A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system

Кузнецов, С. П.; Селезнев, Е. П.

doi:10.1134/s1063776106020166

Cited by 59 publications

(53 citation statements)

References 13 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…An appropriate modification of the above method was described in Refs. [24,25,37]. First, we generate a representative orbit on the attractor by computing Eqs.…”

Section: Amplitude Dynamics In Terms Of Angular Variable and Simpmentioning

confidence: 99%

“…It is worth noticing, however, that the above examples [24][25][26][27] describe specially designed devices rather than systems of natural origin. The search for real-world hyperbolic chaotic attractors and demonstration of their possible significance in, e.g., ecology, physiology and neuroscience, or hydrodynamics remains an interesting and challenging problem [13,21].…”

Section: Introductionmentioning

confidence: 99%

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

2011

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Section: Introductionmentioning

confidence: 99%

“…An appropriate modification of the above method was described in Refs. [24,25,37]. First, we generate a representative orbit on the attractor by computing Eqs.…”

Section: Amplitude Dynamics In Terms Of Angular Variable and Simpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

2011

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“…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.…”

mentioning

confidence: 99%

Hyperbolic Chaos of Turing Patterns

2012

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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...

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“…Subsequently, this method was developed and used as well to chaotic attractors [16,17,[19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Chaos in three coupled rotators: From Anosov dynamics to hyperbolic attractors

Кузнецов¹

2017

IASCS

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Starting from Anosov chaotic dynamics of geodesic flow on a surface of negative curvature, we develop and consider a number of self-oscillatory systems including those with hinged mechanical coupling of three rotators and a system of rotators interacting through a potential function. These results are used to design an electronic circuit for generation of rough (structurally stable) chaos. Results of numerical integration of the model equations of different degree of accuracy are presented and discussed. Also, circuit simulation of the electronic generator is provided using the NI Multisim environment. Portraits of attractors, waveforms of generated oscillations, Lyapunov exponents, and spectra are considered and found to be in good correspondence for the dynamics on the attractive sets of the self-oscillatory systems and for the original Anosov geodesic flow. The hyperbolic nature of the dynamics is tested numerically using a criterion based on statistics of angles of intersection of stable and unstable subspaces of the perturbation vectors at a reference phase trajectory on the attractor.

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A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system

Cited by 59 publications

References 13 publications

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

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

Hyperbolic Chaos of Turing Patterns

Chaos in three coupled rotators: From Anosov dynamics to hyperbolic attractors

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