The shadowing chain lemma for singular Hamiltonian systems involving strong forces

We consider a planar autonomous Hamiltonian system :q+?V(q) = 0, where the potential V: ?² \{??}?? ? has a single well of infinite depth at some point ?? and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ?? we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits ?? the shadowing chain lemma ?? via minimization of action integrals and using simple geometrical arguments.     

Critical homoclinic orbits lead to snap-back repellers

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.     

Divergence-free vector fields with average and asymptotic average shadowing property

Let X be a divergence-free vector field on a three-dimensional compact connected Riemannian manifold. In this paper, we show that if X is in the C¹-interior of the set of divergence-free vector fields which satisfy the average shadowing property then X is Anosov. We also obtain similar result for asymptotic average shadowing property.     

Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations

The structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property is studied. The fundamental difference between the problem under consideration and its counterpart for discrete dynamical systems generated by diffeomorphisms is the reparameterization of shadowing orbits. Depending on the type of reparameterization, Lipschitz and oriented shadowing properties are distinguished. As is known, structurally stable vector fields have the Lipschitz shadowing property. Let X be a vector field, and let p and q be its points of rest or closed orbits. Suppose that the stable manifold of p and the unstable manifold of q have a nontransversal intersection point. It is shown that, in this case, the vector field X does not have the Lipschitz shadowing property. If one of the orbits p and q is closed, then X does not have the oriented shadowing property. These assertions imply that the C ¹-interior of the set of vector fields with the Lipschitz shadowing property coincides with the set of structurally stable vector fields. If the dimension of the manifold under consideration is at most 3, then a similar result is valid for the oriented shadowing property. We study the structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property. It is shown that, in the case of the Lipschitz shadowing property, it coincides with the set of structurally stable systems. For manifolds of dimension at most 3, a similar result is valid for the oriented shadowing property.     

Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point

We consider self-diffeomorphisms of the plane of the class C ^r (1 ?? r < ??) with a fixed hyperbolic point and a nontransversal point homoclinic to it. We present a method for constructing a set of diffeomorphisms for which the neighborhood of a homoclinic point contains countably many stable periodic points with characteristic exponents bounded away from zero.     

Structural stability of vector fields with shadowing

Let X be a C¹ vector field without singularities. In this paper, we show that X is in the C¹ interior of the set of vector fields with the shadowing property if and only if X satisfies both Axiom A and the strong transversality condition; that is, X is structurally stable.     

The existences of transverse homoclinic solutions and chaos for parabolic equations

By using Lyapunov-Schmidt reduction and exponential dichotomies, the persistence of homoclinic orbit is considered for parabolic equations with small perturbations. Bifurcation functions are obtained, where d is the dimension of the intersection of the stable and unstable manifolds. The zeros of H 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 the homoclinic solution for the perturbed system is transversal under the applicable conditions and hence the perturbed system exhibits chaos. The basic tools are shadowing lemma which was obtained by Blazquez (see [C.M. Blazquez, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal. 10 (1986) 1277-1291]).     

Shilnikov lemma for a nondegenerate critical manifold of a Hamiltonian system

Let M be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose M consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the λ-lemma) describing the behavior of trajectories near M. Using this result, trajectories shadowing chains of homoclinic orbits to M are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s second species solutions of the 3 body problem with 2 masses small of order µ. As µ → 0, double collisions of small bodies correspond to a symplectic critical manifold M of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted 3 body problem.     

Stably average shadowable homoclinic classes

Let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold M, and let H_f(p) be the homoclinic class of f containing p. In this paper, we show that if H_f(p) is locally maximal and every hyperbolic periodic point in H_f(p) is uniformly far away from being nonhyperbolic, and H_f(p) has the average shadowing property, then H_f(p) is hyperbolic.     

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

We prove that the chain-transitive sets of C¹-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C¹-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C¹-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff¹(M).     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

Chaos and the shadowing property

We present, as a simpler alternative for the results of [P. Ko?cielniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl. 310 (2005) 188-196; P. Ko?cielniak, M. Mazur, On C⁰ genericity of various shadowing properties, Discrete Contin. Dynam. Syst. 12 (2005) 523-530], an elementary proof of C⁰ genericity of the periodic shadowing property. We also characterize chaotic behavior (in the sense of being semiconjugated to a shift map) of shadowing systems.     

Expansive homoclinic classes of generic <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-vector fields

Let γ be a hyperbolic closed orbit of a C ¹ vector field X on a compact C ^∞ manifold M of dimension n ≥ 3, and let H _X(γ) be the homoclinic class of X containing γ. In this paper, we prove that C ¹-generically, if H _X(γ) is expansive and isolated, then it is hyperbolic.     

Resonance bifurcation from homoclinic cycles

Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R⁴ there exists a classification of different robust homoclinic cycles for which moreover eigenvalue conditions for asymptotic stability are known. We study resonance bifurcations that destroy the asymptotic stability of robust ‘simple homoclinic cycles’ in four-dimensional differential equations. We establish that typically a periodic trajectory near the cycle is created, asymptotically stable in the supercritical case.     

Periodic shadowing and Ω-stability

We show that the following three properties of a diffeomorphism f of a smooth closed manifold are equivalent: (i) f belongs to the C ¹-interior of the set of diffeomorphisms having the periodic shadowing property; (ii) f has the Lipschitz periodic shadowing property; (iii) f is Ω-stable.     

Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.     

Simple homoclinic cycles in low-dimensional spaces

The problem of a classification of robust homoclinic cycles in low-dimensional spaces has been frequently asked in recent years. In this paper, we resume the results in R³ and R⁴ and we solve the problem in R⁵ in the case of orientation-preserving group actions.     

Homoclinic tangencies versus uniform hyperbolicity for conservative 3-flows

We prove that a volume-preserving three-dimensional flow can be C¹-approximated by a volume-preserving Anosov flow or else by another volume-preserving flow exhibiting a homoclinic tangency. This proves the conjecture of Palis for conservative 3-flows and with respect to the C¹-topology.     

A unified approach to theories of shadowing

This paper introduces the notion of a general approximation property, which encompasses many existing types of shadowing. It is proven that there exists a metric space X such that the sets of maps with many types of general approximation properties (including the classic shadowing, the L _p -shadowing, limit shadowing, and the s-limit shadowing) are not dense in C(X), S(X), and H(X) (the space of continuous self-maps of X, continuous surjections of X onto itself, and self-homeomorphisms of X) and that there exists a manifold M such that the sets of maps with general approximation properties of nonlocal type (including the average shadowing property and the asymptotic average shadowing property) are not dense in C(M), S(M), and H(M). Furthermore, it is proven that the sets of maps with a wide range of general approximation properties (including the classic shadowing, the L _p -shadowing, and the s-limit shadowing) are dense in the space of continuous self-maps of the Cantor set. A condition is given that guarantees transfer of general approximation property from a map on X to the map induced by it on the hyperspace of X. It is also proven that the transfer in the opposite direction always takes place.