Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Concerns double homoclinic loops with orbit flips and two resonant eigen- values in a four-dimensional system.We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium,and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops,then compose them to get the important Poincarémap.A simple cal- culation gives explicitly an expression of the associated successor function.By a delicate analysis of the bifurcation equation,we obtain the condition that the original double homoclinic loops are kept,and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface,2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions.We also locate the large periodic orbits and large homoclinic orbits and their number.     

Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields

We study bifurcations of two types of homoclinic orbits—a homoclinic orbit with resonant eigenvalues and an inclination-flip homoclinic orbit. For the former, we prove thatN-homoclinic orbits (N3) never bifurcate from the original homoclinic orbit. This answers a problem raised by Chow-Deng-Fiedler (J. Dynam. Diff. Eq. 2, 177–244, 1990). For the latter, we investigate mainlyN-homoclinic orbits andN-periodic orbits forN=1, 2 and determine whether they bifurcate or not under an additional condition on the eigenvalues of the linearized vector field around the equilibrium point.     

Homoclinic Twist Bifurcation in a System of Two Coupled Oscillators

We analyse the dynamics of two identical Josephson junctions coupled through a purely capacitive load in the neighborhood of a degenerate symmetric homoclinic orbit. A bifurcation function is obtained applying Lin's version of the Lyapunov–Schmidt reduction. We locate in parameter space the region of existence of n-periodic orbits, and we prove the existence of n-homoclinic orbits and bounded nonperiodic orbits. A singular limit of the bifurcation function yields a one-dimensional mapping which is analyzed. Numerical computations of nonsymmetric homoclinic orbits have been performed, and we show the relevance of these computations by comparing the results with the analysis.     

On the Homoclinic Orbits in a Class of Two-Degree-of-Freedom Systems under the Resonance Conditions

ＩｎｔｒｏｄｕｃｔｉｏｎＴｗｏ＿ｄｅｇｒｅｅ＿ｏｆ＿ｆｒｅｅｄｏｍｓｙｓｔｅｍｓｈａｖｉｎｇｃｕｂｉｃｎｏｎｌｉｎｅａｒｉｔｉｅｓａｒｅｅｘｔｅｎｓｉｖｅｌｙｕｓｅｄｉｎｐｈｙｓｉｃｓ,ｍｅｃｈａｎｉｃｓ.Ｆｏｒｅｘａｍｐｌｅ :ｔｈｅｌａｒｇｅ＿ａｍｐｌｉｔｕｄｅｖｉｂｒａｔｉｏｎｓｏｆｓｔｒｉｎｇｓ,ｂｅａｍｓ,ｍｅｍｂｒａｎｅｓａｎｄｐｌａｔｅｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎ＿ｉｓｏｌａｔｉｏｎｓｙｓｔｅｍｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎａｂｓｏｒｂｅｒｓ,ｔｈｅｍｏｔｉｏｎｏｆｓｐｈｅ…     

Bifurcations of heteroclinic loop accompanied by pitchfork bifurcation

In this paper, using the local coordinate moving frame approach, we investigate bifurcations of generic heteroclinic loop with a hyperbolic equilibrium and a nonhyperbolic equilibrium which undergoes a pitchfork bifurcation. Under some generic hypotheses, the existence of homoclinic loop, heteroclinic loop, periodic orbit and three or four heteroclinic orbits is obtained. In addition, the non-coexistence conditions for homoclinic loop and periodic orbit are also given. Note that the results achieved here can be extended to higher dimensional systems.     

2<Emphasis Type="Italic">T</Emphasis>-periodic solution for<Emphasis Type="Italic">m</Emphasis> order neutral type differential equations with time delays

Periodic solution of m order linear neutral equations with constant coefficient and time delays was studied. Existence and uniqueness of 2 T-periodic solutions for the equation were discussed by using the method of Fourier series. Some new necessary and sufficient conditions of existence and uniqueness of 2T-periodic solutions for the equation are obtained. The main result is used widely. It contains results in some correlation paper for its special case, improves and extends the main results in them. Existence of periodic solution for the equation in larger number of particular case can be checked by using the result, but cannot be checked in another paper. In other words, the main result in this paper is most generalized for (1), the better result cannot be found by using the same method. Foundation item: the Natural Science Foundation of Yunnan Education Committee (990002) Biography: ZHANG Bao-sheng (1962-)     

N-Homoclinic bifurcations for homoclinic orbits changing their twisting

We study bifurcations, calledN-homoclinic bifurcations, which produce homoclinic orbits roundingN times (N2) in some tubular neighborhood of original homoclinic orbit. A family of vector fields undergoes such a bifurcation when it is a perturbation of a vector field with a homoclinic orbit.N-Homoclinic bifurcations are divided into two cases; one is that the linearization at the equilibrium has only real principal eigenvalues, and the other is that it has complex principal eigenvalues. We treat the former case, espcially that linearization has only one unstable eigenvalue. As main tools we use a topological method, namely, Conley index theory, which enables us to treat more degenerate cases than those studied by analytical methods.     

The existence of infinitely many homoclinic doubling bifurcations from some codimension 3 homoclinic orbits

An inclination-flip homoclinic orbit of weak type on ³ is a homoclinic orbit given as the intersection of a special one-dimensionalC ²-weak stable manifold and the one-dimensional unstable manifold of a hyperbolic singularity with three real eigenvalues. In this paper, we show that in a generic unfolding of such a homoclinic orbit, there appear curves in the parameter space that correspond to ordinary inclination-flip homoclinic orbit of orderN for any integerN. As a consequence, there exist infinitely many homoclinic doubling bifurcation curves emanating from the codimension three degenerate point corresponding to the inclination flip homoclinic orbit of weak type.     

Homoclinic Shadowing

A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in Rⁿ is presented. It is a computer-assisted technique with two main components. First, a global Newton’s method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.     

Exponential Dichotomies,the Fredholm Alternative,and Transverse Homoclinic Orbits in Partial Functional Differential Equations

In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral. Ams Subject Classifications: 35R10; 58F14.     

The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｅｘｉｓｔｅｎｃｅｏｆ2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｔｏｔｈｅｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｏｆｆｉｒｓｔ＿ｏｒｄｅｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈａｄｅｖｉａｔｉｎｇａｒｇｕｍｅｎｔ ｘ(ｔ) =Ｂｘ(ｔ) Ｆ(ｘ(ｔ-τ) ) ｐ(ｔ) ,( 1 )ｗｈｅｒｅｘ(ｔ)∈Ｒ2 , ｘ(ｔ) =ｄｄｔｘ(ｔ) ,τ∈Ｒ ,Ｂ∈Ｒ2×2 ,Ｆ :Ｒ2 →Ｒ2 ｉｓｂｏｕｎｄｅｄａｎｄｐ∈Ｃ(…     

Homoclinic flip bifurcation with a nonhyperbolic equilibrium

In this paper, the problem of homoclinic bifurcation accompanied by a transcritical bifurcation is investigated for high-dimensional systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. Under certain nongeneric conditions (orbit flip and inclination flip homoclinic orbits), the existence, nonexistence, coexistence and uniqueness of homoclinic and periodic orbits are studied. Some known results are extended.     

Exponential dichotomies and transversal homoclinic orbits in degenerate cases

In this paper, we first give a sufficient condition which assures that a linear differential equation depending on a small parameter admits an exponential dichotomy onR, then we use the result obtained here on exponential dichotomies to investigate the existence of transversal homoclinic orbits of perturbed differential systems in two degenerate cases and obtain a Melnikov-type vector. The results on exponential dichotomies of this paper provide us a tool of proving the transversality of homoclinic orbits in studying degenerate bifurcations.This work is supported by NSF of China.     

Constrained multiobjective games in locally convex<Emphasis Type="Italic">H</Emphasis>-spaces

A new class of constrained multiobjective games with infinite players in noncompact locally convex H-spaces without linear structure are introduced and studied. By applying a Fan-Glicksberg type fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values and a maximum theorem, several existence theorems of weighted Nath-equilibria and Pareto equilibria for the constrained multiobjective games are proved in noncompact locally convex H-spaces. These theorems improve, unify and generalize the corresponding results of the multiobjective games in recent literatures. Contributed by Ding Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province ([2000]25) Biography: Ding Xie-ping (1938-)     

Convergence of a Modified SLP Algorithm for the Extended Linear Complementarity Problem

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｔｈｅｅｘｔｅｎｄｅｄｌｉｎｅａｒｃｏｍｐｌｅｍｅｎｔａｒｉｔｙｐｒｏｂｌｅｍ (ＸＬＣＰ)ｉｎｔｒｏｄｕｃｅｄｂｙＭａｎｇａｓａｒｉａｎａｎｄＰａｎｇ[1](ａｌｓｏｓｅｅＹｅ[2 ]) :Ｆｉｎｄａｖｅｃｔｏｒ(ｘ ,ｙ) ∈Ｒ2ｎｓｕｃｈｔｈａｔＭｘ-Ｎｙ∈Ｋ ,ｘＴｙ=0 ,　ｘ≥ 0 ,ｙ≥ 0 ,( 1 )ｗｈｅｒｅＭ ,Ｎ ∈Ｒｍ×ｎａｒｅｇｉｖｅｎ ,ａｎｄＫｉｓａｐｏｌｙｈｅｄｒａｌｓｅｔｉｎＲｍｄｅｆｉｎｅｄｂｙＫ :=ｕ∈Ｒｍ｜Ｇｕ≥ｇ ,　　Ｇ∈Ｒｌ×ｍ,ｇ∈Ｒｌ.Ｔｈｉｓｐｒｏ…     

Chaos in three-dimensional hybrid systems and design of chaos generators

The existence of horseshoes is proved in a class of 3-dim piecewise linear systems, in which a homoclinic orbit connecting the origin to itself is explicitly given. Based on these results, a mathematically rigorous methodology for design of chaos generators is proposed. Implementation of such chaos generators by circuit is easy and the chaotic attractor is robust under small perturbations.     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅱ)—SINGLE TIME-DELAY CASE

The controllability for switched linear systems with time-delay in controls is first investigated. The whole work contains three parts. This is the second part. The definition and determination of controllability of switched linear systems with single time-delay in control functions is mainly investigated. The sufficient and necessary conditions for the oneperiodic, multiple-periodic controllability of periodic-type systems and controllability of periodic systems are presented, respectively. Contributed by YE Qing-kai Foundation items: the National Natural Science Foundation of China (69925307, 60274001); the National Key Basic Research and Development Program (2002CB312200); the Postdoctoral Program Foundation of China Biography: XIE Guang-ming (1972 ∼), Doctor (E-mail: xiegming@mech.pku.edu.cn)     

Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip

In this paper, the bifurcations of heterodimensional cycles are investigated by setting up a suitable local coordinate system in a four-dimensional system. Under some ungeneric conditions—one orbit flip and one inclination flip—the persistence of heterodimensional cycles, the existence of homoclinic orbit and a family of periodic orbits, the coexistence of heterodimensional loop and periodic orbit are obtained. Also, the relevant bifurcation surfaces and their existing regions are given.     

Bifurcation analysis for spherically symmetric systems using invariant theory

We study a degenerate steady state bifurcation problem with spherical symmetry. This singularity, with the five dimensional irreducible action ofO(3), has been studied by several authors for codimensions up to 2. We look at the case where the topological codimension is 3, theC -codimension is 5. We find a tertiary Hopf bifurcation and a heteroclinic orbit. Our analysis does not use any specific properties of the five dimensional representation and can in principle be used for higher representations as well. The computations are based on invariant theory and orbit space reduction.