Finite-dimensional spatial disorder

Rabinovich, M.I.; Fabrikant, A.L.; Tsimring, L.Sh.

doi:10.3367/ufnr.0162.199208a.0001

Cited by 20 publications

(2 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Swift -Hohenberg equation, which was a simplification of the Proctor -Sivashinsky equations (instead of vector nonlinearity, the scalar one was used [4]) discussed the first kind phase transition in this description model. In this work and in work [5] the process of phase transition was discussed when from amorphous state of disorderly convection, the structure of convective rolls was formed, which in turns turned out to be unstable with formation of hexagonal convective cells.…”

Section: Introductionmentioning

confidence: 94%

A Demonstration Bench for Representing the Character of Phase Transitions of the First and Second Kind

Gushchin¹

2022

East Eur. J. Phys.

View full text Add to dashboard Cite

The paper presents the description of a demonstration bench, which includes a mathematical model and analysis tools for understanding the features of phase transitions of the first and second kind. The advantage of this demonstration bench is the rejection of all phenomenology and the obvious limitation of the application of various approximations and hypotheses. The description is formed on the well-known equations of hydrodynamics, which are well-tested and are a reliable basis for the construction of realistic models. The Proctor-Sivashinsky model, which was used to describe the process of convection development in a thin layer of liquid with poorly conductive heat boundaries, is the basis for the demonstration bench. Exactly this model allows to observe phase transitions of the first and second kind. The feature of the model is that it allocates one spatial scale of interaction, leaving for the evolution of the system the possibility to choose the nature of symmetry. All spatial disturbances of the same size but of different orientation interact with each other. This allows us not to distract from the main task of this work, which is to demonstrate the process of structure formation as a result of a cascade of phase transitions. The mechanism of phase transitions associated with the presence of minimums of the interaction coefficients of modes of the spectrum of the instability. There are a large number of structural defects, which appear as attributes of phase transition. The instability spectrum modes interference is the reason of the high rate of correlations in the propagation of a new phase.

show abstract

Section: Introductionmentioning

confidence: 94%

A Demonstration Bench for Representing the Character of Phase Transitions of the First and Second Kind

Gushchin¹

2022

East Eur. J. Phys.

View full text Add to dashboard Cite

show abstract

“…[9][10][11][12][13] AE derivation for the Swift-Hohenberg model 14 has been widely used for a qualitative description of the convective structures originated from Benard-Marangoni instability or non-Boussinesq Benard convection. 15,16 In this paper, we have derived the AE for the reversible Sel'kov model 17,18 a kinetic model of glycolytic 19,20 D-R system, in which the steady state from the reaction kinetics is required to be obtained by the analytical solution of a cubic equation using Cardon's method, 21 which is rather complicated. For such a D-R model, 17,18 we have attempted to derive the AE, which interprets the stability of various forms of Turing patterns as well as the structural transitions between them.…”

Section: Introductionmentioning

confidence: 99%

Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model

Dutt

2012

AIP Advances

View full text Add to dashboard Cite

For a model glycolytic diffusion-reaction system, an amplitude equation has been derived in the framework of a weakly nonlinear theory. The linear stability analysis of this amplitude equation interprets the structural transitions and stability of various forms of Turing structures. This amplitude equation also conforms to the expectation that time-invariant amplitudes in Turing structures are independent of complexing reaction with the activator species, whereas complexing reaction strongly influences Hopf-wave bifurcation.

show abstract

Chaos in Traveling Waves of Lattice Systems of Unbounded Media

Orendovici

Pesin

2000

The IMA Volumes in Mathematics and Its Applications

View full text Add to dashboard Cite

Abstract. We describe coupled map lattices (CML) of unbounded media corresponding to some well-known evolution partial differential equations (including reaction-diffusion equation, KuramotoSivashinsky, Swift-Hohenberg and Ginzburg-Landau equation). Following Kaneko we view CML also as phenomenological models of the medium and present the dynamical system approach to study the global behavior of solutions of CML. In particular, we establish spatio-temporal chaos associated with the set of traveling wave solutions of CML as well as describe the dynamics of the evolution operator on this set. Several examples are given to illustrate the appearance of Smale horseshoes and the presence of the dynamics of Morse-Smale type.

show abstract

Finite-dimensional spatial disorder

Cited by 20 publications

References 40 publications

A Demonstration Bench for Representing the Character of Phase Transitions of the First and Second Kind

A Demonstration Bench for Representing the Character of Phase Transitions of the First and Second Kind

Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model

Chaos in Traveling Waves of Lattice Systems of Unbounded Media

Contact Info

Product

Resources

About