Numerical and experimental exploration of phase control of chaos

Abstract

A well-known method to suppress chaos in a periodically forced chaotic system is to add a harmonic perturbation. The phase control of chaos scheme uses the phase difference between a small added harmonic perturbation and the main driving to suppress chaos, leading the system to different periodic orbits. Using the Duffing oscillator as a paradigm, we present here an in-depth study of this technique. A thorough numerical exploration has been made focused in the important role played by the phase, from which new interesting patterns in parameter space have appeared. On the other hand, our novel experimental implementation of phase control in an electronic circuit confirms both the well-known features of this method and the new ones detected numerically. All this may help in future implementations of phase control of chaos, which is globally confirmed here to be robust and easy to implement experimentally.

Diffusion dynamics and first passage time in a two-coupled pendulum system

“…Earlier, the role of phase on controlling of chaos has been explored. [38][39][40][41] We have paid our attention to the effect of phase on diffusion dynamics. Diffusion can be controlled by varying the phase .…”

“…[38][39][40][41] For example, Qu et al 38 showed the possibility of control of chaos in Duffing oscillator by tuning the phase factor. Also they reported the occurrence of new type of intermittency called breathing effect, in which the regular and chaotic motions appeared periodically, when the phase difference is a time-dependent function.…”

“…Also they reported the occurrence of new type of intermittency called breathing effect, in which the regular and chaotic motions appeared periodically, when the phase difference is a time-dependent function. Zambrano et al 40 considered parametrically driven Duffing oscillator with the addition of a periodic force. They analyzed the suppression of chaos in the ͑⑀ , ͒ ͑where ⑀ is the amplitude of the parametric force͒ parameter space both numerically solving the equation of motion and through its analog electronic circuit simulation.…”

A Unified Approach to Chaos Suppressing and Inducing in a Periodically Forced Family of Nonlinear Oscillators

“…Nonfeedback control methods have been mainly used to suppress chaos in periodically driven dynamical systems. The key idea of these nonfeedback methods is to apply a harmonic perturbation either to some of the parameters or as an additional forcing [31], and its effectiveness is shown numerically and experimentally in [32]- [36]. If a small external control force is applied in the direction of the unstable manifold, the system's trajectory can converge to the target orbit along the stable manifold.…”

“…However, in experimental implementations, the fast response that these methods require cannot usually be provided. In such cases, nonfeedback methods might be more useful [31]. Nonfeedback control methods have been mainly used to suppress chaos in periodically driven dynamical systems.…”