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

TL;DR

This paper establishes an upper bound on the number of limit cycles in a symmetric three-zone piecewise linear differential system, showing it is finite and can be exactly one or three, with implications for the conjecture of a maximum of five.

Contribution

It proves the finiteness of limit cycles and determines exact maximum counts for certain parameters, advancing understanding of limit cycle bounds in such systems.

Findings

Number of limit cycles is finite, independent of parameters.

Maximum limit cycles can be exactly one or three for certain parameters.

For small perturbations, the system has exactly one, three, or five limit cycles.

Abstract

The study of the dynamics of a continuous observable and non-controllable three-dimensional symmetric piecewise linear system with three zones can be reduced to the study of the existence of limit cycles for the piecewise differential equation $x'=ax+(b-a)\mathop{\rm sat}(x)+\mu\sin t$, where $\operatorname{sat}$ stands for the normalized saturation function. This paper proves that the number of limit cycles of the equation is finite independently of $a,b,\mu$. Moreover, it is proven that the maximum number of limit cycles is exactly one or three for certain values of the parameters. Using Melnikov theory for a certain deformation of the equation, it is proven that for small values of the perturbation parameter, there are exactly three, five or one limit cycles, depending on the values of $\mu$. This strengthens the conjecture that the equation has at most five limit cycles.