Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

“…The first few examples of feasible continuous-time dynamical systems with attractors of Smale-Williams type in their Poincaré maps were suggested in a number of recent papers by Kuznetsov et al [23][24][25][26][27]. Here, the role of angular variable was played by the phase of some oscillating process.…”

Hyperbolic Chaos of Turing Patterns

“…As seen from the diagram, all the exponents depend on the parameter smoothly, without sharp spikes or dips. This is a manifestation of robustness of the hyperbolic chaos [7,9,10]. The Kaplan-Yorke dimension of the attractor varies slightly, see the solid line in Fig.…”

“…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]).…”

“…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]).…”

“…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.…”