Homoclinic Tangency Research Articles

Homoclinic tangles in a Kaldor-like business cycle model

In this paper a nonlinear discrete-time business cycle model of Kaldor-type is considered, to illustrate particular global bifurcations which determine the appearance or disappearance of attracting and repelling closed invariant curves. Such bifurcations sequences, which involve homoclinic tangencies and transversal intersection of the stable and unstable manifolds of saddle cycles, may increase the complexity of the basins of attraction of multiple, coexisting attractors. Particularly, for the business cycle model examined here, such dynamic phenomena explain the co-existence of two stable steady states and an attracting closed curve, with an intricate basin structure, for wide ranges of the parameters.

Boundary crisis bifurcation in two parameters

The boundary crisis bifurcation is well known as a mechanism for destroying (or creating) a strange attractor by variation of one parameter: at the moment of the boundary crisis bifurcation the strange attractor touches the boundary of its own basin of attraction. Here we follow this codimension-one bifurcation in two parameters. One might expect that this leads to a differentiable curve in the two-parameter plane. Mathematically, a boundary crisis is effectively a homoclinic or heteroclinic tangency, the locus of which is a well-defined differentiable curve in a two-parameter system. However, instead of a boundary crisis, the transition through this tangency curve may lead to a basin boundary metamorphosis or an interior crisis bifurcation, in which the attractor persists. This phenomenon is again well known: at the point where the type of transition changes, the boundary crisis switches to another branch of homoclinic or heteroclinic tangencies, associated with manifolds of a periodic point with a different period than before. The curve of boundary crisis bifurcations is not differentiable at such points. In this paper, we show that the curve of boundary crisis bifurcations is, in fact, not even well defined as a piecewise-smooth curve. We illustrate that there are infinitely many gaps in much the same way as the one-parameter bifurcation diagram of the attractor contains infinitely many windows where the attractor is periodic and not strange or chaotic. Throughout, we use the Hénon map to illustrate our findings.

Homoclinic tangencies of arbitrarily high order in conservative two-dimensional maps

Homoclinic tangencies of arbitrarily high order in conservative two-dimensional maps

How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency

Here we study an amazing phenomenon discovered by Newhouse [S. Newhouse, Non-density of Axiom A(a) on S 2 , in: Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 191–202; S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9–18; S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 50 (1979) 101–151]. It turns out that in the space of C r smooth diffeomorphisms Diff r ( M ) of a compact surface M there is an open set U such that a Baire generic diffeomorphism f ∈ U has infinitely many coexisting sinks. In this paper we make a step towards understanding “how often does a surface diffeomorphism have infinitely many sinks.” Our main result roughly says that with probability one for any positive D a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those whose period is relatively large compared to its cyclicity. It verifies a particular case of Palis' Conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have probability zero. One of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II, preprint, 85 pp.].

Rigorous Computations of Homoclinic Tangencies

In this paper, we propose a rigorous computational method for detecting homoclinic tangencies and structurally unstable connecting orbits. It is a combination of several tools and algorithms, including the interval arithmetic, the subdivision algorithm, the Conley index theory, and the computational homology theory. As an example, we prove the existence of generic homoclinic tangencies in the Hénon family.

Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies

We study the hyperbolicity of a class of horseshoes exhibiting an <i>internal</i> tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly <i>not</i> uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.

Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system

Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies.

MULTIPLIERS OF HOMOCLINIC TANGENCIES AND A THEOREM OF GONCHENKO AND SHILNIKOV ON AREA PRESERVING MAPS

We investigate differential invariants for homoclinic tangencies and discuss the role of these invariants in the Hausdorff dimension of invariant sets associated with the tangency and its unfoldings. Further, we give a streamlined proof of a theorem of Goncheko and Shilnikov on the case of a tangency in an area preserving map. Specifically, the invariants determine whether or not a hyperbolic invariant set is formed near the tangency.

Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation

We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers $(\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma)$, where $0<\lambda<1<|\gamma|$ and $|\lambda^2\gamma|=1$. We show that in a three-parameter family, $g_{\eps}$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\eps =0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional H\'enon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional H\'enon maps occupy in the class of quadratic volume-preserving diffeomorphisms.

Classification of homoclinic tangencies for periodically perturbed systems

Abstract Classification of homoclinic tangencies for periodically perturbed systems is discussed. A relationship between the order of Melnikov function’s zeros and the harmonic components of a dynamical system is derived. By applying the singularity theory to the Melnikov function, possible types of homoclinic tangencies are studied for realization of the classification. In addition, certain multi-harmonically perturbed systems are investigated, showing the corresponding homoclinic bifurcation with their bifurcation diagrams.

ON THE RELATION BETWEEN PREDICTABILITY AND HOMOCLINIC TANGENCIES

The predictability of chaotic systems is investigated using paradigmatic models for the conservative and the dissipative cases. Local Lyapunov exponents are used to quantify predictability for short time scales. It is shown that, in both cases, regions of enhanced predictability have been found around homoclinic tangencies. In the dissipative case, we demonstrate that the length of these regions shrinks exponentially with increasing time of prediction.

On Two-Dimensional Area-Preserving Maps with Homoclinic Tangencies that Have Infinitely Many Generic Elliptic Periodic Points

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.

On Bifurcations of Three-Dimensional Diffeomorphisms with a Homoclinic Tangency to a “Neutral” Saddle Fixed Point

Bifurcations of single-round periodic orbits of three-dimensional diffeomorphisms with a quadratic homoclinic tangency of manifolds of a saddle fixed point with saddle value equal to 1 are studied. Bibliography: 8 titles.

A complement to the connecting lemma of Hayashi

Here we show that a C2 surface diffeomorphism that satisfies the hypothesis of the Hayashi connecting lemma either can be approximated, in the C1 topology, by a diffeomorphism exhibiting a homoclinic tangency or the diffeomorphism already presents transversal homoclinic orbits.

On dynamic properties of diffeomorphisms with homoclinic tangency

We study dynamic properties of systems in Newhouse domains near a diffeomorphism having a saddle fixed point with a homoclinic tangency in the following cases: one-dimensional, two-dimensional, where a fixed point is a saddle-focus with one real and two complex-conjugate multipliers, and four-dimensional saddle-focus with two pairs of complex-conjugate multipliers.

On dynamic properties of diffeomorphisms with homoclinic tangency

On dynamic properties of diffeomorphisms with homoclinic tangency

Generalized Hénon Map and Bifurcations of Homoclinic Tangencies

We study two-parameter bifurcation diagrams of a generalized Henon map (GHM) that is known to describe dynamics of iterated maps near homoclinic and heteroclinic tangencies. We prove the nondegeneracy of codimension (codim) 2 bifurcations of fixed points of the GHM analytically and compute its various global and local bifurcation curves numerically. Special attention is given to the interpretation of the results and their application to the analysis of bifurcations of the homoclinic tangency of a neutral saddle in two-parameter families of planar diffeomorphisms. In particular, an infinite cascade of homoclinic tangencies of neutral saddle cycles is shown to exist near the homoclinic tangency of the primary neutral saddle.

Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles

The C 1 density conjecture of Palis asserts that diffeomorphisms exhibiting either a homoclinic tangency or a heterodimensional cycle are C 1 dense in the complement of the C 1 closure of hyperbolic systems. In this paper we prove some results towards the conjecture.

New phenomena associated with homoclinic tangencies

We survey some recently obtained generic consequences of the existence of homoclinic tangencies in diffeomorphisms of surfaces. Among other things, it has been shown that they give rise to invariant topologically transitive sets with maximal Hausdorff dimension, that they prohibit the existence of various kinds of symbolic extensions and that they form an impediment to the existence of Sinai–Ruelle–Bowen (SRB) measures. The main new result described here, together with a positive answer to an as yet unproved conjecture of Palis, would prove that generically on surfaces, SRB measures only exist on uniformly hyperbolic attractors.

Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation.

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.