**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

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it