Double autoresonance in two-dimensional dynamical systems

The phenomenon of double autoresonance in dynamical systems with two degrees of freedom is examined. We analyze the motion of a particle in a two-dimensional centrally symmetric potential, subject to a homogeneous quasiperiodic external field of elliptical polarization. It is shown that if the particle has a sufficiently small initial energy, a double resonance is established when the slowly varying driving frequency approaches the linear resonance frequency. As the driving frequency is changed further, the double resonance is maintained, causing a continuous excitation of the oscillator. When nonlinearity becomes significant, the motion transforms into nearly circular oscillations. The necessary conditions for the persistence of the autoresonance are studied in detail.