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The aim of this paper is to investigate two classical problems related tonilpotent center conditions and bifurcation of limit cycles in switchingpolynomial systems. Due to the difficulty in calculating the Lyapunov constantsof switching polynomial systems at non-elementary singular points, it isextremely difficult to use the existing Poincar\'e-Lyapunov method to studythese two problems. In this paper, we develop a higher-orderPoincar\'e-Lyapunov method to consider the nilpotent center problem inswitching polynomial systems, with particular attention focused on cubicswitching Li\'enard systems. With proper perturbations, explicit centerconditions are derived for switching Li\'enard systems at a nilpotent center,which is characterized as global. Moreover, with Bogdanov-Takens bifurcationtheory, the existence of five limit cycles around the nilpotent center isproved for a class of switching Li\'enard systems, which is a new lower boundof cyclicity for such polynomial systems around a nilpotent center.

Nilpotent center conditions in cubic switching polynomial Li\'enard systems by higher-order analysis