Maintenance and Suppression of Chaos by Weak Harmonic Perturbations: A Unified View

Abstract

General results concerning maintenance or enhancement of chaos are presented for dissipative systems subjected to two harmonic perturbations (one chaos inducing and the other chaos enhancing). The connection with previous results on chaos suppression is also discussed in a general setting. It is demonstrated that, in general, a second harmonic perturbation can reliably play an enhancer or inhibitor role by solely adjusting its initial phase. Numerical results indicate that general theoretical findings concerning periodic chaos-inducing perturbations also work for aperiodic chaos-inducing perturbations, and in arrays of identical chaotic coupled oscillators.

May chaos always be suppressed by parametric perturbations?

“…1 and 2, where it was proposed to perturb periodically the system parameters. Later, this approach gained analytical substantiation in a series of publications [3][4][5][6][7][8][9] ͑as a review, see Ref. 10͒.…”

“…14,21 Modifications of the Duffing oscillator have also attracted a considerable amount of interest as an appropriate model for the investigation of the chaos suppression phenomenon ͑see Refs. 3,8,19, and 20 and references cited therein͒. One of the modified Duffing equations is known as the Duffing-Holmes system:…”

Recent Developments in Dynamical Systems: Three Perspectives

“…Let us start with an example coming from using the Poincaré map to understand the behavior of orbits in non-linear and non-autonomous differential equations of second order (see for example [Chacón, 2001]), namely…”

Dynamics of two coupled chaotic systems driven by external signals

“…[23,24], appearing new states, including chaos suppression, bifurcations, intermittences, etc., as it was mentioned. A very clear analysis of the effects of signal injection to low dimensional systems (Lorentz, Rössler, Van der Pol and Takens Bogdanov) related with symmetries, can be found in [25] the parameter space).…”