| Abstrakt: |
We consider the existence of response solutions for the quasi-periodic perturbation of degenerate reversible harmonic oscillators ẍ − λxn = ∈ƒ(ωt, x, x, ∈), x ∈ R, where λ = ±1, n > 1, is an integer and ƒ(−ωt,x,−x,∈) = ƒ(ωt,x,x,∈). With satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and ε* sufficiently small, response solutions exist for each satisfying a weak non-resonant condition; (2) For and sufficiently small, there exists a Cantor set E ∈ (0,ε*) with almost full Lebesgue measure such that response solutions exist for each ε ∈ E if ω satisfies a Diophantine condition. Non-existence of response solutions is also discussed when ƒ fails to satisfy the non-degenerate conditions. [ABSTRACT FROM AUTHOR] |
