ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 58 Cheng, Chong-Qing, Nanjing University, China Aboundance of Lower Dimensional Invariant Tori in Hamiltonian Systems Consider a Hamiltonian system with n degrees of freedom (n ≥ e3). Assume it is of KAM type, H(p, q) = N(p) + P (p, q), i. e. N is the main part and P is a small perturbation. Also the Hessian of N is nondegenerate. If ω = \frac{∂N}{∂p} |_{p_{o}} satisfies one resonance condition ⟨I, ω⟩ = 0, (I ∈ ℤ^{n}), then the invariant torus of unperturbed system p = p_{o} has a foliation into a family of (n − 1) dimensional tori. Assume the frequence of the flow on these lower dimensional tori are of diophantine. We show that there are infinitely many (n − 1)-dimensional invariant tori emerging from the break down of the resonance torus under small perturbations. Generically, these tori are in infinitely many different homological classes. References:

1. D. Berstein and A. Katok, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians, Invent. Math. 88 (1987) 224–241

2. C. -Q. Cheng, Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems, Commun. Math. Phys. 177 (1996) 529–559