Bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit

In this paper, we investigate bifurcation phenomena, such as those of the periodic solutions, for the ‘unperturbed’ nonlinear system Gð _ xÞ€CbxZ 0, with Gð _ xÞZ ðaC _ x 2 Þ= ð1C _ x 2 Þ and aO1, bO0, when we add the two competing termsKf ðtÞCg _ x, with f (t )a time-periodic analytic ‘forcing’ function and gO0 the dissipative parameter. The resulting differential equation Gð _ xÞ€ xCbxCg _ xKf ðtÞZ 0 describes approximately an electronic system known as the saturating inductor circuit. For any periodic orbit of the unperturbed system, we provide conditions which give rise to the appearance of subharmonic solutions. Furthermore, we show that other bifurcation phenomena arise as there is a periodic solution with the same period as the forcing function f (t) which branches off the origin when the perturbation is switched on. We also show that such a solution, which encircles the origin, attracts the entire phase space when the dissipative parameter becomes large enough. We then compute numerically the basins of attraction of the attractive periodic solutions by choosing specific examples of the forcing function f(t), which are dictated by experience. We provide evidence showing that all the dynamics of the saturating inductor circuit is organized by the persistent subharmonic solutions and by the periodic solution around the origin.