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

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 $x^{\mathrm{\prime }}=ax+(b-a)(x)+\mu \sin t$, where $\mathrm{sat}$ 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.