Homoclinic Tangency Research Articles

Homoclinic Bifurcation of a Quadratic Family of Real Functions with Two Parameters

In this work the homoclinic bifurcation of the family H={h(a,b)(x)=ax2+b:a∈R/{0},b∈R} is studied. We proved that this family has a homoclinic tangency associated to x=0 of P1 for b=-2/a. Also we proved that Wu(P1) does not intersect the backward orbit of P1 for b>-2/a, but has intersection for b0. So H has this type of the bifurcation.

Homoclinic tangency and variation of entropy

In this paper we study the e¤ect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the C^{\infty} -topology. We also prove results about variation of entropy in other topologies and when the tangency does not correspond to a basic set with maximal entropy. We also show an example of discontinuity of the entropy among C^{\infty} diffeomorphisms of three dimensional manifolds.

Newhouse Laminations of polynomials on ℂ2

It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these laminations, called Newhouse laminations, occur also in the holomorphic context. In the space of polynomials of [Formula: see text], with bounded degree, there are Newhouse laminations.

Robust Degenerate Unfoldings of Cycles and Tangencies

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index $c>0$ and homoclinic tangencies of arbitrary codimension $c>0$. These sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite we also construct robust homoclinic tangencies of large codimension which cannot be inside a strong partially hyperbolic set.

Entropy formula and continuity of entropy for piecewise expanding maps

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.

Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors

We prove that heterodimensional cycles can be created by unfolding a pair of homoclinic tangencies in a certain class of Cr-diffeomorphisms . This implies the existence of a C2-open domain in the space of dynamical systems with a certain type of symmetry where systems with heterodimensional cycles are dense in Cr. In particular, we describe a class of three-dimensional flows with a Lorenz-like attractor such that an arbitrarily small time-periodic perturbation of any such flow can belong to this domain—in this case the corresponding heterodimensional cycles belong to a chain-transitive attractor of the perturbed flow.

Finiteness of partially hyperbolic attractors with one-dimensional center

We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-generic diffeomorphisms far from homoclinic tangencies in dimension $3$ either have at most finitely many attractors, or satisfy Newhouse phenomenon.

A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies

We prove a Ergodic Closing Lemma for nonsingular \begin{document}$ C^1 $\end{document} endomorphisms, claiming that the set of eventually strongly closable points is a total probability set. The forward means that the closing perturbation is involved along a finite part of the orbit of a point in a total probability set, which is the same perturbation as in Mane's Ergodic Closing Lemma for \begin{document}$ C^1 $\end{document} diffeomorphisms. As an application, Shub's Entropy Conjecture for nonsingular \begin{document}$ C^1 $\end{document} endomorphisms away from homoclinic tangencies is proved, extending the result for \begin{document}$ C^1 $\end{document} diffeomorphisms by Liao, Viana and Yang.

Moduli of Surface Diffeomorphisms with Cubic Tangencies

In this paper, we study conjugacy invariants for 2-dimensional diffeomorphisms with homoclinic cubic tangencies (two-sided tangencies of the lowest order) under certain open conditions. Ordinary arguments used in past studies of conjugacy invariants associated with one-sided tangencies do not work in the two-sided case. We present a new method which is applicable to the two-sided case.

Existence and Non-existence of Length Averages for Foliations

Since the pioneering work of Ghys, Langevin and Walczak among others, it has been known that several methods of dynamical systems theory can be adopted to study of foliations. Our aim in this paper is to investigate complexity of foliations, by generalising existence problem of time averages in dynamical systems theory to foliations: It has recently been realised that a positive Lebesgue measure set of points without time averages only appears for complicated dynamical systems, such as dynamical systems with heteroclinic connections or homoclinic tangencies. In this paper, we introduce the concept of length averages to singular foliations, and attempt to collect interesting examples with/without length averages. In particular, we demonstrate that length averages exist everywhere for any codimension one $\mathscr{C}^1 $ orientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi-minimal sets of the foliation, which is in strong contrast to time averages of surface flows.

An investigation of the parameter space for a family of dissipative mappings.

The parameter plane investigation for a family of two-dimensional, nonlinear, and area contracting map is made. Several dynamical features in the system such as tangent, period-doubling, pitchfork, and cusp bifurcations were found and discussed together with cascades of period-adding, period-doubling, and the Feigeinbaum scenario. The presence of spring and saddle-area structures allow us to conclude that cubic homoclinic tangencies are present in the system. A set of complex sets such as streets with the same periodicity and the period-adding of spring-areas are observed in the parameter space of the mapping.

The onset of transient turbulence in minimal plane Couette flow

The onset of transient turbulence in minimal plane Couette flow has been identified theoretically as homoclinic tangency with respect to a simple edge state for the Navier–Stokes equation, i.e., the gentle periodic orbit (the lower branch of a saddle-node pair) found by Kawahara & Kida (J. Fluid Mech., vol. 449, 2001, pp. 291–300). The first tangency of a pair of distinct homoclinic orbits to this periodic edge state has been discovered at Reynolds number $Re\equiv Uh/\unicode[STIX]{x1D708}=Re_{T}\approx 240.88$ ($U$, $h$, and $\unicode[STIX]{x1D708}$ being half the difference of the two wall velocities, half the wall separation, and the kinematic viscosity of fluid, respectively). At $Re>Re_{T}$ a Smale horseshoe appears on the Poincaré section through transversal homoclinic points to generate a transient chaos that eventually relaminarises. In numerical experiments a sustaining chaos, which is a consequence of period-doubling cascade stemming from the upper branch of another saddle-node pair of periodic orbits, is observed in a narrow range of the Reynolds number, $Re\approx 240.40$–240.46. At the upper edge of this $Re$ range it is found that the chaotic set touches the lower branch of this pair, i.e., another edge state. The corresponding chaotic attractor is replaced by a chaotic saddle at $Re\approx 240.46$, and subsequently this saddle touches the gentle periodic edge state on the boundary of the laminar basin at the tangency Reynolds number $Re=Re_{T}$. After this crisis on the boundary of the laminar basin, for $Re>Re_{T}$, chaotic transients that eventually relaminarise can be observed.

On some attractors of a two-dimensional quadratic map

In this paper, we study the appearance, evolution and neighbourhood of two attractors of a dynamical system defined by a quadratic polynomial map T: R2 → R2. The first is a Cantor-type attractor located on an invariant straight line. Thus, it suffices to study the restriction of the map T to this invariant line. The second is a closed curves cycle of period 2. We show, by a numerical approach, that when a parameter of the system varies, the evolution of the orbits in the region close to this second attractor is dependent on the evolution of the stable and unstable sets (homoclinic tangency) of a saddle cycle of period 2 located in this region.

Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large “chaotic sea” with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the so-called LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized Hénon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of four-dimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018].

On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

Gorodetski et al. [Nonremovability of zero Lyapunov exponents. Funktsional. Anal. i Prilozhen. 39(1) (2005), 27–38 (in Russian); Engl. Transl. Funct. Anal. Appl. 39(1) (2005), 21–30] and Bochi et al. [Robust criterion for the existence of nonhyperbolic ergodic measures. Comm. Math. Phys. 344(3) (2016), 751–795] propose two very different ways for building non-hyperbolic measures, Gorodetski et al. (2005) building such a measure as the limit of periodic measures and Bochi et al. (2016) as the $\unicode[STIX]{x1D714}$ -limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in Gorodetski et al. (2005) has been used in a generic setting in Bonatti et al. [Non-hyperbolic ergodic measures with large support. Nonlinearity 23(3) (2010), 687–705] and Díaz and Gorodetski [Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1479–1513], as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in Bochi et al. (2016) are accumulated by periodic measures. In this paper we use a shadowing lemma from Gan [A generalized shadowing lemma. Discrete Contin. Dyn. Syst. 8(3) (2002), 527–632]: ∙for getting the periodic orbits in Gorodetski et al. (2005) without perturbing the dynamics;∙for recovering the compact set in Bochi et al. (2016) with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset ${\mathcal{U}}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in {\mathcal{U}}$ , there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset ${\mathcal{V}}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in {\mathcal{V}}$ , there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

Robustly non-hyperbolic transitive symplectic dynamics

We construct symplectomorphisms in dimension d ≥ 4 having a semi-local robustly transitive partially hyperbolic set containing C2-robust homoclinic tangencies of any codimension $c$ with 0

Robust tangencies of large codimension

We construct C2-robust homoclinic and heterodimensional tangencies of large codimension inside transitive partially hyperbolic sets.

Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders

ABSTRACTConsider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold Ws(O), one-dimensional unstable manifold Wu(O), and with the saddle value σ different from 1. We assume that Ws(O) and Wu(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when σ > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.

Homoclinic points of 2D and 4D maps via the parametrization method

An interesting problem in solid state physics is to compute discrete breather solutions in coupled 1D Hamiltonian particle chains and investigate the richness of their interactions. One way to do this is to compute the homoclinic intersections of invariant manifolds of a saddle point located at the origin of a class of -dimensional invertible maps. In this paper we apply the parametrization method to express these manifolds analytically as series expansions and compute their intersections numerically to high precision. We first carry out this procedure for a two-dimensional (2D) family of generalized Hénon maps (), prove the existence of a hyperbolic set in the non-dissipative case and show that it is directly connected to the existence of a homoclinic orbit at the origin. Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond which the homoclinic intersection disappears. Proceeding to , we use the same approach to accurately determine the homoclinic intersections of the invariant manifolds of a saddle point at the origin of a 4D map consisting of two coupled 2D cubic Hénon maps. For small values of the coupling we determine the homoclinic intersection, which ceases to exist once a certain amount of dissipation is present. We discuss an application of our results to the study of discrete breathers in two linearly coupled 1D particle chains with nearest-neighbor interactions and a Klein–Gordon on site potential.