Global center and limit cycles in generalized piecewise cubic Liénard systems

This paper aims to investigate two classical problems related to global center conditions and bifurcation of small-amplitude limit cycles in piecewise Liénard systems of the form $\dot{x}=y-F\left(x\right)$, $\dot{y}=-g\left(x\right)$, where $F\left(x\right)$ and $g\left(x\right)$ are both piecewise cubic polynomials. The explicit global center conditions at the origin are derived from this piecewise Liénard system. Furthermore, utilizing Poincaré-Lyapunov theory, the existence of nine limit cycles (isolate periodic solutions) around the origin is proved. As recognized to now, it is a new lower bound of the maximum number of limit cycles for such Liénard systems.