Upper bound for the number of limit cycles for symmetric piecewise linear differential equations with three zones

Abstract

The study of the periodic dynamics of the family of continuous observable and non-controllable three-dimensional symmetric piecewise linear systems with three zones considered here can be reduced to the study of periodic solutions of the piecewise differential equation $x^{\prime }=ax+(b-a)\mathrm{sat}(x)+\mu \sin t$, where $\mathrm{sat}$ stands for the normalized saturation function. This paper proves that the number of limit cycles (that is, isolated periodic solutions) of the equation is uniformly bounded independently of $a,b,\mu$. Moreover, when $ab>0$, the equation has exactly one limit cycle for every $\mu >0$. When $ab<0$, it has exactly three limit cycles for sufficiently small $\mu >0$ and exactly one for sufficiently large $\mu$. When $ab<0$, using Melnikov theory for a deformation of the equation, we prove that, for each fixed $\mu >0$ away from two critical values and for sufficiently small nonzero values of the perturbation parameter, there are exactly three, five or one limit cycles, depending on $\mu$. This strengthens the conjecture that the equation has at most five limit cycles.

arXiv comment25 pages, 7 figures