COMPLETELY DEGENERATE LOWER-DIMENSIONAL INVARIANT TORI IN REVERSIBLE SYSTEMS

In this paper, we consider the persistence of completely degenerate lower-dimensional invariant tori of the following reversible system {(x) over dot = omega + Q(x)z + epsilon P-1(x, z, epsilon), (z) over dot = H(z) + epsilon P-2(x, z, epsilon), where (x, z) = (x, y, u, v) is an element of T-n x R-m x R x R with m >= n + 2, H(z) = (0, v(2p+1) + y(m)(l), uy(m-1)(q))T with y = (y(1), center dot center dot center dot , y(m-1), y(m)), p, q >= 0, l > 0 are integers, the involution G is (x, y, u, v) -> (-x, y,-u, v), Q(x) is a n x m matrix function, omega is a Diophantine frequency, epsilon is a small positive parameter and epsilon P-1, epsilon P-2 are analytic perturbation terms. By the Kolmogorov-ArnoldMoser method, we prove that for sufficiently small epsilon the above reversible system admits lower-dimensional invariant tori with prescribed frequency omega if average of a part of Q(x) is non-singular. This should be the first persistence result of lower-dimensional invariant tori in completely degenerate reversible systems.