Multiresonant control of two-dimensional dynamical systems

Abstract

It is shown that many two degree of freedom (2D) nonlinear dynamical systems can be controlled by continuous phase-locking (double autoresonance) between the two canonical angle variables of the system and two independent external oscillating perturbations having slowly varying frequencies. Conditions for stability of the 2D autoresonance and classification of systems with doubly autoresonant solutions in the vicinity of a stable equilibrium are outlined in terms of the Hessian matrix elements of the unperturbed system. The doubly autoresonant states in a generic, driven 2D system can be accessed by starting in equilibrium and simultaneous passage through two linear resonances in the system, provided that the driving amplitudes exceed a threshold scaling as alpha(3/4) , alpha being the characteristic chirp rate of the driving frequencies. The formation of nearly periodic trajectories in linearly nondegenerate, 2D driven systems with a single stable equilibrium is suggested as an application. Examples of autoresonant excitation and formation of nearly periodic states in other types of driven systems are presented, including a three-particle Toda chain, a particle in a 2D double-well potential, and a 3D oscillator.