Response Solutions for Completely Degenerate Oscillators Under Arbitrary Quasi-Periodic Perturbations

In this paper, we consider a one-dimensional completely degenerate oscillator subjected to an analytically e-dependent quasi-periodic perturbation, whose frequencies satisfy a Diophantine condition. By the KAM method, we show that one of the following results holds true: 1. For all sufficiently small e and all initial values ? E T1, there exists a family of analytically (e,?)-parameterized response solutions, which corresponds to the persistence of the resonant Lagrangian torus of the equivalent Hamiltonian system. 2. For all sufficiently small e, there exists a response solution, moreover, for an uncountable number of sufficiently small e, there exists another response solution. In this case, the resonant Lagrangian torus of the equivalent Hamiltonian system is destroyed and it splits into a hyperbolic or hyperbolic-type degenerate lower dimensional torus for all sufficiently small e, and another (possibly elliptic) lower dimensional torus for an uncountable number of sufficiently small e.