Nonlinear dynamics of quasiperiodically driven oscillators: Knotted torus attractors and their bifurcations

The influence of quasiperiodic parametric excitation on a nonlinear ordinary differential equation (ODE) is investigated. The equation of motion for axial oscillations of charged particles in quadrupole ion traps, z&d3;+uz&d2; +4g+acos2t-ecos 2wft-z+b z3=0, 0.1 is considered as a particular example. This ODE has the form of a nonlinear quasiperiodic Mathieu equation with damping. To capture the non-periodic driving, we develop a geometric perspective by suspending (0.1) in a 4-dimensional phase space, R2 x T2. Solutions to (0.1) in this space are attracted to invariant 2-tori. Interestingly, these torus attractors are non-trivially knotted as surfaces. Cross-sections of these dynamically and topologically intriguing objects have the form of braids. Using pairs of braids obtained by constructing two essentially orthogonal Poincare sections, the topology of the invariant 2-tori can be captured. The braid pairs are most useful for observing a new bifurcation through which knotted torus attractors change knot type, called a topological torus bifurcation (TTB). In a TTB, an attracting parent torus loses stability and becomes of saddle type, while shedding a new daughter attractor. Braids associated with the daughter attractor have double the number of strands of their now saddle-type parent braids.; TTBs strongly influence the dynamics of the ODE. They produce increasingly complicated power spectra, adding spectral lines in a predictable way that can be directly related to the topology of the torus attractor and the associated braids. TTBs also lead to a loss of normal hyperbolicity at the bifurcation where the principal Lyapunov exponent passes through zero. By computing the principal Lyapunov exponent over a range of epsilon values, a geometric sequence of TTBs is observed. After sufficiently many bifurcations, the principal Lyapunov exponent becomes positive indicating the transition to chaos. Following the transition to chaos, knotted torus attractors are replaced by strange attractors. The cross sections of these strange attractors have the topology of fractal braids, and they preserve in the chaotic dynamics ghosts of quasiperiodic behavior.; TTBs also play a role in resonance behavior for (0.1). Pairs of TTBs bound sharp, large-amplitude resonances for appropriate choices of o f. A perturbation technique based on the familiar method of multiple scales is developed to analyze these resonances. In the modification developed here, a sequence of linear Mathieu equations is solved order-by-order, in place of the standard simple harmonic oscillator. The method gives resonance conditions, provides approximations to solutions, and captures the topology of invariant tori in the phase space.; Finally, some interesting paths for further research are sketched. These include an investigation of chaotic transport in the phase space mediated by heteroclinic intersection of stable and unstable manifolds of different knotted tori, and a study of the influence of stochastic driving on knotted tori.