Switching between orbits in a periodic window

Triggering and enhancing chaos with a prescribed target Lyapunov exponent using optimized perturbations of minimum power

The objective of the present work is to propose a method for nonfeedback anticontrol of chaos with perturbations of minimum power for a preset control goal. The noted Lorenz system is employed as the test model for chaotification with the target state specified by a prescribed positive value of the largest Lyapunov exponent (LLE), lambda[over ]>0 . Periodic and quasiperiodic perturbations are used as control signals, and the signals parameters are optimized using a genetic algorithm under restriction of minimum power. Performance of the optimized signals in triggering chaos at an ordered state, fixed point or periodic, as well as further enhancing chaoticity at a chaotic state is explored. The present numerical experiments reveal the following interesting physics about chaotification. In general, the power for chaotification increases with the preset value of lambda and quasiperiodic signals can achieve the control goal with a lower power than periodic ones. Given the same increment of LLE from that of the uncontrolled state (lambda1,0) , i.e., Deltalambda=lambda-lambda1,0, the further enhancement of chaoticity in a chaotic state needs a higher control power than the triggering of chaos from an ordered state. The minimum power required for chaotification of an ordered state increases relatively slowly for lower lambda[over ] but increases drastically as the preset target LLE reaches a certain critical value. Most strikingly, the numerical experiments demonstrate that this critical value of lambda corresponds to LLE of the nearest chaotic state in the neighborhood of the uncontrolled state. Robustness of applying the present method in the presence of external noise is also demonstrated.

Maintenance and suppression of chaos by weak harmonic perturbations: a unified view

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.

Fluctuations and the energy-optimal control of chaos

The energy-optimal entraining of the dynamics of a periodically driven oscillator, moving it from a chaotic attractor to a coexisting stable limit cycle, is investigated via analysis of fluctuational transitions between the two states. The deterministic optimal control function is identified with the corresponding optimal fluctuational force, which is found by numerical and analog simulations.