Design of coupled Andronov–Hopf oscillators with desired strange attractors

Abstract

This paper develops a design method for the interconnections of a network of Andronov-Hopf oscillators such that the system exhibits a desired strange attractor. Because of the structure of the oscillators, the desired behavior can be achieved via weak linear coupling, which destabilizes the oscillators' phase difference. First, a set of sufficient conditions are established that result in phase destabilization, and thus instability, of a desired periodic solution. Then, an additional condition is determined to ensure that all harmonic periodic orbits will be unstable. Finally, additional numerical properties are assessed, where tuning of a small parameter can result in chaos.

Distributed Network of Coupled Oscillators With Multiple Limit Cycles

“…where M ∈ C n×n is the coupling matrix consisting of g kl and ω k . It should be noted that the couplings in (4) are nonlinear, unlike the linearly coupled Andronov-Hopf oscillators 3 considered in the literature [26], [30], [50]. It turns out that the nonlinearity of the couplings is important for encoding the oscillation frequency into the network, rather than into the individual oscillator unit.…”

Control and anti-control of chaos based on the moving largest Lyapunov exponent using reinforcement learning

“…Li and Xu proposed that chaotification of the quasi-zero-stiffness system can mask line spectrum characteristics of acoustic noise of machinery vibration, so as to enhance the concealment capability of underwater vehicles [18]. In the neural network of biological systems with periodic solutions, Kohannim and Iwasaki achieved an expected strange attractor by tuning a small parameter to destabilize oscillators' phase difference [19]. In this sense, chaos provides great flexibility for the system performance because the chaotic attractor is embedded in infinite unstable periodic orbits and we can accomplish different goals with the aid of these unstable orbits [20].…”