Persistence of Multiscale Degenerate Invariant Tori for Reversible Systems with Multiscale Degenerate Equilibrium Points

In this paper, we focus on the persistence of degenerate lower-dimensional invariant tori with a normal degenerate equilibrium point in reversible systems. Based on the Herman method and the topological degree theory, it is proved that if the frequency mapping has nonzero topological degree and the frequency $\omega_0$ satisfies the Diophantine condition, then the lower-dimensional invariant torus with the frequency $\omega_0$ persists under sufficiently small perturbations. Moreover, the above result can also be obtained when the reversible system is Gevrey smooth. As some applications, we apply our theorem to some specific examples to study the persistence of multiscale degenerate lower-dimensional invariant tori with prescribed frequencies.