Mode-locking in heavily damped, periodically and quasiperiodically driven nonlinear oscillators
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In the case of certain nonlinear oscillators, both elapsed time t and the system’s primary state variable θ may be thought of as periodic, meaning that the state space for the oscillator may be projected on to a torus. Each orbit may therefore be characterised by a winding number. Such systems exhibit a tendency to mode-lock: that is, to tend to orbits whose winding numbers are the sums of rational multiples of the driving frequencies. Mode-locking is influenced by the amplitudes of the components of such driving, and may be induced or suppressed by adjusting these. The phenomenon of strange nonchaotic attractors is associated with mode-locking, and its absence, in quasiperiodically driven oscillators. Various driving regimes, such as harmonic (i.e. sinusoidal) driving, biharmonic quasiperiodic driving (that is quasiperiodic driving consisting of superposition of two sinusoids), and biharmonic periodic driving, give rise to systems belonging to the same family, for which a general method of deploying second-order perturbation theory has been developed and applied. In all cases, the shape of regions in parameter space associated with mode-locking may be approximated using Bessel functions, or infinite sums that involve them. These results have been compared with numerical findings resulting from simple boundary location using interval bisection. Under certain conditions, all three of the types of driving exhibit “pinching”: regions in parameter space associated with mode-locking, themselves of positive measure, have zero width along certain paths. At such points, the behaviour of these non-integrable systems becomes surprisingly regular: in the case of periodic driving, for example, all orbits become, themselves, periodic (and not merely asymptotically so). Robust numerical evidence has been collected, using algorithms developed in Mathematica, for the reality of pinching in some important regions of parameter space, but the phenomenon is also very easy to break down (either by leaving such regions or through introducing certain kinds of harmonic).
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