Subharmonic solutions in the restricted three-body problem
Hildeberto E. Cabral Zhihong Xia
Discrete & Continuous Dynamical Systems - A 1995, 1(4): 463-474 doi: 10.3934/dcds.1995.1.463
In this paper, we study the subharmonic bifurcations in the restricted three-body problem. By study the Melnikov integrals for the subharmonic solutions, we obtain the precise bifurcation scenario nearby the circular solutions when one of the two primaries is small.
keywords: subharmonic bifurcations Melnikov integrals. restricted three-body problem
Heteroclinic orbits and chaotic invariant sets for monotone twist maps
Tifei Qian Zhihong Xia
Discrete & Continuous Dynamical Systems - A 2003, 9(1): 69-95 doi: 10.3934/dcds.2003.9.69
We consider the monotone twist map $\bar f$ on $(\mathbb R/\mathbb Z)\times R$, itslift $f$ on $R^2$ and its associated variational principle $h:\mathbb R^2\to\mathbb R$ through its generating function. By working with the variationalprinciple $h$, we first show that for an adjacent minimal chain$\{(u^k, v^k)\}_{k=s}^t$ of fixed points of $f$, if there exists abarrier $B_k$ for each adjacent minimal pair $u^k < u^{k+1}$, $ s \le k \le {t-1} $, then there exists a heteroclinic orbit between $(u^s, v^s)$ and$(u^t, v^t)$, then by assuming that there is a barrier for any twoneighboring globally minimal critical points and $m$ is sufficientlylarge, we construct an invariant set $\Lambda^m\subset (\mathbb R/\mathbb Z)\times\mathbb R$ such that the shift map of the $n$-symbol space is a factor of$\bar f^m|_{\Lambda^m}$, where $n$ is the total number of the globallyminimal fixed points of $\bar f$.
keywords: chaotic invariant sets. Hamiltonian dynamics variational method Twist maps
Hyperbolic invariant sets with positive measures
Zhihong Xia
Discrete & Continuous Dynamical Systems - A 2006, 15(3): 811-818 doi: 10.3934/dcds.2006.15.811
In this short paper we prove some results concerning volume-preserving Anosov diffeomorphisms on compact manifolds. The first theorem is that if a $C^{1 + \alpha}$, $\alpha >0$, volume-preserving diffeomorphism on a compact connected manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. The same result had been obtained by Bochi and Viana [2]. This result is not necessarily true for $C^1$ maps. The proof uses a Pugh-Shub type of dynamically defined measure density points, which are different from the standard Lebesgue density points. We then give a direct proof of the ergodicity of $C^{1+\alpha}$ volume preserving Anosov diffeomorphisms, without using the usual Hopf arguments or the Birkhoff ergodic theorem. The method we introduced also has interesting applications to partially hyperbolic and non-uniformly hyperbolic systems.
keywords: Ergodic theory hyperbolic invariant set. volume-preserving diffeomorphisms Anosov map
Homoclinic points and intersections of Lagrangian submanifold
Zhihong Xia
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 243-253 doi: 10.3934/dcds.2000.6.243
In this paper, we prove certain persistence properties of the homoclinic points in Hamiltonian systems and symplectic diffeomorphisms. We show that, under some general conditions, stable and unstable manifolds of hyperbolic periodic points intersect in a very persistent way and we also give some simple criteria for positive topological entropy. The method used is the intersection theory of Lagrangian submanifolds of symplectic manifolds.
keywords: Lagrangian submanifold. Homoclinic points

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