COEXISTENCE OF HYPERBOLIC AND ELLIPTIC INVARIANT TORI FOR COMPLETELY DEGENERATE QUASI-PERIODICALLY FORCED MAPS

Consider the following completely degenerate quasi-periodically forced skew-product maps of the form {(x) over bar = x + y(m) + epsilon f(1)(x, y, theta, epsilon) + h(1)(x, y, theta, epsilon), (y) over bar = y + lambda x(n) + epsilon f(2)(x, y, theta, epsilon) + h(2)(x, y, theta, epsilon), (theta) over bar = theta + omega, where (x, y, theta) is an element of R x R x T-d, lambda = +/- 1, omega is an element of R-d, n and m are positive integers satisfying n >= m, mn > 1, f(1), f(2), h(1), h(2) are real analytic on (x, y, theta) and C-1-Whitney smooth on epsilon, and h(1), h(2) = O(vertical bar(x, y)vertical bar(n+1)). The existence of weak-hyperbolic (weak-elliptic) invariant tori for the above maps with lambda = 1 (lambda = -1) has been proved in [25, 30]([25]). In this paper, we prove the above completely degenerate skew-product map both in the case lambda = 1 and in the case lambda = -1 not only admits weak-hyperbolic invariant tori but also weak-elliptic invariant tori under the suitable conditions. Moreover, the number of invariant tori is investigated in both cases. See Theorem 2.1 and Theorem 2.2. The results of this paper are the situations that are not discussed in the existing literature.

Automated summary

This paper investigates the existence and quantitative properties of completely degenerate quasi-periodically forced skew-product maps of specific forms, and proves that under certain conditions, these skew-product maps not only have weak-hyperbolic invariant tori, but also weak-elliptic invariant tori. The number of invariant tori is also examined in both cases. These results address situations that have not been discussed in existing literature.