Arnold’s cat map dynamics in a system of coupled nonautonomous van der Pol oscillators

Abstract

An example of a flow system is presented with an attractor concentrated mostly at a surface of a two-dimensional torus, the dynamics on which is governed by the Arnold cat map. The system is composed of four coupled nonautonomous van der Pol oscillators. Three of them have equal characteristic frequencies, and in the other one the frequency is twice as large. The parameters controlling excitation of the two pairs of oscillators are forced to undergo a slow counterphase periodic modulation in time. At the end of the active stage for one pair of the oscillators, the excitation is passed to another pair, than back, and so on. In terms of a stroboscopic Poincaré section, the respective eight-dimensional (8D) mapping, due to strong phase volume compression, reduces approximately to a 2D map for the phases of one pair of the oscillators that corresponds approximately to the Arnold cat map. The largest two Lyapunov exponents (one positive and one negative) are close to those predicted with the cat map model. Estimates for the fractal dimension of the attractor of the Poincaré map are close to 2.

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. ).…”

Autonomous coupled oscillators with hyperbolic strange attractors

“…Later, such a system was designed as an electronic device and studied experimentally [9]. Similar constructions of coupled nonautonomous self-oscillators were used to implement dynamics associated with the Arnold cat map [10] and with the robust strange nonchaotic attractor [11].…”

Chaotic communication with robust hyperbolic transmitter and receiver

“…x r = εs(t), y r − (A cos ω 0 t/N − y 2 r )ẏ r + ω 2 0 y r = εw r , z r + (A cos ω 0 t/N + z 2 r )ż r + 4ω 2 0 z r = εx r y r , w r + (A cos ω 0 t/N + w 2 r )ẇ r + ω 2 0 z r = εx r , (13) where s(t) = z d cos(ω 0 t + ξ(t)). The parameter values are chosen as in [10]: A = 2.0, ε = 0.4, ε = 2π, N = 20. From the viewpoint of confidentiality, such scheme has an advantage due to higher dimension of the phase space; that makes difficulties for the decoding of the potentially intercepted signal.…”