Taming chaos in damped driven systems by incommensurate excitations

The possibility of reducing and even suppressing chaos in dissipative nonautonomous systems by additional incommensurate chaos-suppressing excitations is theoretically demonstrated through the universal example of a perturbed Duffing oscillator by considering rational approximations to the irrational ratio Omega/omega between the chaos-suppressing and chaos-inducing frequencies. For each chosen rational approximation, analytical predictions for the suitable amplitudes and initial phases of the chaos-suppressing excitation are numerically confirmed for small amplitudes and at the predicted suitable initial phases. For the significant case of the golden ratio Phi = (1 + root 5) /2, we study the structural stability of the suppressory scenario as the respective convergents approximate the irrational ratio Omega/omega = 1/Phi. Our theory predicts and numerical simulations confirm that the values of the suitable amplitudes are rather insensitive to high-order convergents. On the contrary, the number and values of the suitable initial phases critically depend on each particular convergent in order to satisfy two requirements: (1) Maximum approximation to the frustration of the homoclinic bifurcation existing in the absence of the chaos-suppressing excitation and (2) maximum survival of a relevant spatio-temporal symmetry of the dynamical equation. (c) 2019 Elsevier B.V. All rights reserved.