Abstract
The nonlinear second order differential equation which describes the driven series RL-diode circuit, biased to operate only in the depletion region is given by X + γX + Xμ = α - βsin&tau, where differentiation is with respect to time τ and γ > 0, α > |β| > 0, μ > 1 and X ≥ 0. Without these restrictions and specifically with μ = 2, this differential equation has also been found to exist in a couple of other research contexts, namely in the studies of ship roll and capsize, as well as in a perturbed Korteweg-de Vries equation, where stationary wave solutions are described by a special case of the differential equation. Particular attention is paid firstly to the case μ = 2, which in contrast to Duffing’s equation (μ = 3) has received little attention, and secondly, the case μ = 1.67, which arises from measured values made on a practical diode. The main aim here has been to give a detailed analysis of some properties of the system’s solutions. A rigorous phase plane analysis establishes solution behaviour and a criterion for which solutions grow without bound, before subharmonic solutions of various orders are exhibited. By partitioning the phase plane into regions in which only certain solution behaviour occurs, a variety of invariant sets can be constructed. A numerical scheme which detects unstable periodic orbits is applied to the system, resulting in detection of a set of unstable periodic solutions. This detailed analysis goes some way towards understanding the dynamics of this system.