Parabolic invariant tori in quasi-periodically forced skew-product maps

We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps phi : R-n x T-d -> R-n x T-d: phi((z)(theta)) = ((theta + omega) (z + phi(z) + h(z, theta) + epsilon f(z, theta))), where phi : R-n -> R-n is a homogeneous function of degree 1 with l >= 2 and h = O(vertical bar z vertical bar(l+1)) We obtain the following results: (a) For n = 1, l being odd and epsilon sufficiently small, parabolic invariant tori exist if omega satisfies the Brjuno-Russmann's non-resonant condition. (b) For n = 1, and epsilon sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency omega does not need any arithmetic condition. (ii) First order average is non-zero and omega satisfies the Brjuno-type weak non-resonant condition; (iii) l = 2, first order average is zero, both first and second order non-average parts are small enough and omega satisfying Brjuno-type weak non-resonant condition; (iv) l > 2, first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and omega satisfies the Brjuno-type weak non-resonant condition. (c) In the case n > 1, if first order average belongs to the interior of the range of phi, Spec(D phi)boolean AND iR = empty set, first order non-average part is small enough and the forcing frequency omega does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for epsilon sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This study investigates the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps. Different conditions are considered for different scenarios, and it is shown that parabolic invariant tori exist under certain conditions.