Simple small fixed-amplitude feedback perturbation controls complex dynamics

Many nonlinear dynamical systems exhibit complex periodic motions and chaos, which can be undesirable. We demonstrate that these complex periodic dynamics and chaos can be controlled by feedback perturbations of a small fixed amplitude (ϵ) on the relevant parameter. In particular, period-2 dynamics can be suppressed by creating a small confining attractor, higher-period states can be controlled to lower ones, and chaos can be tamed to the desired periodic states. In addition, the critical values of ϵ can be calculated analytically. Results of the control on the logistic map are presented with exact explicit expressions for critical ϵ c for period-2 suppression and period-4 to period-2 control. The phase diagram for stable periodic regions under the control is obtained analytically and verified numerically. Furthermore, the control method is demonstrated experimentally, without prior knowledge of the equation of motion, for the complex motions of a compass under an oscillating magnetic field. The robustness of the method suggests that small regulatory effects can suppress undesirable higher-period and chaotic dynamics in complex dynamical systems, and the relevant biological or physiological implications are discussed.