Three Problems In Nonlinear Dynamics: Time Delay, Fractionality And Synchronization

Abstract

Three problems in nonlinear dynamics are studied with concern to the effects of time history dependent functions on steady state behavior. In each problem we consider either a single oscillator or a system of oscillators. The appearance of the time history dependence varies over the problems, and is be introduced either as a delayed term or a fractional derivative. In our first problem, a van der Pol type system with delayed feedback is explored by employing a two variable expansion perturbation method. The resulting amplitude-delay relation predicts two Hopf bifurcation curves, such that in the region between these two curves oscillations will be quenched. The perturbation results are verified by comparison with numerical integration. The second and third problems are on the subject of fractional derivatives. In analyzing these problems we look to extend classic perturbation methods to the treatment of fractional derivatives. In the second problem we also consider a single oscillator. The oscillator may be described as a damped Mathieu type where the damping term has been replaced by a fractional derivative. The order of the fractional derivative considered ranges from 0 to 1. Both lowest order and higher order approximations for the n = 1 transition curves, which separate regions of stability from instability, are found using the method of harmonic balance. An approximation for the n = 0 transition curve is also obtained. In the limiting cases of the fractional derivative's order, [alpha], being 0 or 1, the fractional Mathieu equation being considered respectively reduces to the familiar undamped and damped Mathieu equations. The undamped and damped Mathieu equations are well studied and our results may be compared with the known results in these cases. Through these comparisons conclusions are drawn as to the validity of assumptions made in applying the method of harmonic balance as well as the effect of the fractional derivative. In the third problem, the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators is studied. A two variable perturbation method is applied to the system's corresponding variational equations to obtain expressions for the transition curves separating regions of stability from instability. The perturbation results are validated with numerics and, as in the second problem, through direct comparison with known results in the limiting cases of fractional derivative order taking on the values of [alpha] = 0 and [alpha] = 1.