FM 06-10 (mp_arc 06-201)

Authors: Michele V. Bartuccelli, Jonathan H.B. Deane, and Guido Gentile

Title: Bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit
 
Abstract: In this paper we investigate bifurcation phenomena, such as those of the periodic solutions, for the ``unperturbed'' nonlinear system $G(\dot{x}) \ddot{x} + \beta x =0$, with $G(\dot{x}) = (\alpha + \dot{x}^2)/(1+\dot{x}^2)$ and $\alpha > 1$, $\beta > 0$, when we add the two competing terms $-f(t) + \gamma \dot{x}$, with $f(t)$ a time-periodic analytic ``forcing'' function and $\gamma>0$ the dissipative parameter. The resulting differential equation $G(\dot{x}) \ddot{x} + \beta x + \gamma \dot{x} - f(t) = 0$ 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 organised by the persistent subharmonic solutions and by the periodic solution around the origin.

Keywords: Nonlinear dynamics; Bifurcation theory; Subharmonic solutions; Basin of attraction; Dissipative systems; Electric circuits.

Michele Bartuccelli
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: m.bartuccelli@surrey.ac.uk

Jonathan Deane
Department of Mathematics and Statistics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: j.deane@surrey.ac.uk

Guido Gentile
Dipartimento di Matematica
UniversitÓ di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it