Hyperbolic Chaos of Turing Patterns

Abstract

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions. PACS numbers: 89.75.Kd, 05.45.Jn In nonlinear dynamics the notion of structural stability, or robustness, is one of the key tools allowing one to specify systems and effects that are really significant for theoretical and numerical researches, and especially for practical applications [1,2]. Among chaotic attractors, structural stability is intrinsic to those possessing the uniform hyperbolicity ("the systems with axiom A"), mathematical examples of which were advanced already since 60's -70's [3][4][5][6]. That time, such attractors were expected to be relevant for various physical situations (such as hydrodynamic turbulence), but later it became clear that the chaotic attractors, which normally occur in applications, do not relate to the class of structurally stable ones. This is an obvious contradiction to the principle of significance of the robust systems mentioned above.Recently, this inconsistency has been partially resolved by introducing a number of physically realizable systems with hyperbolic chaotic attractors [7][8][9][10]. It has been shown that simple systems of coupled oscillators that are excited alternately (in time) possess hyperbolic attractors of Smale-Williams type (for experimental realizations, see [9][10][11]). Hyperbolic chaos in these systems is related to the dynamics of the phases of the oscillators, evolution of which on the successive stages of activity is governed by a Bernoulli-type expanding circle map.In this letter we develop a similar approach, but deal with the spatial phases of patterns in a spatially extended system. We demonstrate the occurrence of hyperbolic chaos in dynamics resulting from an interplay of two Turing patterns of different wave lengths arising in succession. This advance, first, extends a toolbox for design of models manifesting robust chaos. Second, it suggests a novel direction for search of situations associated with hyperbolic chaos in the context e.g. of fluid turbulence, convection, and reaction-diffusion systems. Third, the description in terms of truncated equations for amplitudes of spatial modes leads to new prototypical lowdimensional model systems with hyperbolic attractors. (Note analogy with the Lorenz equations, which were derived originally as a finite-dimensional model for fluid convection.)Let us illustrate the approach with a concrete example based on the one-dimensional Swift...

Hyperbolic chaos at blinking coupling of noisy oscillators

“…With this interpretation, our results demonstrate that an interaction between blinking patterns results in a chaotic relocation of their positions, moreover, this chaos is hyperbolic. A kind of such behavior was reported recently in a model of interacting Turing patterns with different wave numbers [24]. The presented model based on the nonlinear Fokker-Plank equation provides another indication that the above mechanism of the hyperbolic chaos is realizable in quite generic circumstances.…”

Some Lattice Models with Hyperbolic Chaotic Attractors

“…Replacing the spatial differentiation operator in equation of Ref. [4] with a difference operator, we obtain , ) 1 where u j is dynamical variable related to the j-th spatial cell, k, A, ε are parameters. The quantities d j define the spatial non-homogeniety, the role of which will be explained below.…”

“…2 S. P. Kuznetsov ______________________________ CHEBOKSARY, 2-6 JUNE, 2019 ______________________________ the three-dimensional state space, but such attractors can occur in spaces of higher dimension too.Physical examples of systems with attractors of Smale -Williams type can be constructed using oscillators residing in states of excitation and inhibition alternately, while the angular variable has a sense of the oscillator phase [3]. Another approach is based on treatment of patterns arising in an active medium, say, that for Turing structures or standing waves, and the angular variable is a spatial phase [4,5,6]. A disadvantage of the first approach is that it requires, as a rule, a use of rather complex external driving for parameter modulation, combining low-frequency and high-frequency components.…”

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