Transversal Homoclinic Points Research Articles

Chaotic Dynamics at the Boundary of a Basin of Attraction via Non-transversal Intersections for a Non-global Smooth Diffeomorphism

In this paper, we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three-cycle associated with the Secant map. Using Moser’s version of Birkhoff–Smale’s theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of N-symbols for any integer N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document} or infinity.

Homoclinic points of symplectic partially hyperbolic systems with 2D centre

We consider a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with area-preserving diffeomorphisms and prove that every hyperbolic periodic point has transverse homoclinic points.

Chaos analysis for a class of impulse Duffing-van der Pol system

Abstract Chaotic dynamics of an impulse Duffing-van der Pol system is studied in this paper. With the Melnikov method, the existence condition of transversal homoclinic point is obtained, and chaos threshold is presented. In addition, numerical simulations including phase portraits and time histories are carried out to verify the analytical results. Bifurcation diagrams are also given, from which it can be seen that the system may undergo chaotic motions through period doubling bifurcations.

Horseshoes for Singly Thermostated Hamiltonians

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 27 January 2020Accepted: 16 June 2020Published online: 13 October 2020Keywordsthermostats, Nosé--Hoover thermostat, Hamiltonian mechanics, transverse homoclinic points, horseshoesAMS Subject Headings37J30, 37A60, 37J46, 82B03Publication DataISSN (online): 1536-0040Publisher: Society for Industrial and Applied MathematicsCODEN: sjaday

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.

New global bifurcation diagrams for piecewise smooth systems: Transversality of homoclinic points does not imply chaos

In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic γ→(t). We assume that the critical point 0→ lies on the discontinuity surface Ω0. We consider 4 scenarios which differ for the presence or not of sliding close to 0→ and for the possible presence of a transversal crossing between γ→(t) and Ω0. We assume that the systems are subject to a small non-autonomous perturbation, and we obtain 4 new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics.

Continuation and collapse of homoclinic tangles

By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The new bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. These bifurcation equations consist of a finite or infinite set of hilltop normal forms known from singularity theory. For the Henon family we determine numerically the connected components of branches with multi-humped homoclinic orbits that pass through several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.

0501 平板間クエット流における過渡的乱れの発生(OS5-1 壁乱流:統計,構造,動力学の理解,OS5 壁乱流:統計,構造,動力学の理解)

The onset of transient turbulence in plane Couette flow is identified as a homoclinic tangency with respect to a simple edge state for the Navier-Stokes equation, i.e., the gentle periodic orbit found by Kawahara & Kida (J. Fluid Mech., vol. 449, 2001, pp. 291-300). The tangency of a pair of distinct homoclinic orbits is numerically discovered at the critical Reynolds number Re ≡ Uh/ν = 241.3 (U, h, and ν being half the difference of the two wall velocities, half the wall separation, and the kinematic viscosity of fluid, respectively) above which transversal homoclinic points appear on the Poincar´e section to generate a transient chaos through a Smale horseshoe.

Positive-entropy geodesic flows on nilmanifolds

Let Tn be the nilpotent group of real n × n upper-triangular matrices with 1s on the diagonal. The Hamiltonian flow of a left-invariant Hamiltonian on T*Tn naturally reduces to the Euler flow on , the dual of . This paper shows that the Euler flows of the standard Riemannian and sub-Riemannian structures of T4 have transverse homoclinic points on all regular coadjoint orbits. As a corollary, left-invariant Riemannian metrics with positive topological entropy are constructed on all quotients D \\ Tn where D is a discrete subgroup of Tn and n ⩾ 4.

On the existence of chaotic circumferential waves in spinning disks

We use a third-order perturbation theory and Melnikov's method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points lead, respectively, to a random reversal in the traveling direction of circumferential waves and a random phase shift of magnitude pi for both forward and backward wave components. These long-term phenomena occur in imperfect low-speed disks sufficiently far from fundamental resonances.

Transversal homoclinic points of the Hénon map

Using shadowing techniques we prove that the Hénon map $H_{a,b}(x,y)=(a-x^{2}+by,x)$ admits a transversal homoclinic point for a set of parameters which is not small. For the area and orientation preserving Hénon map (corresponding to b=-1) we prove that a transversal homoclinic point exists for a≥0.265625. Applying a computer-assisted version of our scheme we show that the result holds even for a≥-0.866. This supports an old conjecture due to Devaney and Nitecki dating back to 1979, see [4], claiming that the Hénon map in the case b=-1 admits a transversal homoclinic point for a>-1.

Structurally stable but chaotic limit set of E-infinity Cantorian space–time

In the present work, first we give some definitions and theorems on hyperbolic maps, structurally stability and deterministic chaos. The limit set of the Kleinian transformation acting on the E -infinity Cantorian space–time turned out to be a set of periodic continued fractions as shown in [Chaos, Solitons & Fractals, 21 (2004) 9]. That set has a hyperbolic structure and is structurally stable. Subsequently, we show that the appearance of transversal homoclinic points induces a chaotic behavior in that set.

Transversal Homoclinic Orbits in the Collinear Restricted Three-Body Problem

Using the perturbation method of Melnikov, we prove in a simple way the existence of transversal homoclinic points in the collinear restricted three-body problem. As a consequence we can embed a Bernoulli shift on a suitable cross section of the flow, showing easily that this problem possesses chaotic dynamics.

Structurally stable but chaotic limit set of E-infinity Cantorian space–time

In the present work, first we give some definitions and theorems on hyperbolic maps, structurally stability and deterministic chaos. The limit set of the Kleinian transformation acting on the E -infinity Cantorian space–time turned out to be a set of periodic continued fractions as shown in [Chaos, Solitons & Fractals, 21 (2004) 9]. That set has a hyperbolic structure and is structurally stable. Subsequently, we show that the appearance of transversal homoclinic points induces a chaotic behavior in that set.

Homoclinic Points for a Singularly Perturbed System of Differential Equations with Delay

We obtain a representation of the integral manifold of a system of singularly perturbed differential-difference equations with periodic right-hand side. We show that, under certain conditions imposed on the right-hand side, the Poincare map for the perturbed system has a transversal homoclinic point.

Chaotic Invariant Sets of High-Dimensional Hénon-Like Maps

High-dimensional Hénon-like maps have many applications in the research of spatial chaos and traveling waves of extended systems. Meanwhile, they are of great interest in their own right. The aim of this paper is, by applying the implicit function theorem, to show for high-dimensional Hénon-like maps the existence of chaotic invariant sets and the density of homoclinic points and heteroclinic points in them. Our method is motivated by Aubry's “anti-integrability” concept and is rather different from the traditional techniques such as horseshoes, transversal homoclinic points and heteroclinic cycles, and snap-back repellers.

The density of the transversal homoclinic points in the Henon-like strange attractors

We consider the Henon-like strange attractors Λ in a family which is a nonsingular perturbation of a d-modal family. The existence of the Henon-like strange attractors in this family was proved by Diaz et al. [Inventions Math. 125 (1996) 37]. We prove that the transversal homoclinic points are dense in Λ, and that hyperbolic periodic points are dense in Λ. Moreover, the hyperbolic periodic points that are heteroclinically related to the primary periodic point (transversal intersection of stable and unstable manifolds) are dense in Λ.

NUMERICAL COMPUTATION OF HOMOCLINIC ORBITS FOR FLOWS

It is shown that numerical computation of homoclinic orbits for flows will generate transverse homoclinic points when one uses very accurate schemes.

Chaotic patterns in coupled map lattices

In this paper we consider the existence and stability of chaotic patterns (periodic in space and chaotic in time) in coupled map lattice with local map f(x)= sin x . This is equivalent to investigating the stable chaotic set of some high-dimensional dynamical system. It is not easy in high-dimensional space to apply horseshoe technique or the theory of transversal homoclinic point and transversal cycles to show the existence of chaotic set. Motivated by the concept of anti-integrability (Aubry S. Physica D 1995;86:284–96), we prove the corresponding finite-dimensional system F α possesses stable chaotic set Λ α in phase space R N for the nonlinearity strength α appropriately large and the coupling strength small. Consequently, there exist stable chaotic patterns in CML system.

Chaotic dynamics in a two-dimensional overlapping generations model

Abstract This paper investigates the global dynamics of a two-dimensional Diamond-type overlapping generations model extended to allow for government intervention. Using a singular perturbation method, we identify conditions under which transverse homoclinic points to the golden rule steady state are generated. For a parametric example with a CES production function, the occurrence of complicated dynamics (e.g. strange attractors) associated with homoclinic bifurcations is demonstrated.