Michele V. Bartuccelli, Jonathan H.B. Deane, Guido Gentile
Bifurcation phenomena and attractive periodic solutions 
in the saturating inductor circuit
(995K, pdf)

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.