Several complete genomic sequences were analyzed, using visualization of tables of triplet counts in a sliding window. The distribution of 64-dimensional vectors of triplet frequencies displays a well-detectable cluster structure. The structure was found to consist of seven clusters, corresponding to protein-coding information in three possible phases in one of the two complementary strands and in the non-coding regions. Awareness of the existence of this structure allows development of methods for the segmentation of sequences into regions with the same coding phase and non-coding regions. This method may be completely unsupervised or use some external information. Since the method does not need extraction of ORFs, it can be applied even for unassembled genomes. Accuracy calculated on the base-pair level (both sensitivity and specificity) exceeds 90%. This is not worse as compared to such methods as HMM, however, has the advantage to be much simpler and clear. Availability: The software and datasets are available at http://www.ihes.fr/~zinovyev/bullet

Gorban, A.N.;Karlin, I.V.,
Method of invariant manifold for chemical kinetics
, Chem. Eng. Sci.. 58, 2003, 4751-4768  ChemEngSci2003.pdf
In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of the MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics

A. N. Gorban, A. Y. Zinovyev, D.C. Wunsch
Application of The Method of Elastic Maps In Analysis of Genetic Texts, Proceedings of IJCNN2003 
GZW2003.pdf
Method of elastic maps allows to construct efficiently 1D, 2D and 3D non-linear approximations to the principal manifolds with different topology (piece of plane, sphere, torus etc.) and to project data onto it. We describe the idea of the method and demonstrate its applications in analysis of genetic sequences. 

Gorban A. N., Karlin I. V.
Quasi-Equilibrium Closure Hierarchies for The Boltzmann Equation E-print,
http://arXiv.org/abs/cond-mat/0305599 v1 26 May 2003  Triangl2003.pdf
Explicit method of constructing of approximations (Triangle Entropy Method) is developed for strongly nonequilibrium problems of Boltzmann's--type kinetics, i.e. when standard moment variables are insufficient. This method enables one to treat any complicated nonlinear functionals that fit the physics of a problem (such as, for example, rates of processes) as new independent variables. The method is applied to the problem of derivation of hydrodynamics from the Boltzmann equation. New macroscopic variables are introduced (moments of the Boltzmann collision integral, or collision moments). They are treated as independent variables rather than as infinite moment series. This approach gives the complete account of rates of scattering processes. Transport equations for scattering rates are obtained (the second hydrodynamic chain), similar to the usual moment chain (the first hydrodynamic chain). Using the triangle entropy method, three different types of the macroscopic description are considered. The first type involves only moments of distribution functions, and results coincide with those of the Grad method in the Maximum Entropy version. The second type of description involves only collision moments. Finally, the third type involves both the moments and the collision moments (the mixed description). The second and the mixed hydrodynamics are sensitive to the choice of the collision model. The second hydrodynamics is equivalent to the first hydrodynamics only for Maxwell molecules, and the mixed hydrodynamics exists for all types of collision models excluding Maxwell molecules. Various examples of the closure of the first, of the second, and of the mixed hydrodynamic chains are considered for the hard spheres model. It is shown, in particular, that the complete account of scattering processes leads to a renormalization of transport coefficients.
The paper gives English translation of the first part of the paper: Gorban, A. N., Karlin, I. V., Quasi-equilibrium approximation and non-standard expansions in the theory of the Boltzmann kinetic equation, in: "Mathematical Modelling in Biology and Chemistry. New Approaches", ed. R. G. Khlebopros, Nauka, Novosibirsk, P.69-117 (1992) [in Russian].

Gorban A. N.
Neuroinformatics: What are us, where are we going, how to measure our way? The lecture was given at the USA-NIS Neurocomputing opportunities workshop, Washington DC, July 1999 (Associated with IJCNN'99) E-print: 
http://arxiv.org/abs/cond-mat/0308331
What is neuroinformatics? We can define it as a direction of science and information technology, dealing with development and study of the methods for solution of problems by means of neural networks. A field of science cannot be determined only by fixing what it is "dealing with". The main component, actually constituting a scientific direction, is "THE GREAT PROBLEM", around which the efforts are concentrated. One may state even categorically: if there is no a great problem, there is no a field of science, but only more or less skilful imitation. What is "THE GREAT PROBLEM" for neuroinformatics? The problem of effective parallelism, the study of brain (solution of mysteries of thinking), etc are discussed. The neuroinformatics was considered not only as a science, but as a services sector too. The main ideas of generalized technology of extraction of explicit knowledge from data are presented. The mathematical achievements generated by neuroinformatics, the problem of provability of neurocomputations, and benefits of neural network realization of solution of a problem are discussed.

Gorban A. N., Karlin I. V.
Geometry of irreversibility: The film of nonequilibrium states E-print:
http://arxiv.org/abs/cond-mat/0308331 
A general geometrical framework of nonequilibrium thermodynamics is developed. The notion of macroscopically definable ensembles is developed. The thesis about macroscopically definable ensembles is suggested. This thesis should play the same role in the nonequilibrium thermodynamics, as the Church-Turing thesis in the theory of computability. The primitive macroscopically definable ensembles are described. These are ensembles with macroscopically prepared initial states. The method for computing trajectories of primitive macroscopically definable nonequilibrium ensembles is elaborated. These trajectories are represented as sequences of deformed equilibrium ensembles and simple quadratic models between them. The primitive macroscopically definable ensembles form the manifold in the space of ensembles. We call this manifold the film of nonequilibrium states. The equation for the film and the equation for the ensemble motion on the film are written down. The notion of the invariant film of non-equilibrium states, and the method of its approximate construction transform the the problem of nonequilibrium kinetics into a series of problems of equilibrium statistical physics. The developed methods allow us to solve the problem of macro-kinetics even when there are no autonomous equations of macro-kinetics

Iliya V. Karlin, Larisa L. Tatarinova, Alexander N. Gorban, Hans Christian Ottinger
Irreversibility in the short memory approximation Physica A, 327, 2003, 399-424  Online: http://arXiv.org/abs/cond-mat/0305419  v1 18 May 2003  KTGOe2003LANL.pdf
A recently introduced systematic approach to derivations of the macroscopic dynamics from the underlying microscopic equations of motions in the short-memory approximation [Gorban et al, Phys. Rev. E 63 , 066124 (2001)] is presented in detail. The essence of this method is a consistent implementation of Ehrenfest's idea of coarse-graining, realized via a matched expansion of both the microscopic and the macroscopic motions. Applications of this method to a derivation of the nonlinear Vlasov-Fokker-Planck equation, diffusion equation and hydrodynamic equations of the uid with a long-range mean field interaction are presented in full detail. The advantage of the method is illustrated by the computation of the post-Navier-Stokes approximation of the hydrodynamics which is shown to be stable unlike the Burnett hydrodynamics.
 

2002

Alexander N. Gorban, Iliya V. Karlin
Family of additive entropy functions out of thermodynamic limit, Physical Review E 67, 016104, 2003. Online:
http://arXiv.org/abs/cond-mat/0205511 24 May 2002PRE162003.pdf
We derive a one-parametric family of entropy functions that respect the additivity condition, and which describe effects of finiteness of statistical systems, in particular, distribution functions with long tails. This one-parametric family is different from the Tsallis entropies, and is a convex combination of the Boltzmann- Gibbs-Shannon entropy and the entropy function proposed by Burg. An example of how longer tails are described within the present approach is worked out for the canonical ensemble. We also discuss a possible origin of a hidden statistical dependence, and give explicit recipes on how to construct corresponding generalizations of the master equation.
 

Gorban A. N., Karlin I. V.
Geometry of irreversibility, in: Recent Developments in Mathematical and Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F. Uribe (Kluwer, Dordrecht, 2002), pp. 19-43. 
GeoNeo02.pdf
A general geometrical setting of nonequilibrium thermodynamics is developed. The approach is based on the notion of the natural projection which generalizes Ehrenfests' coarse-graining. It is demonstrated how derivations of irreversible macroscopic dynamics from the microscopic theories can be addressed through a study of stability of quasiequilibrium manifolds.
 

A. Gorban, A. Rossiev, N. Makarenko, Y. Kuandykov, V. Dergachev
Recovering data gaps through neural network methods, International Journal of Geomagnetism and Aeronomy vol. 3, no. 2, pages 191-197, December 2002 
geomag02.pdf
A new method is presented to recover the lost data in geophysical time series. It is clear that gaps in data are a substantial problem in obtaining correct outcomes about phenomenon in time series processing. Moreover, using the data with irregular coarse steps results in the loss of prime information during analysis. We suggest an approach to solving these problems, that is based on the idea of modeling the data with the help of small-dimension manifolds, and it is implemented with the help of a neural network. We use this approach on real data and show its proper use for analyzing time series of cosmogenic isotopes. In addition, multifractal analysis was applied to the recovered 14C concentration in the Earth's atmosphere.
 

Gorban A.N., Karlin I.V.
Methods of nonlinear kinetics, Contribution to the "Encyclopedia of Life Support Systems" (EOLSS Publishers, Oxford). 
encboltz02.pdf E-print: http://arxiv.org/abs/cond-mat/0306062
Nonlinear kinetic equations are reviewed for a wide audience of specialists and postgraduate students in physics, mathematical physics, material science, chemical engineering and interdisciplinary research.
Contents:
1. The Boltzmann equation
2. Phenomenology of the Boltzmann equation
3. Kinetic models
4. Methods of reduced description
4.1. The Hilbert method
4.2. The Chapman-Enskog method
4.3. The Grad moment method
4.4. Special approximations
4.5. The method of invariant manifold
4.6. Quasi-equilibrium approximations
5. Discrete velocity models
6. Direct simulation
7. Lattice Gas and Lattice Boltzmann models
8. Other kinetic equations
8.1. The Enskog equation for hard spheres
8.2. The Vlasov equation
8.3. The Smoluchowski equation
 

Gorban A.N., Karlin I.V.
Method of invariant manifold for chemical kinetics Online:
http://arXiv.org/abs/cond-mat/0207231 v1 9 Jul 2002  InvManLANL2002.pdf
In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.
 

Gorban A.N., Karlin I.V.
Hydrodynamics from Grad's equations: What can we learn from exact solutions?
Annalen der Physics, 2002. Online: http://arXiv.org/abs/cond-mat/0209560 v1 24 Sep 2002.  annphys02.pdf
A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the Navier-Stokes approximation. Various techniques, such as the method of partial summation, Pad_e approximants, and invariance principle are compared both in linear and nonlinear situations.
 

Karlin I.V., Grmela M., Gorban A.N.
Duality in nonextensive statistical mechanics. Physical Review E, 2002, Volume 65, 036128. P.1-4. 
PRE362002.pdf
We revisit recent derivations of kinetic equations based on Tsallis’ entropy concept. The method of kinetic functions is introduced as a standard tool for extensions of classical kinetic equations in the framework of Tsallis’ statistical mechanics. Our analysis of the Boltzmann equation demonstrates a remarkable relation between thermodynamics and kinetics caused by the deformation of macroscopic observables.
 

Gorban A.N., Karlin I.V., Ottinger H.C.
The additive generalization of the Boltzmann entropy, Physical Review E, 2003, Volume 67, 067104,. Online: http://arXiv.org/abs/cond-mat/0209319 v1 13 Sep 2002  ProofMS2003.pdf
There exists only one generalization of the classical Boltzmann-Gibbs-Shannon entropy functional to a one-parametric family of additive entropy functionals. We find analytical solution to the corresponding extension of the classical ensembles, and discuss in some detail the example of the deformation of the uncorrelated state.
 

Gorban A.N., Karlin I.V.
Macroscopic dynamics through coarse-graining: A solvable example, Physical Review E, 2002, Volume 65, 026116, P.1-5. 
PREEhr02.pdf
The recently derived fluctuation-dissipation formula (A. N. Gorban et al., Phys. Rev. E 63, 066124. 2001) is illustrated by the explicit computation for McKean’s kinetic model (H. P. McKean, J. Math. Phys. 8, 547. 1967). It is demonstrated that the result is identical, on the one hand, to the sum of the Chapman-Enskog expansion, and, on the other hand, to the exact solution of the invariance equation. The equality between all three results holds up to the crossover from the hydrodynamic to the kinetic domain.
 

Gorban' A., Braverman M., and Silantyev V.
Modified Kirchhoff flow with a partially penetrable obstacle and its application to the efficiency of free flow turbines, Mathematical and Computer Modelling, Volume 35, Issue 13, June 2002, P. 1371-1375.
MCM2002-2.pdf
An explicitly solvable analog of the Kirchhoff flow for the case of a semipenetrable obstacle is considered. Its application to estimating the efficiency of free flow turbines is discussed.
 

Gorban' A., Silantyev V.
Riabouchinsky flow with partially penetrable obstacle, Mathematical and Computer Modelling, Volume 35, Issue 13, June 2002, P. 1365-1370. MCM2002-1.pdf
An explicitly solvable Riabouchinsky model with a partially penetrable obstacle is introduced. This model applied to the estimation of the efficiency of free flow turbines allows us to take into account the pressure drop past the lamina.
 

2001

Gorban' A.N., Gorlov A.N., Silantyev V.M.
Limits of the Turbine Efficiency for Free Fluid Flow, Journal of Energy Resources Technology - December 2001 - Volume 123, Issue 4, pp. 311-317. 
Gorlov2001.pdf
An accurate estimate of the theoretical power limit of turbines in free fluid flows is important because of growing interest in the development of wind power and zero-head water power resources. The latter includes the huge kinetic energy of ocean currents, tidal streams, and rivers without dams. Knowledge of turbine efficiency limits helps to optimize design of hydro and wind power farms. An explicitly solvable new mathematical model for estimating the maximum efficiency of turbines in a free (nonducted) fluid is presented. This result can be used for hydropower turbines where construction of dams is impossible (in oceans) or undesirable (in rivers), as well as for wind power farms. The model deals with a finite two-dimensional, partially penetrable plate in an incompressible fluid. It is nearly ideal for two-dimensional propellers and less suitable for three-dimensional cross-flow Darrieus and helical turbines. The most interesting finding of our analysis is that the maximum efficiency of the plane propeller is about 30 percent for free fluids. This is in a sharp contrast to the 60 percent given by the Betz limit, commonly used now for decades. It is shown that the Betz overestimate results from neglecting the curvature of the fluid streams. We also show that the three-dimensional helical turbine is more efficient than the two-dimensional propeller, at least in water applications. Moreover, well-documented tests have shown that the helical turbine has an efficiency of 35 percent, making it preferable for use in free water currents.
 

Gorban A.N., Zinovyev A.Yu.
Visualization of Data by Method of Elastic Maps and its Applications in Genomics, Economics and Sociology, Institut des Hautes Etudes Scientifiques Preprint. IHES M/01/36. Online: 
http://www.ihes.fr/PREPRINTS/M01/Resu/resu-M01-36.html elmap.pdf
Technology of data visualization and data modeling is suggested. The basic of the technology is original idea of elastic net and methods of its construction and application. A short review of relevant methods has been made. The methods proposed are illustrated by applying them to the real biological, economical, sociological datasets and to some model data distributions.
 

Gorban A.N., Karlin I.V., Ilg P., Ottinger H.C.
Corrections and enhancements of quasi-equilibrium states, J. Non-Newtonian Fluid Mech. 2001, 96, P. 203-219.
NonNew01.pdf
We give a compact non-technical presentation of two basic principles for reducing the description of nonequilibrium systems based on the quasi-equilibrium approximation. These two principles are: construction of invariant manifolds for the dissipative microscopic dynamics, and coarse-graining for the entropy-conserving microscopic dynamics. Two new results are presented: first, an application of the invariance principle to hybridization of micro–macro integration schemes is introduced, and is illustrated with non-linear dumbbell models; second, Ehrenfest’s coarse-graining is extended to general quasi-equilibrium approximations, which gives the simplest way to derive dissipative equations from the Liouville equation in the short memory approximation.
 

Gorban A.N., Karlin I.V., Ottinger H.C., Tatarinova L.L.
Ehrenfest’s argument extended to a formalism of nonequilibrium thermodynamics, Physical Review E, 2001. Volume 63, 066124, P.1-6.
PREEhr01.pdf
A general method of constructing dissipative equations is developed, following Ehrenfest’s idea of coarse graining. The approach resolves the major issue of discrete time coarse graining versus continuous time macroscopic equations. Proof of the H theorem for macroscopic equations is given, several examples supporting the construction are presented, and generalizations are suggested.
 

Gorban A.N., Zinovyev A.Yu., Popova T.G.
Self-organizing approach for automated gene identification in whole genomes, Institut des Hautes Etudes Scientifiques Preprint. IHES. December 12, 2001, Online:
http://arXiv.org/abs/physics/0108016 v1 10 Aug 2001  lanlgpz01.pdf
An approach based on using the idea of distinguished coding phase in explicit form for identi cation of protein-coding regions in whole genome has been proposed. For several genomes an optimal window length for averaging GC-content function and calculating codon frequencies has been found. Self-training procedure based on clustering in multidimensional space of triplet frequencies is proposed.
 

Gorban A.N., Zinovyev A.Yu., Popova T.G.
Statistical approaches to automated gene identification without teacher. Institut des Hautes Etudes Scientifiques Preprint. IHES M/01/34.  Online: http://www.ihes.fr/PREPRINTS/M01/Resu/resu-M01-34.html geneid.pdf
Overview of statistical methods of gene identification is made. Particular attention is given to the methods which need not a training set of already known genes. After analysis several statistical approaches are proposed for computational exon identification in whole genomes. For several genomes an optimal window length for averaging GC-content function and calculating codon frequencies has been found. Self-training procedure based on clustering in multidimensional codon frequencies space is proposed.
 

A. N. Gorban, K. O. Gorbunova, D. C. Wunsch II
Liquid Brain: Kinetic Model of Structureless Parallelism,
liquidbrain.pdf
A new formal model of parallel computations, the Kirdin kinetic machine, is suggested. It is expected that this model will play the role for parallel computations similar to Markov normal algorithms, Kolmogorov and Turing machine or Post schemes for sequential computations. The basic ways in which computations are realized are described; correctness of the elementary programs for the Kirdin kinetic machine is investigated. It is proved that the determined Kirdin kinetic machine is an effective calculator. A simple application of the Kirdin kinetic machine, heap encoding, is suggested. Subprograms similar to usual programming enlarge the Kirdin kinetic machine.
 

2000

Gorban A.N., Karlin I.V., Zmievskii V.B., Dymova S.V.
Reduced description in the reaction kinetics, Physica A, 2000, 275, P.361-379.  
GKZD2000.pdf
Models of complex reactions in thermodynamically isolated systems often demonstrate evolution towards low-dimensional manifolds in the phase space. For this class of models, we suggest a direct method to construct such manifolds, and thereby to reduce the effective dimension of the problem. The approach realizes the invariance principle of the reduced description, it is based on iterations rather than on a small parameter expansion, it leads to tractable linear problems, and is consistent with thermodynamic requirements. The approach is tested with a model of catalytic reaction.
 

Gorban A.N., Popova T.G., Sadovsky M.G.
Classification Of Symbol Sequences Over Thier Frequency Dictionaries: Towards The Connection Between Structure And Natural Taxonomy, Open Sys. & Information Dyn. 7: 1-17, 2000. 
opsygps00.pdf
The classifications of bacterial 16S RNA sequences developed over the real and transformed frequency dictionaries have been studied. Two sequences considered to be close each other, when their frequency dictionaries were close in Euclidean metrics. A procedure to transform a dictionary is proposed that makes clear some features of the information pattern of a symbol sequence. A comparative study of classifications developed over the real frequency dictionaries vs. the transformed ones has been carried out. A correlation between an information pattern of nucleotide sequences and taxonomy of the bearer of the sequence was found. The sites with high information value are found, that were the main factors of the difference between the classes in a classification. The classification of nucleotide sequences developed over the real frequency dictionaries of the thickness 3 reveals the best correlation to a gender of bacteria. A set of sequences of the same gender is included entirely into one class, as a rule, and the exclusions occur rarely. A hierarchical classification yields one or two taxonomy groups on each level of the classification. An unexpectedly often (in comparison to the expected), or unexpectedly rare occurrence of some sites within a sequence makes a basic difference between the structure patterns of the classes yielded; a number of those sites is not too great. Further investigations are necessary in order to compare the sites revealed with those determined due to other methodology.
 

1999

A. N. Gorban, I.V. Karlin, and V.B. Zmievskii
Two-Step Approximation of Space-Independent Relaxation, TRANSPORT THEORY AND STATISTICAL PHYSICS, 28(3) (1999), 271-296.
GorKarZmiTTSP99.pdf Local copy

 

In this paper we introduce a new method of constructing approximate trajectories for space independent kinetic equations confirming to the second law of thermodynamics. Classical examples are the space independent Boltzmann equation and chemical kinetics equations for closed homogeneous systems. This family of kinetic equations is characterized by the following general properties:

(1). There exists a set of functions which remain constant on a solution (these are density, momentum and energy in context of the Boltzmann equation).

(ii). There exists a convex function which monotonically decreases along any solution from its value in the initial state to an absolute minima in the final equilibrium state (this is the H-theorem for the Boltzmann equation) .

 

Usually we do know only the initial and the final (equilibrium) states, and the kinetic equation neither can be solved exactly, nor contains small parameters to develop a reliable perturbation theory. Still, we would like to get (perhaps a rather rough but a simple) approximation of the relaxation trajectory.

 

An express method to approximate trajectories of space independent kinetic equations is developed. It involves a two-step treatment of relaxation through a quasiequilibria located on a line emerging from the initial state in the direction prescribed by the kinetic equation. A test for the Boltzmann equation shows the validity of the method.

A.N. Gorban, A.A. Rossiev, D. C. Wunsch II
Neural Network Modeling of Data with Gaps: Method of Principal Curves, Carleman's Formula, and Other, The talk was given at the USA-NIS Neurocomputing opportunities workshop, Washington DC, July 1999 (Associated with IJCNN'99).
Online:
http://arXiv.org/abs/cond-mat/0305508 21 May 2003  gaps.pdf
A method of modeling data with gaps by a sequence of curves has been developed. The new method is a generalization of iterative construction of singular expansion of matrices with gaps. Under discussion are three versions of the method featuring clear physical interpretation:
1) linear: modeling the data by a sequence of linear manifolds of small dimension;
2) quasilinear: constructing "principal curves": (or "principal surfaces"), univalently projected on the linear principal components;
3) essentially non-linear, based on constructing "principal curves": (principal strings and beams) employing the variation principle; the iteration implementation of this method is close to Kohonen self-organizing maps.
The derived dependencies are extrapolated by Carleman’s formulas. The method is interpreted as a construction of neural network conveyor designed to solve the following problems:
1) to fill gaps in data;
2) to repair data, to correct initial data values in such a way as to make the constructed models work best;
3) to construct a calculator to fill gaps in the data line fed to the input.
 

Gorban A. N.
Neuroinformatics: What are us, where are we going, how to measure our way? The lecture was given at the USA-NIS Neurocomputing opportunities workshop (
http://phy025.lubb.ttuhsc.edu/wldb/Witali/WWW/NSF_NOW.html ), Washington DC, July 1999 (Associated with IJCNN'99 ( http://www.cas.american.edu/~medsker/ijcnn99/ijcnn99.html). neurolec.pdf
What is neuroinformatics? For me here and now neuroinformatics is a direction of science and information technology, dealing with development and study of the methods for solution of problems by means of neural networks. A base example of artificial neural network, which will be referred to below, is a feed-forward network from standard neurons.
 

Alexander N. Gorban, Eugeniy M. Mirkes and Victor G. Tsaregorodtsev
Generation of Explicit Knowledge from Empirical Data through Pruning of Trainable Neural Networks, International Joint Conference on Neural Networks, Washington, DC July 10-16, 1999. (
http://www.cas.american.edu/~medsker/ijcnn99/ijcnn99.html ). know.pdf E-print: http://arxiv.org/abs/cond-mat/0307083
This paper presents a generalized technology of extraction of explicit knowledge from data. The main ideas are:
1) maximal reduction of network complexity (not only removal of neurons or synapses, but removal all the unnecessary elements and signals and reduction of the complexity of elements),
2) using of adjustable and flexible pruning process (the pruning sequence shouldn't be predetermined - the user should have a possibility to prune network on his own way in order to achieve a desired network structure for the purpose of extraction of rules of desired type and form),
3) extraction of rules not in predetermined but any desired form.
Some considerations and notes about network architecture and training process and applicability of currently developed pruning techniques and rule extraction algorithms are discussed. This technology, being developed by us for more than 10 years, allowed us to create dozens of knowledge-based expert systems.
 

1998

Karlin I.V., Gorban A.N., Dukek G., Nonnenmacher T. F.
Dynamic correction to moment approximations. Physical Review E, February 1998 Volume 57, Number 2, P.1668-1672. 
KGDN98.pdf
Considering the Grad moment ansatz as a suitable first approximation to a closed finite-moment dynamics, the correction is derived from the Boltzmann equation. The correction consists of two parts, local and nonlocal. Locally corrected thirteen-moment equations are demonstrated to contain exact transport coefficients. Equations resulting from the nonlocal correction give a microscopic justification to some phenomenological theories of extended hydrodynamics.
 

Gorban A. N.
Approximation of Continuos Functions of Several Variables by an Arbitrary Nonlinear Continuous Function of One Variable, Linear Functions, and Their Superpositions, Appl. Math. Lett., Vol. 11, No. 3, pp 45-49, 1998 
approx98.pdf
 

Karlin I.V., Gorban A.N., Succi S., Boffi V.
Maximum Entropy Principle for Lattice Kinetic Equations. Physical Review Letters Volume 81, Number 1, 6 July 1998, P.6-9. 
p6_11998.pdf
The entropy maximum approach to constructing equilibria in lattice kinetic equations is revisited. For a suitable entropy function, we derive explicitly the hydrodynamic local equilibrium, prove the H theorem for lattice Bhatnagar-Gross-Krook models, and develop a systematic method to account for additional constraints.
 

1997

Gorban A.N., Shokin Yu.I., Verbitskii V.I.
Simultaneously dissipative operators and the infinitesimal Moore effect in interval spaces, Online:
http://arXiv.org/abs/physics/9702021 , 1997.  gorvershok.pdf
In solving a system of ordinary differential equations by an interval method the approximate solution at any considered moment of time t represents a set (called interval) containing the exact solution at the moment t. The intervals determining the solution of a system are often expanded in the course of time irrespective of the method and step used.
The phenomenon of interval expansion, called the Moore sweep effect, essentially decreases the efficiency of interval methods. In the present work the notions of the interval and the Moore effect are formalized and the Infinitesimal Moore Effect (IME) is studied for autonomous systems on positively invariant convex compact. With IME the intervals expand along any trajectory for any small step, and that means that when solving a system by a stepwise interval numerical method with any small step the interval expansion takes place for any initial data irrespective of the applied method. The local conditions of absence of IME in terms of Jacoby matrices field of the system are obtained. The relation between the absence of IME and simultaneous dissipativity of the Jacoby matrices is established, and some sufficient conditions of simultaneous dissipativity are obtained. (The family of linear operators is simultaneously dissipative, if there exists a norm relative to which all the operators are dissipative.)

M.Yu. Senashova, A.N. Gorban, D. C. Wunsch II
Back-propagation of accuracy,  The talk given on ICNN97 (The 1997 IEEE  International Conference on Neural Networks, Houston, USA), Online: 
http://arXiv.org/abs/cond-mat/0305527   gorsenwu.pdf
In this paper we solve the problem: how to determine maximal allowable errors, possible for signals and parameters of each element of a network proceeding from the condition that the vector of output signals of the network should be calculated with given accuracy? "Back-propagation of accuracy" is developed to solve this problem.

A. N: Gorban, Ye. M. Mirkes, D.C. Wunsch II
High order ortogonal tensor networks: information capacity and reliability. The talk given on ICNN97 (The 1997 IEEE  International Conference on Neural Networks, Houston, USA), 
gomirwu1.pdf
Neural networks based on construction of ortogonal projectors in the tensor power of space of signals are described. A sharp estimate of their ultimate information capacity is obtained. The numbers of stored prototype patterns (prototypes) can many times exceed the number of neurons. A comparison with the error control codes is made.
 

1996

Gorban A.N., Karlin I.V.
Short-Wave Limit of Hydrodynamics: A Soluble Example. Physical Review Letters, Volume 77, Number 2, 8 July 1996. P. 282-285.
p282_11996.pdf
The Chapman-Enskog series for shear stress is summed up in a closed form for a simple model of Grad moment equations. The resulting linear hydrodynamics is demonstrated to be stable for all wavelengths, and the exact asymptotic of the acoustic spectrum in the short-wave domain is obtained.
 

Gorban A.N., Karlin I.V. Nonnenmacher T. F., Zmievskii V.B.
Relaxation Trajectories: Global approximation. Physica A, 1996, 231, P.648-672.
GKZNPhA96.pdf
 

Gorban A. N., Karlin I. V.
Scattering rates versus moments: Alternative Grad equations, Physical Review E October 1996 Volume 54, Number 4, P. 3109-3112.
pR3109_11996.pdf
Scattering rates (moments of collision integral) are treated as independent variables, and as an alternative to moments of the distribution function, to describe the rarefied gas near local equilibrium. A version of the entropy maximum principle is used to derive the Grad-like description in terms of a finite number of scattering rates. The equations are compared to the Grad moment system in the heat nonconductive case. Estimations for hard spheres demonstrate, in particular, some 10% excess of the viscosity coefficient resulting from the scattering rate description, as compared to the Grad moment estimation.
 

1994

Alexander N. Gorban' , Iliya V. Karlin
General approach to constructing models of the Boltzmann equation, Physica A, 1994, 206, P.401-420.
GKPhA94.pdf
The problem of thermodynamic parameterization of an arbitrary approximation of reduced description is solved. On the base of this solution a new class of model kinetic equations is constructed that gives a model extension of the chosen approximation to a kinetic model. Model equations describe two processes: rapid relaxation to the chosen approximation along the planes of rapid motions, and the slow motion caused by the chosen approximation. The H-theorem is proved for these models. It is shown, that the rapid process always leads to entropy growth, and also a neighborhood of the approximation is determined inside which the slow process satisfies the H-theorem. Kinetic models for Grad moment approximations and for the Tamm-Mott-Smith approximation are constructed explicitly. In particular, the problem of concordance of the ES-model with the H-theorem is solved.
 

1992

Alexander N. Gorban' , Iliya V. Karlin
Thermodynamic parameterization, Physica A, 1992, 190, P.393-404
GKPhA92.pdf
A new method of successive construction of a solution is developed for problems of strongly nonequilibrium Boltzmann kinetics beyond normal solutions. Firstly, the method provides dynamic equations for any manifold of distributions where one looks for an approximate solution. Secondly, it gives a successive procedure of obtaining corrections to these approximations. The method requires neither small parameters, nor strong restrictions upon the initial approximation; it involves solutions of linear problems. It is concordant with the H-theorem at every step. In particular, for the Tamm-Mott-Smith approximation, dynamic equations are obtained, an expansion for the strong shock is introduced, and a linear equation for the first correction is found.

V. I. Verbitskii and A. N. Gorban
Jointly dissipative operators and their applications, Siberian Mathematical Journal, Volume 33, Number 1 (1992), 19-23, DOI: 10.1007/BF00972932
The jointly dissipative operators were introduced by Verbitskii and Gorban' (1989). Let E be an n-dimensional real or complex linear space, and let L(E) be the space of linear operators in E. Let us introduce a norm ||…|| on E and the corresponding norm in L(E). An operator A from L(E) is said to be dissipative if ||exp(tA)||≤1 for all t≥0. It is roughly dissipative if there is ε > 0 such that ||exp(tA)||≤exp(-εt) for all t≥0. For the existence of a norm with respect to which the operator A is be roughly dissipative it is necessary and sufficient that the system (i) be asymptotically stable, i.e., that the matrix of A be stable (i.e., that the spectrum of A Lie in the open left halfplane). A family of operators is said to be jointly dissipative (resp. jointly roughly dissipative) if there exists a norm with respect to which all operators from this family are dissipative (resp., roughly dissipative).  The jointly dissipative operators find application in the analysis of dynamical properties of nonlinear systems of ordinary differential equations and in some applications (chemical kinetics, numerical analysis). In the present paper we discuss the properties of jointly dissipative operators and some of their applications. For example, the following theorems are proved: (Theorem 1) Suppose the family {A} is compact, generates a solvable Lie algebra, and all matrices in {A} are stable. Then {A} is jointly roughly dissipative. (Theorem 2) Suppose the family {A} is finite, generates a nilpotent Lie algebra, and for each operator from {A} there exists a norm with respect to which it is is dissipative. Then the family {A} is jointly dissipative.

1980-1990

Gorban A.N., Bykov V.I.
A model of autooscillations in association reactions, Chemical Engineering Science. 1987, Vol. 42, No. 5. P. 1249-1251. 
BG1987.pdf
The aim of this paper is to show that association reactions can result in the appearance of autooscillations in nonlinear systems.
 

Gorban A.N., Bykov V.I., Yablonskii G.S.
Thermodynamic function analogue for reactions proceeding without interaction of various substances, Chemical Engineering Science, 1986. Vol. 41, No. 11. P. 2739-2745.
BGYa1986.pdf
Function similar to Lyapunov’s function has been constructed for reactions with $a_i A_i \to b_j A_j$ stages. This provides for the quasi-thermodynamics of the appropriate kinetic model, which implies steady-state uniqueness and global stability in the reaction polyhedron. The kinetic law generalizing the Marcelin-de Donder kinetics has been written for a separate stage. Explicit Lyapunov thermodynamic functions have been written for various conditions of the reaction proceeding in closed systems. The matrix of linear approximation close to equilibrium is expressed by means of the introduced scalar product. Particularly, the absence of damped oscillations as equilibrium is approached as shown.
 

Gorban A.N., Bykov V.I.
Macroscopic clusters induced by diffusion in a catalytic oxidation reactions, ChemicaI Engineering Science, 1980. Vol. 35, P. 2351-2352
BG1980.pdf
 

Gorban A.N.
Singularities of Transition Processes In Dynamical Systems.
http://arXiv.org/abs/chao-dyn/9703010 v1 18 Mar 1997, Translation of Candidate (Ph.D) Thesis, 1980  slorelax.pdf
The paper gives the systematic analysis of singularities of transition processes in general dynamical systems. Dynamical systems depending on parameter are studied. A system of relaxation times is constructed. Each relaxation time depends on three variables: initial conditions, parameters k of the system and accuracy \epsilon of relaxation. This system of times contains: the time before the first entering of the motion into \epsilon -neighbourhood of the limit set, the time of final entering in this neighbourhood and the time of stay of the motion outside the \epsilon -neighbourhood of the limit set. The singularities of relaxation times as functions of (x_0; k) under fixed \epsilon are studied. A classification of different bifurcations (explosions) of limit sets is performed. The bifurcations fall into those with appearance of new limit points and bifurcations with appearance of new limit sets at finite distance from the existing ones. The relations between the singularities of relaxation times and bifurcations of limit sets are studied. The peculiarities of dynamics which entail singularities of transition processes without bifurcations are described as well. The peculiarities of transition processes under perturbations are studied. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for smooth two-dimensional structural stable systems.

 

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