Autonomous coupled oscillators with hyperbolic strange attractors

Abstract

We propose several examples of smooth low-order autonomous dynamical systems which have apparently uniformly hyperbolic attractors. The general idea is based on the use of coupled self-sustained oscillators where, due to certain amplitude nonlinearities, successive epochs of damped and excited oscillations alternate. Because of additional, phase sensitive coupling terms in the equations, the transfer of excitation from one oscillator to another is accompanied by a phase transformation corresponding to some chaotic map (in particular, an expanding circle map or Anosov map of a torus). The first example we construct is a minimal model possessing an attractor of the Smale-Williams type. It is a four-dimensional system composed of two oscillators. The underlying amplitude equations are similar to those of the predator-pray model. The other three examples are systems of three coupled oscillators with a heteroclinic cycle. This scheme presents more variability for the phase manipulations: in the six-dimensional system not only the Smale-Williams attractor, but also an attractor with Arnold cat map dynamics near a two-dimensional toral surface, and a hyperchaotic attractor with two positive Lyapunov exponents, are realized.

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

“…The proposed principle can be implemented, for example, in systems of electronics and nonlinear optics for the generation of robust chaos. Moreover, on this principle, one can build many models manifesting various interesting phenomena of complex dynamics, like it was done with systems based on manipulation by phases of successively generated oscillatory trains (e.g., Arnold cat map dynamics [26,38], complex analytic dynamics with Mandelbrot and Julia sets [39], robust strange nonchaotic attractor [40], etc. ).…”

“…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.…”

“…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].…”

Hyperbolic Chaos of Turing Patterns

“…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]).…”