Oscillatory variation of anomalous diffusion in pendulum systems

Abstract

Numerical studies of anomalous diffusion in undamped but periodically-driven and parametrically-driven pendulum systems are presented. When the frequency of the periodic driving force is varied, the exponent μ, which is the rate of divergence of the mean square displacement with time, is found to vary in an oscillatory manner. We show the presence of such a variation in other statistical measures such as variance of position, kurtosis, and exponents in the power-exponential law of probability distribution of position.

Strong anomalous diffusion of the phase of a chaotic pendulum

“…Of course, the identification of just six ballistic regions is a simplification of the real complexity of the phase space structure: it may well be that the velocity-based approach we are adopting here simply does not have enough resolving power to distinguish regions with similar winding number but quite different residence time distributions, and this is reflected in the lack of a definite power law for regions B ± 2 . Note that previous studies of the phase space structure of this same system [20] focused on the extreme velocity regions B ± 2 and B ± 3 , overlooking the role played by B ± 1 which we on the contrary find determinant to describe the p-4 Table 1: Parameters of the kinetic model that mimics the diffusion properties of the phase of the chaotic pendulum for ω = 0.8. The system diffuses during an exponentially distributed time, and then enters with probability pi one of the six ballistic regions characterised by winding number wi and residence time exponent αi.…”

“…The periodically driven undamped pendulum equation we consider herë θ + sin θ = γ sin(ωt) (2) has been studied previously by the authors of refs. [17][18][19][20], who report instances of both regular (in the case γ = 1.2, ω = 0.1) and anomalous (for γ = 1.2, ω = 0.8) diffusion. Actually, we partially correct here those statements, showing explicitly that the diffusion is strongly anomalous in both cases.…”