Persistence Of Lower Dimensional Invariant Tori In Hamiltonian Systems

The persistence of solutions to integrable Hamiltonian systems had puzzled people for long up to the appearance of the celebrated KAM [8, 1, 13] theory, which affirmed the majority of invariant tori. i.e. the solutions to Hamiltonian systems, persist under small perturbations. The classical KAM theorem answered the persistence of invariant tori for standard Hamiltonian systems.Later Melnikov [12] formulated a KAM type persistence result for elliptic lower dimensional tori of integrable Hamiltonian systems under now so-called Melnikov's non-resonance condition. More precisely, for a system with the following HamiltonianMelnikov announced that the majority of invariant tori survive the small perturbations under the following conditionsfor k Zn, l Zm, |k| +|l| 0. The complete proof of his result was later carried out by Eliasson, Kuksin, and P schel [4, 9, 17]. In fact, Moser [14] had already noted the persistence of elliptic lower dimensional tori. He proved that the existence of the quasi-periodic solutions when the toriadmit 2-dimensional elliptic equilibrium point. However the way he used can not be applied to higher dimension, because he requested the tangent frequency be fixed. Later, Eliasson [4] removed the restriction successfully by letting the frequency suffer small perturbations and later P schel [17] simplified the proof of Eliasson [4]. For Hamiltonianwhere (x,y,u) Tn x Rn R2m. Moser [14] obtained the persistence of hyperbolic invariant tori when 0 6 Rn is a fixed Diophantine toral frequency and the eigenvalues of JM (J being the standard symplectic matrix in R2m) are real and distinct. Graff [5] generalized Moser's result by allowing multiple eigenvalues of JM. And the proof of Graff's result was later given by Zehnder [22] using implicit function techniques. More recently, Li and Yi [11] generalized the results of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by allowing the degeneracy of the unperturbed Hamiltonians and they obtain the preservation of part or full components of tangent frequencies. They adopted the Fourier series expansion for normal form N, which is a new technique.Recently, Chow, Li and Yi [2] proved that the majority of the unperturbed tori on sub-manifolds will persist under a non-degenerate condition of R ssmann type for standard Hamiltonian systems. Motivated by their work, in the first part of this paper, we shall show that lower dimensional tori also survive small perturbations on sub-manifolds under some assumptions. The surviving tori might be elliptic, hyperbolic, or of mixed type.We consider a real analytic family of Hamiltonian systems of the following formwhere (x,y,u) lies in a complex neighborhood {(x, y, u) : |Imx| r, dist(y, G) s, |u| s} of Tn G {0} C Tn x Rn x R2m, G Rn(n 2) is a bounded closed region and P is small. Besides these, we also assume that A0)Nu(y,0) = 0, detNw(y,0) 0.To prove the persistence of lower dimensional invariant tori of system (1.1), we first consider the following parameter-dependent, real analytic Hamiltonian systemwhere (x,y,u) lies in a complex neighborhood D(r, s) = {(x, y, u) : |Imx| r, |y| s, |u| s} of Tn {0} {0} Tn Rn R2m, A is a parameter lying in a bounded closed region A Rn0 and M( ) is nonsingular on A. In the above, all A dependency are of class Cl00for some l0 n.We assume the following conditions: A1) rank{ : |a| n - 1} = n for all A.A2) For (k,l) satisfying 0 < |k| K = max1 i 2m,0 r n-1| | and|l| < 2, the following holds:where = ( 1, ... , 2m) , j, j= 1, ..., 2m are eigenvalues of J M, a is a constant which will be determined later and "meas" denotes the Lebesgue measure in Rn0. A3) rankA(X) = d on A, and, there is a smoothly varying, nonsingular,d x d principal minor A(X) of A(X).Under the above assumptions, we obtain the persistence of lower dimensional tori on sub-manifolds. Since the eigenvalues of JM may be real, or pure imaginary, or some are real and others are imaginary, the presived tori may be hyperbol...