Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

We study connected branches of nonconstant 2π-periodic solutions of the Hamilton equation

     

The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

A twist condition and periodic solutions of Hamiltonian systems

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian system

(HS)     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

Summary. Generalizing the degenerate KAM theorem under the Rüssmann nondegeneracy and the isoenergetic KAM theorem, we employ a quasilinear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth submanifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the submanifold, we shall show the following: (a) the majority of the unperturbed tori on the submanifold will persist; (b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; (c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a subisoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.     

Limit cycles near hyperbolas in quadratic systems

In this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane. We study a strip of hyperbolas occurring in some quadratic systems. We deal with the cyclicity of the degenerate graphics DI2a from the programme, set up in [F. Dumortier, R. Roussarie, C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations 110 (1994) 86-133], to solve the finiteness part of Hilbert's 16th problem for quadratic systems. Techniques from geometric singular perturbation theory are combined with the use of the Bautin ideal. We also rely on the theory of Darboux integrability.     

The compact category and multiple periodic solutions of Hamiltonian systems on symmetric starshaped energy surfaces

Dieter Puppe zum 60. Geburtstag gewidmet     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

The existence of periodic solutions of non-autonomous second-order Hamiltonian systems

The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second-order Hamiltonian systems. Some new existence theorems are obtained by using the least action principle and the saddle point theorem.     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

Local invariants of singular surfaces in an almost complex four-manifold

In this paper we define two local invariants, the local self-intersection index and the Maslov index, for singular surfaces in an almost complex four-manifold and prove formulae involving these invariants, which generalize formulae of Lai and Givental.Dedicated to Professor Masahisa Adachi on his 60th birthday.     

The Brake Orbits of Hamiltonian Systems on Positive-type Hypersurfaces

This paper deals with the brake orbits of Hamiltonian system on given energy hypersurfaces Σ = H ⁻¹(1). We introduce a class of contact type but not necessarily star-shaped hypersurfaces in ℝ²ⁿ and call them normalized positive-type hypersurfaces. By using of the critical point theory, we prove that if Σ is a partially symmetric normalized positive-type hypersurface, it must carries a brake orbit of (HS). Furthermore, we obtain some multiplicity results under certain pinching conditions. Our results include the earlier works on this subject given by P. Rabinowitz and A. Szulkin in star-shaped case. An example of partially symmetric normalized positive-type hypersurface in ℝ⁴ that is not star-shaped is also presented Partially supported by NNSF of China (10571085) and Science Foundation of Hohai University.     

Completely degenerate lower-dimensional invariant tori for Hamiltonian system

We study the persistence of lower-dimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.     

Existence of homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary-value problems which are obtained by the minimax methods.     

Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term

In this paper, the boundedness of all solutions of the oscillator

x^″+f(x,x^′)+ω²x+?(x)=p(t)$x^{″}+f(x,x^{\prime })+{\omega }^{2}x+?(x)=p(t)$

Non-autonomous second order Hamiltonian systems

We study the existence of periodic solutions for a second order non-autonomous dynamical system containing variable kinetic energy terms. Our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other. We study sublinear problems and the existence of non-constant solutions.     

Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmann's non-degeneracy condition

In this paper we prove Gevrey-smoothness of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under Rüssmann's non-degeneracy condition by an improved KAM iteration.