Heteroclinic Intersections Research Articles

Outer billiard around a curvilinear triangle with a fixed diameter

We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoes.

The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in R2 through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the piecewise-smooth systems. In the unperturbed case, the system possesses two C0 normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them. These heteroclinic manifolds are foliated by heteroclinic connections between C0 tori located at the same energy levels. By means of the impact map we prove the persistence of these objects under perturbation. In addition, we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of Melnikov-like functions. The heteroclinic manifolds allow us to define the scattering map, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. First order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds. Hence we have an essential tool for the construction of a heteroclinic skeleton which, when followed, can lead to the existence of Arnold diffusion: trajectories that, on large time scales, destabilize the system by further accumulating energy. We validate all the theoretical results with detailed numerical computations of a mechanical system with impacts, formed by the linkage of two rocking blocks with a spring.

On the simple isotopy class of a source-sink diffeomorphism on the 3-sphere

The results obtained in this paper are related to the Palis-Pugh problem on the existence of an arc with finitely or countably many bifurcations which joins two Morse-Smale systems on a closed smooth manifold M n . Newhouse and Peixoto showed that such an arc joining flows exists for any n and, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. For n = 1, this is related to the presence of the Poincare rotation number, and for n = 2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimension n = 3, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse-Smale diffeomorphism on the 3-sphere without heteroclinic intersections to be joined by a simple arc with a “source-sink” diffeomorphism are also found.

Controlling practical stability and safety of mechanical systems by exploiting chaos properties

In this paper, a method for controlling the global nonlinear dynamics of mechanical systems is applied to two models: the model of Augusti and an inverted guyed pendulum. These simplified models represent a large class of structures liable to buckling exhibiting interacting buckling phenomena. These structures may fail at load levels well below the theoretical buckling load due to complex nonlinear phenomena that decrease the safety and the dynamic integrity of the structure; this often occur as a consequence of imperfections and of the erosion of the basins of attraction of the safe pre-buckling solutions. So, it is of paramount practical importance to increase the safety of these structures in a dynamic environment. This can be achieved by increasing the integrity of the basins of attraction of the safe solutions, a goal that is attained by a control method which consists of the (optimal) elimination of homoclinic (or heteroclinic) intersection by properly adding superharmonic terms to a given harmonic excitation. By means of the solution of an appropriate optimization problem, it is possible to select the amplitudes and the phases of the added superharmonics in such a way that the manifolds distance is as large as possible. The results show that this methodology increases the integrity of the basins of attraction of the system and, consequently, the practical safety of the structure.

Classification of Morse-Smale diffeomorphisms with one-dimensional set of unstable separatrices

Let Mn be a closed orientable manifold of dimension n > 3. We study the class G1(Mn) of orientation-preserving Morse-Smale diffeomorphisms of Mn such that the set of unstable separatrices of any f ∈ G1(Mn) is one-dimensional and does not contain heteroclinic intersections. We prove that the Peixoto graph (equipped with an automorphism) is a complete topological invariant for diffeomorphisms of class G1(Mn), and construct a standard representative for any class of topologically conjugate diffeomorphisms.

The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web

A model map $Q$ for the Hopf-saddle-node (HSN) bifurcation of fixed points of diffeomorphisms is studied. The model is constructed to describe the dynamics inside an attracting invariant two-torus which occurs due to the presence of quasi-periodic Hopf bifurcations of an invariant circle, emanating from the central HSN bifurcation. Resonances of the dynamics inside the two-torus attractor yield an intricate structure of gaps in parameter space, the so-called Arnol'd resonance web. Particularly interesting dynamics occurs near the multiple crossings of resonance gaps, where a web of hyperbolic periodic points is expected to occur inside the two-torus attractor. It is conjectured that heteroclinic intersections of the invariant manifolds of the saddle periodic points may give rise to the occurrence of strange attractors contained in the two-torus. This is a concrete route to the Newhouse-Ruelle-Takens scenario. To understand this phenomenon, a simple model map of the standard two-torus is developed and studied and the relations with the starting model map $Q$ are discussed.

Stable and unstable invariant manifolds in a partially chaotic magnetic configuration generated by nonlinear reconnection

A numerical contour dynamics code has been employed to calculate the stable and unstable manifolds related to two interacting magnetic island chains. The magnetic configuration is generated by a nonlinear reconnection process described in D. Borgogno et al. [Phys. Plasmas. 12, 032309 (2005)]. The appearance of the first homoclinic and heteroclinic intersections of the dominant manifolds are shown and one of the associated uniformly hyperbolic orbits is given. The stickiness of the field lines around the island and the eventual development of global stochasticity are discussed. The basic geometry of the magnetic configuration is periodic so that the structure of the manifolds may be compared with the one obtained with Poincaré plots.

Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem

We consider generic properties of Lagrangians. Our main result is aKupka-Smale Theorem for the Lagrangian setting. We show that forconvex and superlinear Lagrangians defined on a compact surface and$k\in \mathbb{R}$, then generically, in Mañé's sense, the energylevel $k$ is regular and all periodic orbits in this level arenondegenerate at all orders (the linearized Poincaré map,restricted to this energy level, does not have roots of unity aseigenvalues). Moreover, all heteroclinic intersections in this levelare transversal. The results that we present are true in dimension$n \geq 2$, with the exception of Theorem 4.5, which we are onlyable to prove in dimension 2.

Bifurcation structure of dissipative solitons

In this paper we analyze in detail the structure of the phase space of a reversible dynamical system describing the stationary solutions of a model for a nonlinear optical cavity. We compare our results with the general picture described in [P.D. Woods, A.R. Champneys, Physica D 129 (1999) 147; P. Coullet, C. Riera, C. Tresser, Phys. Rev. Lett. 84 (2000) 3069] and find that the stable and unstable manifolds of homogeneous and patterned solutions present a much higher level of complexity than predicted, including the existence of additional localized solutions and fronts. This extra complexity arises due to homoclinic and heteroclinic intersections of the invariant manifolds of low-amplitude periodic solutions, and to the fact that these periodic solutions together with the high-amplitude ones constitute a one-parameter family generating a closed line on the symmetry plane.

Symmetric homoclinic tangles in reversible systems

We study the dynamics near transverse intersections of stable and unstable manifolds of sheets of symmetric periodic orbits in reversible systems. We prove that the dynamics near such homoclinic and heteroclinic intersections is not $C^1$ structurally stable. This is in marked contrast to the dynamics near transverse intersections in both general and conservative systems, which can be $C^1$ structurally stable. We further show that there are infinitely many sheets of symmetric periodic orbits near the homoclinic or heteroclinic orbits. We establish the robust occurrence of heterodimensional cycles, that is, heteroclinic cycles between hyperbolic periodic orbits of different index, near the transverse intersections. This is shown to imply the existence of hyperbolic horseshoes and infinitely many periodic orbits of different index, all near the transverse intersections.

GLOBAL AND LOCAL CONTROL OF HOMOCLINIC AND HETEROCLINIC BIFURCATIONS

A comprehensive resonant optimal control method is developed and discussed for suppressing homoclinic and heteroclinic bifurcations of a general one-degree-of-freedom nonlinear oscillator. Based on an adjustable phase shift, the primary resonant optimal control method is presented. By solving an optimization problem for the optimal amplitude coefficients to be used as the control parameters, the force term as the controller can be designed. Three kinds of resonant optimal control methods are compared. The control mechanism of the primary resonant optimal control method is to enlarge to the largest possible degree the control region where homoclinic and/or heteroclinic transversal intersections do not occur, and this is accomplished at lowest cost. It is shown that the primary resonant optimal control method has much better performance than the superharmonic resonant optimal control method, and it works well even when the superharmonic optimal control method fails. In particular, one new global optimal control method is presented, whose central idea is to find a frequency such that the asymmetric homoclinic bifurcations or the multiple homoclinic and heteroclinic bifurcations can attain the same critical values. On the basis of these same critical bifurcation values, chaos resulting from asymmetric homoclinic or multiple homoclinic and heteroclinic bifurcations can be effectively suppressed by the primary resonant optimal control method. This is confirmed by two illustrative examples. The theoretical analyses concerning the suppression of local and global homoclinic and heteroclinic bifurcations are in agreement with the numerical simulations, including the identification of the stable and unstable manifolds and the basins of attraction.

HETEROCLINIC BIFURCATIONS AND OPTIMAL CONTROL IN THE NONLINEAR ROCKING DYNAMICS OF GENERIC AND SLENDER RIGID BLOCKS

A method for controlling nonlinear dynamics and chaos, previously developed by the authors, is applied to the rigid block on a moving foundation. The method consists in modifying the shape of the excitation in order to eliminate, in an optimal way, the heteroclinic intersections embedded in the system dynamics. Two different cases are examined: (i) generic block under small perturbations and (ii) slender block under generic perturbations, and they are investigated analytically either by a perturbation analysis (former case) or exactly (latter case). Two different strategies are proposed: (i) one-side control, which consists in eliminating the intersections of a single heteroclinic connection, and (ii) global control, which consists in simultaneously eliminating the intersections of both heteroclinic connections. The best excitations permitting the maximum distance between stable and unstable manifolds are determined in both cases. Finally, some numerical investigations aimed at highlighting meaningful aspects of system response under controlled (optimal) and noncontrolled (harmonic) excitations are performed.

Kupka-Smale theorem for polynomial automorphisms of ℂ2 and persistence of heteroclinic intersections

A map is Kupka-Smale if all periodic points are hyperbolic and the stable and unstable manifolds of any two saddle points are transverse. Here we prove that Kupka-Smale maps form a residual set of full Lebesgue measure in the space $\mathcal{P}_d$ of polynomial automorphisms of ℂ2 of fixed dynamical degree d≥2. We also prove that a heteroclinic point of two saddle periodic orbits may be continued over (almost) the entire parameter space for this set of maps. This is one of the first persistence theorems proved in holomorphic dynamics in several variables.

Chaos in the case of two fixed black holes

We study the structure of chaos in the relativistic problem of particles moving in the field of two fixed black holes M1 and M2 by considering the asymptotic curves of some simple unstable periodic orbits on a surface of section. These curves consist of infinite arcs reaching the two black holes. Most orbits starting along these curves (asymptotic orbits) escape, i.e., they fall into the black holes M1 (type I orbits) and M2 (type II orbits). The number of the remaining orbits after n iterations (intersections with the surface of section) decreases exponentially with n. The asymptotic curves intersect at infinite homoclinic and heteroclinic points. We study in detail the forms of the asymptotic orbits with emphasis on the homoclinic and heteroclinic orbits. The homoclinic and heteroclinic intersections are confined in certain intervals along the asymptotic curves. Every homoclinic and heteroclinic orbit is the limit of infinite more homoclinic and heteroclinic orbits. Between an asymptotic orbit falling on the black hole M1 and another orbit falling on the black hole M2 there are infinite homoclinic and heteroclinic orbits and infinite transitions between type I and type II orbits. Therefore the orbits of type I and II form fractal sets. The nonasymptotic curves falling on the black holes M1 and M2 also form fractal sets. The black holes act as attractors and the areas on the surface of section are not conserved despite the fact that the system is conservative.

Chaotic self-similar wave maps coupled to gravity

It was shown recently that the $\mathrm{SU}(2)$ sigma model coupled to gravity has periodic spherically symmetric self-similar solutions for small values of the coupling constant. In this article we extend the analysis of Bizo\ifmmode \acute{n}\else \'{n}\fi{}, Szybka, and Wasserman to larger values of the coupling constant $(\ensuremath{\alpha}g~0.426)$ and present numerical evidence for the chaotic solution and the fractal threshold behavior. We explain this phenomenon in terms of horseshoelike dynamics and heteroclinic intersections.

Heteroclinic intersections between invariant circles of volume-preserving maps

We develop a Melnikov method for volume-preserving maps that have normally hyperbolic invariant sets with codimension-one invariant manifolds. The Melnikov function is shown to be related to the flux of the perturbation through the unperturbed invariant surface. As an example, we compute the Melnikov function for a perturbation of a three-dimensional map that has a heteroclinic connection between a pair of invariant circles. The intersection curves of the manifolds are shown to undergo bifurcations in homology.

Exponentially Small Splitting and Arnold Diffusion for Multiple Time Scale Systems

We consider the class of Hamiltonians: [Formula: see text] where [Formula: see text], and the perturbing function f(q) is a rational function of eiq. We prove upper and lower bounds on the splitting for such class of systems, in regions of the phase space characterized by one fast frequency. Finally using an appropriate Normal Form theorem we prove the existence of chains of heteroclinic intersections.

Periodic perturbations of quadratic planar polynomial vector fields

In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.

Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps.

We study families of volume preserving diffeomorphisms in R(3) that have a pair of hyperbolic fixed points with intersecting codimension one stable and unstable manifolds. Our goal is to elucidate the topology of the intersections and how it changes with the parameters of the system. We show that the "primary intersection" of the stable and unstable manifolds is generically a neat submanifold of a "fundamental domain." We compute the intersections perturbatively using a codimension one Melnikov function. Numerical experiments show various bifurcations in the homotopy class of the primary intersections. (c) 2000 American Institute of Physics.

On the Appearance Order of Homoclinic and Heteroclinic Intersections in a Family of Dissipative Maps of the Plane

The appearance order of homoclinic and heteroclinic intersections are analyzed in a family of dissipative maps f μ of the plane which have two saddles and an attractor. We consider a simple situation that stable and unstable manifolds of two saddles are disjoint for μ = 0, and they form a simple heteroclinic cycle for μ = 1. Thirteen routes from the initial state to the final state have been obtained. As a by-product, a new type of the first tangency which may give birth to homoclinic intersection points in conservative maps of the plane has been obtained. Finally, numerical examples are exhibited and the relation between the symmetry of the map and the occurrence of the routes is discussed.