Zhihong Xia

The existence of noncollision singularities in newtonian systems

In this paper we solve a long-standing problem in celestial mechanics proposed by Painleve and Poincare in the last century. The problem, which concerns the nature of the singularities in the n-body problem, asks whether there exists a noncollision singularity in the newtonian n-body problem? Here we give an affirmative answer to this problem by proving the existence of noncollision singularities in the 5-body problem. We consider n point-masses moving in a euclidean space W3. Let the mass of the ith particle be mi > 0, let its position be qj E R3 and let 4i E ]3 be its velocity. According to Newtons law,

Bifurcations of heteroclinic loops

By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop Γ to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1-hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1-per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1-hom and 1-per orbit and the inexistence of the 2-hom and 2-per orbit are also obtained.

Existence of invariant tori in volume-preserving diffeomorphisms

In this paper we consider certain volume-preserving diffeomorphisms on I × T n , where I ∈ ℝ is a closed interval and T n is an n -dimensional torus. We show that under certain non-degeneracy conditions, all of the maps sufficiently close to the integrable maps preserve a large set of n -dimensional invariant tori.

Central configurations with many small masses

Abstract By using the method of analytical continuation, we find the exact numbers of central configurations for some open sets of n positive masses for any choice of n. It turns out that the numbers increase dramatically as n increases; e.g., for some open set of 18 positive masses, some 2.08766 × 1020 classes of distinctive central configurations are found. In the mean time, we obtained some results about the Hausdorff measure for the set of n positive masses where degenerate central configuration arises.

Melnikov method and transversal homoclinic points in the restricted three-body problem

Abstract In this paper we show, by Melnikov method, the existence of the transversal homoclinic orbits in the circular restricted three-body problem for all but some finite number of values of the mass ratio of the two primaries. This implies the existence of a family of oscillatory and capture motion. This also shows the non-existence of any real analytic integral in the circular restricted three-body problem besides the well-known Jacobi integral for all but possibly finite number of values of the mass ratio of the two primaries. This extends a classical theorem of Poincare [10]. Because the resulting singularities in our equation are degenerate, a stable manifold theorem of McGehee [7] is used.

Homoclinic points in symplectic and volume-preserving diffeomorphisms

LetMn be a compactn-dimensional manifold and ω be a symplectic or volume form onMn. Let ϕ be aC1 diffeomorphism onMn that preserves ω and letp be a hyperbolic periodic point of Φ. We show that genericallyp has a homoclinic point, and moreover, the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp. Takens [11] obtained the same result forn=2.

Partial hyperbolicity or dense elliptic periodic points for C 1-generic symplectic diffeomorphisms

We prove that if a symplectic diffeomorphism is not partially hyperbolic, then with an arbitrarily smal C 1 perturbation we can create a totally elliptic periodic point inside any given open set. As a consequence, a C 1 -generic symplectic diffeomorphism is either partially hyperbolic or it has dense elliptic periodic points. This extends the similar results of S. New-house in dimension 2 and M.-C. Arnaud in dimension 4. Another interesting consequence is that stably ergodic symplectic diffeomorphisms must be partially hyperbolic, a conserve to Shub-Pughs stable ergodicity conjecture for the symplectic case.

Topological entropy and partially hyperbolic diffeomorphisms

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of all

The existence of oscillatory and superhyperbolic motion in Newtonian systems

C^\8

Area-Preserving Surface Diffeomorphisms

diffeomorphisms. The proof uses a topological invariant we introduced; Yomdins theorem on upper semi-continuity; Katoks theorem on lower semi-continuity for two dimensional systems and a refined Pesin-Ruelle inequality we proved for partially hyperbolic diffeomorphisms.