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1.
When nondegenerate homoclinic orbits to an expanding fixed point of a map f:X→X,X⊆Rn, exist, the point is called a snap-back repeller. It is known that the relevance of a snap-back repeller (in its original definition) is due to the fact that it implies the existence of an invariant set on which the map is chaotic. However, when does the first homoclinic orbit appear? When can other homoclinic explosions, i.e., appearance of infinitely many new homoclinic orbits, occur? As noticed by many authors, these problems are still open. In this work we characterize these bifurcations, for any kind of map, smooth or piecewise smooth, continuous or discontinuous, defined in a bounded or unbounded closed set. We define a noncritical homoclinic orbit and a homoclinic orbit of an expanding fixed point is structurally stable iff it is noncritical. That is, only critical homoclinic orbits are responsible for the homoclinic explosions. The possible kinds of critical homoclinic orbits will be also investigated, as well as their dynamic role. 相似文献
2.
Dr. F. Rothe 《manuscripta mathematica》1992,74(1):299-319
A theorem of Poincaré guarantees existence of the local conjugacy of an entire analytic mapping with an hyperbolically unstable fixed point to the linearized mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. Especially we focus on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snap back to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. It turns out that in general there exist infinitely many snapback points and for each one of them there exist infinitely many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the Theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, we related the multiplicity of the zeros of the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most multiplicity two. 相似文献
3.
Jian Cheng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(3):400-419
We study in this article the topological entropy of billiard systems on a convex domain
of the Euclidean plane. We restrict our attention to those systems whose boundary curve has
positive curvature and show that for generic billiard ball systems satisfying this condition the
topological entropy is positive. 相似文献
4.
研究较一般的高维退化系统的同宿、异宿轨道分支问题.利用推广的Melnikov函数、横截性理论及奇摄动理论,对具有鞍—中心型奇点的带有角变量的奇摄动系统,在角变量频率产生共振的情况下,讨论其同宿、异缩轨道的扰动下保存和横截的条件.推广和改进了一些文献的结果。 相似文献
5.
In this paper, we use the functional analytic method (theory of exponential dichotomies and Liapunov-Schmidt method) to study the homoclinic bifurcations of higher dimensional difference equations in a degenerate case. We obtain a Melnikov vector mapping for difference equations with the help of which the existence of transversal homoclinic orbits can be detected. 相似文献
6.
A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide. 相似文献
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8.
Tasso J. Kaper Gregor Kovacic 《Transactions of the American Mathematical Society》1996,348(10):3835-3887
We establish the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multi-bump \v{S}ilnikov orbits, connect equilibria and periodic orbits in the resonance band. The main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands.
9.
《Nonlinear Analysis: Theory, Methods & Applications》2010,72(12):6254-6264
The persistence of degenerate homoclinic orbit is considered for parabolic functional differential equations with small periodic perturbations. Bifurcation functions constructed between two finite-dimensional spaces are obtained. The zeros of the function correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover, We show that the homoclinic solution for the perturbed system is transversal and hence the perturbed system exhibits chaos. 相似文献
10.
Eric Paturel 《Calculus of Variations and Partial Differential Equations》2001,12(2):117-143
In this paper, we obtain the existence of at least two nontrivial homoclinic orbits for a class of second order autonomous
Hamiltonian systems. This multiplicity result is obtained by a new variational method based on the relative category: to overcome
the lack of compactness of the problem, we first solve perturbed nonautonomous problems and study the limit of the solutions
as the nonautonomous perturbation goes to 0. This method allows to get rid of some assumptions on the potential used in the
work of Ambrosetti and Coti-Zelati.
Received August 9, 1999 / Accepted September 7, 1999 / Published online September 14, 2000 相似文献
11.
Guangping Luo 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):6254-6264
The persistence of degenerate homoclinic orbit is considered for parabolic functional differential equations with small periodic perturbations. Bifurcation functions constructed between two finite-dimensional spaces are obtained. The zeros of the function correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover, We show that the homoclinic solution for the perturbed system is transversal and hence the perturbed system exhibits chaos. 相似文献
12.
We obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth-order difference equations by using Mountain Pass Theorem, a weak convergence argument and a discrete version of Lieb’s lemma. 相似文献
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14.
In this paper, the existence of homoclinic orbits for the second-order Hamiltonian systems without periodicity is studied and infinitely many homoclinic orbits for both superlinear and asymptotically linear cases are obtained. 相似文献
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16.
Some existence theorems for even homoclinic orbits are obtained for a class of second-order nonautonomous Hamiltonian systems with symmetric potentials under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil-boundary-value problems which are obtained by the minimax methods. 相似文献
17.
We consider the construction of the plateau of the α-function in a hyperbolic and positive definite Lagrangian system, and link the boundries of the α-function's plateau with the distribution of c-minimal homoclinic orbits to Aubry sets. 相似文献
18.
Consider the equation is small. For λ = 0, there is a homoclinic orbit Γ through zero. For λ ≠ 0 and small, there can be “strange” attractors near Γ. The purpose of this paper is to determine the curves in λ-space of bifurcation to “strange” attractors and to relate this to hyperbolic subharmonic bifurcations. 相似文献
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20.
WUSHAOPING YANGHAITAO 《高校应用数学学报(英文版)》1998,13(3):251-262
Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a(t)|s|^2∫^t0f(t,s)ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a(t)dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too. 相似文献