The transversal homoclinic point of a saddle-node implies horseshoe

A method of identifying the existence of horseshoe for a two-dimension diffeomorphism is introduced and utilized to generalize the Birkhoff-Smale Theorem to the saddle-node case. Project supported by the National Natural Science Foundation of China (Grant No. 19531070).     

The transversal homoclinic solutions and chaos for stochastic ordinary differential equations

We consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a Brownian force. By Lyapunov–Schmidt reduction, the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces. It is shown that, for almost all sample paths of the Brownian motion, the perturbed system exhibits chaos.     

Variations of maximum-clique transversal sets on graphs

A maximum-clique transversal set of a graph G is a subset of vertices intersecting all maximum cliques of G. The maximum-clique transversal set problem is to find a maximum-clique transversal set of G of minimum cardinality. Motivated by the placement of transmitters for cellular telephones, Chang, Kloks, and Lee introduced the concept of maximum-clique transversal sets on graphs in 2001. In this paper, we introduce the concept of maximum-clique perfect and some variations of the maximum-clique transversal set problem such as the {k}-maximum-clique, k-fold maximum-clique, signed maximum-clique, and minus maximum-clique transversal problems. We show that balanced graphs, strongly chordal graphs, and distance-hereditary graphs are maximum-clique perfect. Besides, we present a unified approach to these four problems on strongly chordal graphs and give complexity results for the following classes of graphs: split graphs, balanced graphs, comparability graphs, distance-hereditary graphs, dually chordal graphs, doubly chordal graphs, chordal graphs, planar graphs, and triangle-free graphs.     

The density of the transversal homoclinic points in the Henon-like strange attractors

We consider the Henon-like strange attractors Λ in a family which is a nonsingular perturbation of a d-modal family. The existence of the Henon-like strange attractors in this family was proved by Diaz et al. [Inventions Math. 125 (1996) 37]. We prove that the transversal homoclinic points are dense in Λ, and that hyperbolic periodic points are dense in Λ. Moreover, the hyperbolic periodic points that are heteroclinically related to the primary periodic point (transversal intersection of stable and unstable manifolds) are dense in Λ.     

Balanced sets and circuits in a transversal space

The study of fully dependent sets (unions of circuits) has played a part in characterizing transversal spaces. In fact, the fully dependent sets satisfy |Δ(F)| = ?(F) in any deltoid representation, and it is with a consideration of this property that we begin the present paper. We study “balanced” sets and from our results draw conclusions about fully dependent sets and circuits in a transversal space. These include upper bounds for the number of circuits, and the result that a non-trivial transversal space can be neither a hereditary circuit space nor the dual of a geometric hereditary circuit space. The paper is reasonably self-contained; all unusual terms are defined as they are encountered.     

A generalization of hadwiger's transversal theorem to intersecting sets

Given an ordered family of compact convex sets in the plane, if every three sets can be intersected by some directed line consistent with the ordering, then there exists a common transversal of the family. This generalizes Hadwiger's Transversal Theorem to families of compact convex sets which are not necessarily pairwise disjoint. If every six sets can be intersected by some directed line consistent with the ordering, then there exists a common transversal which is consistent with the ordering. If the family is pairwise disjoint and every four sets can be intersected by some directed line consistent with the ordering, then there exists a common transversal which is consistent with the ordering.     

Transversal homoclinic points and hyperbolic sets for non-autonomous maps II

The method of Melnikov is generalized to non-autonomous maps. If the Melnikov function has infinitely many zeros with derivatives bounded away from zero then the system admits a generalized hyperbolic set as it was introduced in part I. The developed theory is applied to almost periodically perturbed differential equations.

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Zusammenfassung Die Methode von Melnikov wird verallgemeinert für nichtautonome Abbildungen. Falls die Melnikov-Funktion unendlich viele Nullstellen mit von Null weg beschränkten Ableitungen hat, dann enthält das System eine verallgemeinerte hyberbolische Menge, wie sie in Teil I eingeführt wurde. Die entwickelte Theorie wird auf fast periodisch gestörte Systeme angewandt.

     

Transversal homoclinic points and hyperbolic sets for non-autonomous maps I

A concept of generalized hyperbolic sets for non-autonomous maps is developed. Starting from transversal homoclinic orbits such generalized hyperbolic sets are constructed. The Shadowing Lemma is proven for maps admitting a generalized hyperbolic set. Time dependent symbolic dynamics is introduced and related to non-autonomous maps.

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Zusammenfassung Das Konzept von verallgemeinerten hyperbolischen Mengen für nicht-autonome Abbildungen wird entwickelt. Ausgehend von transversalen homoklinen Bahnen werden solche verallgemeinerte hyperbolische Mengen konstruiert. Das Shadowing Lemma wird für Abbildungen bewiesen, welche eine verallgemeinerte hyperbolische Menge haben. Es wird zeitabhängige symbolische Dynamik eingeführt und der Zusammenhang mit nicht-autonomen Abbildungen dargestellt.

     

Hunting for homoclinic orbits in reversible systems: A shooting technique

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are Hamiltonian systems with quadratic potential energy. In such systems, homoclinic orbits that are invariant under the reversible transformation are typically not destroyed as a parameter is varied. A strategy is proposed for the direct numerical approximation to paths of such homoclinic orbits, exploiting the special properties of reversible systems. This strategy incorporates continuation using a simplification of known methods and a shooting approach, based on Newton's method, to compute starting solutions for continuation. For Hamiltonian systems, the shooting uses symplectic numerical integration. Strategies are discussed for obtaining initial guesses for the unknown parameters in Newton's method. An example system, for which there is an infinity of symmetric homoclinic orbits, is used to test the numerical techniques. It is illustrated how the orbits can be systematically located and followed. Excellent agreement is found between theory and numerics.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4–14 July 1992 with support from the SERC under grant reference number GR/H03964.     

The transversal homoclinic points are dense in the codimension-1 Hénon-like strange attractors

We consider the codimension-1 Hénon-like strange attractors . We prove that the transversal homoclinic points are dense in , and that hyperbolic periodic points are dense in . Moreover the hyperbolic periodic points that are heteroclinically related to the primary fixed point ( transversal intersection of stable and unstable manifolds) are dense in .

     

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line. All authors are partially supported by CONACYT research grant 5040017.     

Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems

In this paper, we study the problem of homoclinic orbits to Aubry sets for time-periodic positive definite Lagrangian systems. We show that there are infinitely many homoclinic orbits to some Aubry set under the conditions that the associated Mather set is uniquely ergodic and the first relative homology group of the projection of this Aubry set is nonzero.     

Invariant sets and symmetric periodic motions of reversible mechanical systems

A method of constructing and classifying all symmetric periodic motions of a reversible mechanical system is proposed. The principal solution of the above problem is given for the Hill problem, the restricted three-body problem (including the photogravitational problem), the problem of a heavy rigid body with a fixed point, and that of a heavy rigid body on a rough plane. In particular, problems requiring a systematic numerical study are therby formulated.     

Topologically Algebraic Algebras

We show that a locally convex algebra is topologically algebraic if, and only if, it is algebraic.Thanks are offered to Professor M. Oudadess for many remarks which allowed an improvement of the first version of this paper. We are also indebted to Professor A. Mallios for valuable suggestions.     

拓扑遍历映射

本文研究紧致度量空间上拓扑遍历自映射的性质、判定及与混沌的关系．