Hyperbolic Invariant Sets Research Articles

Chaotic Dynamics in Nonautonomous Maps: Application to the Nonautonomous Hénon Map

In this paper, we analyze chaotic dynamics for two-dimensional nonautonomous maps through the use of a nonautonomous version of the Conley–Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley–Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous Hénon map to have a hyperbolic chaotic invariant set.

Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps

In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems – a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, area-preserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors explicitly for a certain class of norms. This enables us to provide a rigorous setting for the notion that the “singular sets” of the Lagrangian descriptors correspond to the stable and unstable manifolds of hyperbolic invariant sets, as well as to understand how this depends upon the particular norms that are used. Finally we analyze, from the computational point of view, the performance of this tool for general nonlinear maps, by computing the “chaotic saddle” for autonomous and nonautonomous versions of the Hénon map.

Uniform hyperbolicity of locally maximal chaotic invariant sets in a system with only one stable equilibrium

By the methods of Poincaré map and the cone field theory, we present a rigorous computer-assisted method to verify the existence of a locally maximal uniformly hyperbolic chaotic invariant set for an autonomous chaotic system which has only one stable equilibrium. At the same time, our arguments show that the second return Poincaré map restricted to the hyperbolic set is indeed topologically conjugate to the 2-shift map.

Perturbations of weakly hyperbolic invariant sets of two-dimension periodic systems

The question of structural stability is one of the most important areas in a present-day theory of differential equations. In this paper, we study small C1 perturbations of a systems of differential equations. We introduce the concepts of a weakly hyperbolic invariant set K and leaf Y for a system of ordinary differential equations. The Lipschitz condition is not assumed. We show that, if the perturbation is small enough, then there is a continuous mapping h, i.e., K → KY, where KY is a weakly hyperbolic set of the perturbed equation system. Moreover, we show that h(Y) is a leaf of the perturbed system.

On bounded positive stationary solutions for a nonlocal Fisher–KPP equation

We study the existence of stationary solutions for a nonlocal version of the Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) equation. The main motivation is a recent study by Berestycki et al. (2009) where the nonlocal FKPP equation has been studied and it was shown for the spatial domain R and sufficiently small nonlocality that there are only two bounded non-negative stationary solutions. Here we provide a similar result for Rd using a completely different approach. In particular, an abstract perturbation argument is used in suitable weighted Sobolev spaces. One aim of the alternative strategy is that it can eventually be generalized to obtain persistence results for hyperbolic invariant sets for other nonlocal evolution equations on unbounded domains with small nonlocality, i.e., to improve our understanding in applications when a small nonlocal influence alters the dynamics and when it does not.

Global Stable Manifolds in Holomorphic Dynamics Under Bunching Conditions

We prove that the stable manifold of every point in a compact hyperbolic invariant set of a holomorphic automorphism of a complex manifold is biholomorphic to a complex vector space, provided that a bunching condition, which is weaker than the classical bunching condition for linearizability, holds.

Computability, noncomputability, and hyperbolic systems

In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyperbolic system, having many applications in physical sciences and other fields. We show that (i) locally these manifolds can be computed, but (ii) globally they cannot, since their degree of computational unsolvability lies on the second level of the Borel hierarchy. We also show that Smale’s horseshoe, the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit, is computable.

Chaotic Dynamics and Bifurcations in Impact Systems

Bifurcations of dynamical systems described byseveral second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to the occurrence of a hyperbolic chaotic invariant set.

Hyperbolicity of the invariant sets for the real polynomial maps

In this paper, the conditions under which there exits a uniformly hyperbolic invariant set for the map fa(x)=ag(x) are studied, where a is a real parameter, and g(x) is a monic real-coefficient polynomial. It is shown that for certain parameter regions, the map has a uniformly hyperbolic invariant set on which it is topologically conjugate to the one-sided subshift of finite type for A, where ∣a∣ is sufficiently large, A is an eventually positive transition matrix, and g has at least two different real zeros or only one real zero. Further, it is proved that there exists an invariant set on which the map is topologically semiconjugate to the one-sided subshift of finite type for a particular irreducible transition matrix under certain conditions, and one type of these maps is not hyperbolic on the invariant set.

Hyperbolic dynamics in graph-directed IFS

A cone space is a complete metric space ( X , d ) with a pair of functions c s , c u : X × X → R , such that there exists K > 0 satisfying 1 K d ( x , x ′ ) ⩽ max ( c s ( x , x ′ ) , c u ( x , x ′ ) ) ⩽ K d ( x , x ′ ) for x , x ′ ∈ X . For a partial map f between cone spaces X and Y we introduce | f | s which measures the stable contraction rate and 〈 f 〉 u which measures the unstable expansion rate. We say that f is cone-hyperbolic if | f | s < 1 < 〈 f 〉 u . Using cone field and graph-directed IFS we build an abstract metric model which describes the dynamics of the hyperbolic-like systems. This allows to obtain estimations from below and above of the fractal dimension of the hyperbolic invariant set.

Hyperbolic invariant sets of the real generalized Hénon maps

In this paper, the conditions under which there exists a uniformly hyperbolic invariant set for the generalized Hénon map F( x, y) = ( y, ag( y) − δx) are investigated, where g( y) is a monic real-coefficient polynomial of degree d ⩾ 2, a and δ are non-zero parameters. It is proved that for certain parameter regions the map has a Smale horseshoe and a uniformly hyperbolic invariant set on which it is topologically conjugate to the two-sided fullshift on two symbols, where g( y) has at least two different non-negative or non-positive real zeros, and ∣ a∣ is sufficiently large. Moreover, it is shown that if g( y) has only simple real zeros, then for sufficiently large ∣ a∣, there exists a uniformly hyperbolic invariant set on which F is topologically conjugate to the two-sided fullshift on d symbols.

Chaotic quasi-collision trajectories in the 3-centre problem

We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin and Negrini, J Differ Equ 190:539–558, 2003): this result has been obtained by the use of the Poincaré-Melnikov theory. Here we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the perturbing centre in \({\mathbb{R}^2}\) , we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use of a general result of Bolotin and MacKay (Celest Mech Dyn Astron 77:49–75, 77:49–75, 2000; Celest Mech Dyn Astron 94(4):433–449, 2006). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic orbits of the regularised problem passing through the third centre.

Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space

LYAPUNOV EXPONENTS AND INVARIANT MANIFOLD FOR RANDOM DYNAMICAL SYSTEMS IN A BANACH SPACE Zeng Lian Department of Mathematics Ph.D of Mathematics Abstract We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

Horseshoes in the forced van der Pol system

The forced van der Pol equation was introduced in the 1920s as a model of an electrical circuit. Cartwright and Littlewood established the existence of invariant sets with complex topology in this system. This paper contains, for the first time, a full description of the nonwandering set for an open set of parameters near the singular limit of the forced van der Pol equation. In particular, we prove the existence of a hyperbolic chaotic invariant set. We also prove that the system is structurally stable. The analysis is conducted from the perspective of geometric singular perturbation theory. Verifying the hypothesis is done by implementing self-validating numerical algorithms.

Stickiness in three-dimensional volume preserving mappings

We investigate the orbital diffusion and the stickiness effects in the phase space of a 3-dimensional volume preserving mapping. We first briefly review the main results about the stickiness effects in 2-dimensional mappings. Then we extend this study to the 3-dimensional case, studying for the first time the behavior of orbits wandering in the 3-dimensional phase space and analyzing the role played by the hyperbolic invariant sets during the diffusion process. Our numerical results show that an orbit initially close to a set of invariant tori stays for very long times around the hyperbolic invariant sets near the tori. Orbits starting from the vicinity of invariant tori or from hyperbolic invariant sets have the same diffusion rule. These results indicate that the hyperbolic invariant sets play an essential role in the stickiness effects. The volume of phase space surrounded by an invariant torus and its variation with respect to the perturbation parameter influences the stickiness effects as well as the development of the hyperbolic invariant sets. Our calculations show that this volume decreases exponentially with the perturbation parameter and that it shrinks down with the period very fast.

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C 1-residual (dense G δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C 1 zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ⊆ M there exists Y arbitrarily close to X such that Y t has an elliptical closed orbit through U.

Covering relations and the existence of topologically normally hyperbolic invariant sets

We present a topological method for the detection of normally hyperbolictype invariant sets for maps. The invariant set covers a sub-manifold without a boundary in $\mathbb{R}^k$. For the method to hold weonly need to assume that the movement of the system transversal to themanifold has directions of topological expansion and contraction. Themovement in the direction of the manifold can be arbitrary. The result isbased on the method of covering relations and local Brouwer degree theory.

Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter

We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows and Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from linear to exponential in time. Our theory also applies to non-Hamiltonian systems with a first integral.

Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions

For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits C1-persistent homoclinic tangency. It provides an open subset of the space of C 1 -diffeomorphisms in which generic diffeomorphisms have arbitrary given growth of the number of attracting periodic orbits and admit no symbolic extensions.

Persistence of nonhyperbolic measures for C 1-diffeomorphisms

In the space of diffeomorphisms of an arbitrary closed manifold of dimension ≥ 3, we construct an open set such that each difteomorphism in this set has an invariant ergodic measure with respect to which one of its Lyapunov exponents is zero. These difteomorphisins are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with the circle as the fiber. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products.