Nonstationary dissipative structures and diffusion-induced chaos in nonlinear media

Akhromeyeva, T. S.; Kurdyumov, S. P.; Malinetskii, Georgii Gennadyevich; Samarskiî, A. A.

doi:10.1016/0370-1573(89)90001-x

Cited by 58 publications

(41 citation statements)

References 121 publications

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“…The details of the mathematical models and nonlinear analysis can be found in recent publications of S.P. Kurdyumov and his collaborators 9 .…”

Section: Synergetics As a New Methodology Of Futures Studiesmentioning

confidence: 99%

Synergetics and the images of future

Knyazeva

1999

Futures

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The hope of nding new methods of predicting the course of historical processes, could be connected with the recent d e v elopments of the theory of self-organization, called also synergetics. It provides us with knowledge of constructive principles of coevolution of the complex social systems, coevo l u t i o n o f c o u n tries and geopolitical regions being at di erent stages of development, integration of the East and the West, the North and the South. Due to the growth of population on the Earth in blow-up regime, the general and local instability of development is increasing. The social world seems to go towards a unity, a sustainable commonwealth through the pulsations rather than monotonously. I t g o e s via alternation of disintegration, even if partial, and getting more powerful integrations of structures. There is a path of multiple reduction of temporal and material expenses, path of a resonant excitation of desirable and -that is not less important -feasible in a given social system structures. In the case of right topology of integration of geopolitical structures an arising whole organization can accelerate its tempo of evolution.

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“…The details of the mathematical models and nonlinear analysis can be found in recent publications of S.P. Kurdyumov and his collaborators 9 .…”

Section: Synergetics As a New Methodology Of Futures Studiesmentioning

confidence: 99%

Synergetics and the images of future

Knyazeva

1999

Futures

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“…In particular, for the case of small domain size, the hierarchy of simplified models has been proposed [6], which enabled qualitative features of its solutions to be studied. These features proved to be typical of reaction-diffusion systems far from the bifurcation point [17].…”

Section: U T=(a +Tzai)u+f(u) U~r Nu(o)=u Omentioning

confidence: 99%

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

et al. 1994

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Summary. We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusion u~ = eDUxx + (A + eAt)u + F(u), u R2. These equations (quasinormal forms) describe the behaviour of solutions of the original equation for 8 ~ 0.One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called a sensitivity to small parameters: they contain a new parameter 0(e)---(ae -1/2 rood l) that influences the behaviour Of solutions, but changes infinitely many times when e ~ 0. This does not create problems'in numerical analysis of quasinormal forms, but makes numerical study of the original problem involving e almost impossible.

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“…In the present paper we study in detail the dynamic behavior of a pair of oscillators, and also make several generalizations to the case of a system of several coupled oscillators. The theoretical foundations and methods of studying this model are based on the approaches discussed in detail in [11][12][13][14][15]. …”

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confidence: 99%

A mathematical model of oxidation of Co in a thin layer of a Pd zeolite catalyst

Kurkina¹,

Tolstunova²

1999

Comput Math Model

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We construct and study a distributed mathematical model of oxidation of CO Introduction. The present paper is devoted to simulation of the oxidation of CO on a Pd zeolite catalyst. Experimental study of this reaction has revealed a complicated dynamic behavior of the chemical system, which includes both regular and chaotic oscillations of the reaction rate [ 1, 2]. Despite the fact that oxidation of CO is one of the most studied self-oscillatory systems of heterogeneous catalysis [3][4][5], the problem of modeling the complicated and chaotic oscillations continues to be of great interest. Slinko, Jaeger, and Svensson [6] conjectured that taking account of the diffusion of CO in the pores of the zeolite matrix could explain the complication of the shape of the oscillations and the appearance of chaos. As inhomogeneities arise in the concentration of CO inside the crystallites of zeolite, clusters of palladium located at different distances from the surface of the zeolite crystal wind up in unequal conditions. In this case the natural frequencies of oscillation of the reaction rate on the surfaces of individual Pd clusters may differ considerably, and the oscillations of the total reaction rate acquire a complicated character.The fn'st author and others [7] have constructed and studied a mathematical model of this reaction, taking account of the diffusion of CO gas in the pores of the zeolite catalyst in the case when all the crystallites that make up the zeolite catalyst are regarded as identical porous balls with palladium clusters uniformly distributed in them. The model is a chain of parametrically coupled nonlinear catalytic oscillators obtained by partitioning the crystallite into several spherical layers of equal volume. Depending on the values of the parameters, both regular and quasiperiodic and chaotic oscillations of the reaction rate were detected in the system. However, the spherical geometry of the system greatly increased the differences between the oscillators, and it remained unexplained to what extent this is essential for the formation of chaotic modes.In the present paper we construct and study an analogous model of the oxidation of CO in a zeolite catalyst; but, in contrast to [7], here all the crystallites are regarded as a single layer of porous material, which in order to take account of diffusion is divided into several identical planar layers. The mathematical model so constructed is also a chain of oscillators locally coupled through the gaseous phase. However, in this case the oscillators are perfectly identical and can differ only due to the appearance of nonuniform CO pressure inside a layer of the catalyst. Thus this model makes it possible to clarify the role of the internal diffusion of CO "in pure form" and better understand both the model of [7] and the causes of the complicated dynamic behavior of the system. In the present paper we study in detail the dynamic behavior of a pair of oscillators, and also make several generalizations to the case of a system of several coupled ...

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Nonstationary dissipative structures and diffusion-induced chaos in nonlinear media

Cited by 58 publications

References 121 publications

Synergetics and the images of future

Synergetics and the images of future

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

A mathematical model of oxidation of Co in a thin layer of a Pd zeolite catalyst

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