共查询到20条相似文献,搜索用时 218 毫秒
1.
I. S. Shilin 《Doklady Mathematics》2016,94(1):415-417
We show that the Lyapunov instability of Milnor attractors is a locally topologically generic dynamical phenomenon that accompanies persistent homoclinic tangencies associated with sectionally dissipative periodic saddles. 相似文献
2.
We study the semilocal dynamics of two-dimensional symplectic diffeomorphisms with homoclinic tangencies. Conditions for the existence of infinitely many generic elliptic periodic orbits of successive periods starting with some integer are found. Bibliography: 14 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 155–166. 相似文献
3.
B. Al-Hdaibat D. L. van Kekem Yu. A. Kuznetsov 《Journal of Difference Equations and Applications》2018,24(4):575-587
It is known that in the Bogdanov–Takens map there exists a zone of transversal homoclinic intersections bounded by two curves of homoclinic tangencies. In this paper, we derive an improved asymptotic formula for the homoclinic parameter values of the BT map. We compare two methods to approximate the Bogdanov–Takens map by the time-1 flow of a vector-field, and find that they are equivalent. We show that it is essential to include the second-order terms w.r.t. the parameters to obtain a more accurate asymptotic for the homoclinic zone. We show how to use this new homoclinic asymptotic to compute branches of homoclinic tangencies in the BT map numerically, obtaining the whole homoclinic structure of the Bogdanov–Takens map. 相似文献
4.
The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer–van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (which are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies. 相似文献
5.
Anton Gorodetski Vadim Kaloshin 《Proceedings of the Steklov Institute of Mathematics》2009,267(1):76-90
We study generic unfoldings of homoclinic tangencies of two-dimensional area-preserving diffeomorphisms (conservative New
house phenomena) and show that they give rise to invariant hyperbolic sets of arbitrarily large Hausdorff dimension. As applications,
we discuss the size of the stochastic layer of a standard map and the Hausdorff dimension of invariant hyperbolic sets for
certain restricted three-body problems. We avoid involved technical details and only concentrate on the ideas of the proof
of the presented results. 相似文献
6.
We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds. 相似文献
7.
This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of Thom's transversality theorem, as well as the stratification of real-algebraic sets. Optimal results on the order of nondegeneracy are given. 相似文献
8.
We study bifurcations of two-dimensional symplectic maps with quadratic homoclinic tangencies and prove results on the existence
of cascade of elliptic periodic points for one and two parameter general unfoldings.
相似文献
9.
Jens D.M. Rademacher 《Journal of Differential Equations》2010,249(2):305-348
This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered. 相似文献
10.
We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransverse heteroclinic cycles. We show that
bifurcations under consideration lead to the birth of wild-hyperbolic Lorenz attractors. These attractors can be viewed as
periodically perturbed classical Lorenz attractors, however, they allow for the existence of homoclinic tangencies and, hence,
wild hyperbolic sets.
相似文献
11.
Devaney has shown that an autonomous Hamiltonian system in dimension 4, with an orbit homoclinic to a saddle-focus equilibrium, admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds. In this paper we deal with two classes of saddle-focus systems: Lagrangian systems defined on a two-manifold in the presence of a gyroscopic force, and fourth-order systems arising in water-wave theory. We first establish, by a standard variational method, the existence of a homoclinic orbit. Then, under a weak nondegeneracy condition, we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic. Our condition is much easier to check than transversality. For example, it is automatically satisfied for gyroscopic systems on a two-torus, for topological reasons. © 1996 John Wiley & Sons, Inc. 相似文献
12.
Ian Melbourne 《Transactions of the American Mathematical Society》2007,359(5):2421-2441
We show that superpolynomial decay of correlations (rapid mixing) is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which Young proved exponential decay of correlations. The proof combines techniques of Dolgopyat and operator renewal theory.
It follows from our results that planar periodic Lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing.
13.
P. Tracqui 《Journal of Nonlinear Science》1994,4(1):69-103
Summary A well-supported self-oscillating eight-compartment model has been proposed by Staub et al. to account for thein vivo rat calcium metabolism (Staub et al.,Am. J. Physiol.
254, R134–139, 1988). The nonlinear nucleus of this model is a three-compartment subunit which represents the dynamic autocatalytic
processes of phase transition at the interface between bone and extracellular fluids. The organization of the temporal mixed-mode
oscillations which successively appear as the calcium input is varied is analyzed. On one side of the bifurcation diagram,
the generation of periodic trajectories with a single large amplitude oscillation is governed by homoclinic tangencies to
small amplitude limit cycles and follows the universal sequence (U-sequence) given for the periodic solutions of unimodal
transformations of the unit interval into itself. On the other side, the progressive appearance and interweaving of trajectories
with multiple large amplitude oscillations per period is linked to homoclinic tangencies to large amplitude unstable cycles.
The bifurcation sequence responsible for the temporal pattern generation has been analyzed by modeling the first return map
of the differential system associated with the compartmental subunit. We establish that this genealogy does not follow the
usual Farey treelike organization and that a comprehensive view of the resulting fractal bifurcation structure can be obtained
from the unfolding of singular points of bimodal maps. These theoretical features can be compared with those reported in experiments
on dissolution processes, and the extent to which the knowledge of the subunit bifurcation structure provides new conceptual
insights in the field of bone and calcium metabolism is discussed. 相似文献
14.
Marcelo Viana 《Bulletin of the Brazilian Mathematical Society》1993,24(1):13-62
We consider generic one-parameter families of diffeomorphisms on a manifold of arbitrary dimension, unfolding a homoclinic tangency associated to a sectionally dissipative saddle point (the product of any pair of eigenvalues has norm less than 1). We prove that such families exhibit strange attractors in a persistent way: for a positive Lebesgue measure set of parameter values. In the two-dimensional case this had been obtained in a joint work with L. Mora, based on and extending the results of Benedicks-Carleson on the quadratic family in the plane. 相似文献
15.
Ru Song ZHENG 《数学学报(英文版)》2021,(7):1023-1040
We study bi-Lyapunov stable homoclinic classes for a C~1 generic flow on a closed Riemannian manifold and prove that such a homoclinic class contains no singularity. This enables a parallel study of bi-Lyapunov stable dynamics for flows and for diffeomorphisms. For example, we can then show that a bi-Lyapunov stable homoclinic class for a C~1 generic flow is hyperbolic if and only if all periodic orbits in the class have the same stable index. 相似文献
16.
Bernhard Lamel 《Journal of Geometric Analysis》2001,11(4):627-633
We give an invariant nondegeneracy condition for CR-maps between generic submanifolds in different dimensions and use it to
prove a reftection principle for these maps. 相似文献
17.
本文研究具有非双曲奇点的高维系统在小扰动下的同宿轨道分支问题,通过在未扰同宿轨道邻域建立局部坐标系,导出系统在新坐标系下的Poincare映射,对伴随超临界分支的通有同宿轨道的保存及分支出周期轨道的情况进行了讨论,推广和改进了一些文献的结果. 相似文献
18.
利用局部活动坐标架法,讨论了四维空间中连接两个鞍点的异维环分支问题,在一些通有的假设下,分别得到了异维环保存、同宿环、周期轨存在的充分条件以及保存的异维环与分支出的周期轨共存(或不共存)的结果. 相似文献
19.
Valeriia Starichkova 《Regular and Chaotic Dynamics》2018,23(6):767-784
In this paper we prove that generic one-parameter families of vector fields on \(\mathbb{S}^2\) in the neighborhood of the fields of classes AH, SN, HC, SC (Andronov–Hopf, saddle-node, homoclinic curve, saddle connection) are structurally stable. We provide a classification of bifurcations in these families. 相似文献
20.
Bifurcations of heterodimensional cycles with two saddle points 总被引:1,自引:0,他引:1
The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed. 相似文献