Heteroclinic Tangency Research Articles

Matching geometric and expansion characteristics of wild chaotic attractors

Wild chaotic attractors exhibit chaotic dynamics with a robustness property that cannot be destroyed with small perturbations. We consider a discrete-time system with the smallest possible dimension, namely, defined by a non-invertible map on the complex plane. For this map, wild chaos has been proven to exist in a small parameter region. Recently, it was conjectured to exist in a much larger region of parameter space, past a so-called backward critical tangency, at which a sequence of pre-images of a critical point converges to a saddle fixed point. Geometrically, a backward critical tangency leads to an abundance of homoclinic and heteroclinic tangencies between invariant manifolds of different dimensions, generating precisely what are believed to be the necessary ingredients for wild chaos. In this paper, we present corroborating evidence for this conjecture by computing Lyapunov exponents associated with the attractor. When the sum of the two (largest) Lyapunov exponents is positive, the dynamics is wild chaotic for this non-invertible map. We find that the zero-sum locus matches the locus of backward critical tangency, confirming its role as a boundary of existence of wild chaos.

Strange attractors and wandering domains near a homoclinic cycle to a bifocus

In this paper, we explore the three-dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits. Under an additional hypothesis about the coexistence of two heteroclinically related periodic points (one without dominated splitting into one-dimensional sub-bundles), the heteroclinic tangencies can be slightly modified in order to satisfy Tatjer's conditions for a generalized tangency of codimension two. This configuration may be seen as the organizing center, by which one can obtain Bogdanov-Takens bifurcations and therefore, strange attractors, infinitely many sinks and non-trivial contracting wandering domains.

Robust Heteroclinic Tangencies

We construct diffeomorphisms in dimension $$d\ge 2$$ exhibiting $$C^1$$ -robust heteroclinic tangencies.

A special type of explosion of basin boundary

We study a special type of explosion of a basin boundary set in an archetypal oscillator. A typical feature is that the basin boundaries change the number of basins separating at the same time. Before the explosion, a basin boundary contains some Wada points of ten basins. After the explosion, the basin boundary contains some Wada points of eighteen basins. The underlying mechanism for the explosion is investigated by the heteroclinic tangency and Lambda lemma. Basin entropy and boundary basin entropy are also used to describe the nature of basins of attraction and the basin boundary explosion.

Wada boundary bifurcations induced by boundary saddle–saddle collision

In this letter, a Wada boundary bifurcation (WBB) induced by a boundary saddle touching another boundary saddle is first found through the study of a forced damped pendulum. The WBB can be quantitatively described by the change both in the number of basins involved and in the geometrical size of the boundary. We perceive the manifold structures of the two saddles, that is, a pre-existence of heteroclinic crossing and the other nearly forming heteroclinic tangency exist before the WBB. So we schematically construct the equivalent topological structure of the manifolds of arbitrary two saddles, and rigorously prove two theorems that indicate the existence of the heteroclinic tangency and thus generically confirm the mechanism of such WBB.

Kupka–Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

We construct an explicit example of a family of non-uniformly hyperbolic diffeomorphisms, at the boundary of a set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a Cantor set of choices of the second one. For a non-countable subset of these choices, the second leaf is not periodic and the diffeomorphism is Kupka–Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. The bifurcating system is Hölder-conjugated to a subshift of finite type; thus every Hölder potential admits a unique equilibrium state associated with it.

Parameter-dependent behaviour of periodic channels in a locus of boundary crisis

A boundary crisis occurs when a chaotic attractor outgrows its basin of attraction and suddenly disappears. As previously reported, the locus of a boundary crisis is organised by homo- or heteroclinic tangencies between the stable and unstable manifolds of saddle periodic orbits. In two parameters, such tangencies lead to curves, but the locus of boundary crisis along those curves exhibits gaps or channels, in which other non-chaotic attractors persist. These attractors are stable periodic orbits which themselves can undergo a cascade of period-doubling bifurcations culminating in multi-component chaotic attractors. The canonical diffeomorphic two-dimensional Hénon map exhibits such periodic channels, which are structured in a particular ordered way: each channel is bounded on one side by a saddle-node bifurcation and on the other by a period-doubling cascade to chaos; furthermore, all channels seem to have the same orientation, with the saddle-node bifurcation always on the same side. We investigate the locus of boundary crisis in the Ikeda map, which models the dynamics of energy levels in a laser ring cavity. We find that the Ikeda map features periodic channels with a richer and more general organisation than for the Hénon map. Using numerical continuation, we investigate how the periodic channels depend on a third parameter and characterise how they split into multiple channels with different properties.

Boundary crisis for degenerate singular cycles

The term boundary crisis refers to the destruction or creation of a chaotic attractor when parameters vary. The locus of a boundary crisis may contain regions of positive Lebesgue measure marking the transition from regular dynamics to the chaotic regime. This article investigates the dynamics occurring near a heteroclinic cycle involving a hyperbolic equilibrium point E and a hyperbolic periodic solution P, such that the connection from E to P is of codimension one and the connection from P to E occurs at a quadratic tangency (also of codimension one). We study these cycles as organizing centers of two-parameter bifurcation scenarios and, depending on properties of the transition maps, we find different types of shift dynamics that appear near the cycle. Breaking one or both of the connections we further explore the bifurcation diagrams previously begun by other authors. In particular, we identify the region of crisis near the cycle, by giving information on multipulse homoclinic solutions to E and P as well as multipulse heteroclinic tangencies from P to E, and bifurcating periodic solutions, giving partial answers to the problems (Q1)–(Q3) of Knobloch (2008 Nonlinearity 21 45–60). Throughout our analysis, we focus on the case where E has real eigenvalues and P has positive Floquet multipliers.

On Takens’ last problem: tangencies and time averages near heteroclinic networks

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields C2-close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Takens’ last problem (Takens 2008 Nonlinearity 21 T33–6). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens’ problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure.The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers.We illustrate our results with an explicit example where historic behaviour arises C2-close of a -equivariant vector field.

Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies

Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies

Global bifurcations close to symmetry

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection — the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies — trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions.

Dynamics of conservative Bykov cycles: Tangencies, generalized Cocoon bifurcations and elliptic solutions

This paper presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a conservative Bykov cycle where trajectories turn in opposite directions near the two saddle-foci. We show that within the class of divergence-free vector fields that preserve the cycle, tangencies of the invariant manifolds of two hyperbolic saddle-foci densely occur. The global dynamics is persistently dominated by heteroclinic tangencies and by the existence of infinitely many elliptic points coexisting with non-uniformly hyperbolic suspended horseshoes. A generalized version of the Cocoon bifurcations for conservative systems is obtained.

Dense heteroclinic tangencies near a Bykov cycle

This article presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes — we say that the nodes have different chirality. We show that in the set of vector fields defined on a three-dimensional manifold, there is a class where tangencies of the invariant manifolds of two hyperbolic saddle-foci occur densely. The class is defined by the presence of the Bykov cycle, and by a condition on the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality.We illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three-dimensional sphere.

Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping

In this paper, homoclinic bifurcations and chaotic dynamics of a piecewise linear system subjected to a periodic excitation and a viscous damping are investigated by the Melnikov analysis for nonsmooth systems in detail. The piecewise linear system can be seen as a simple linear feedback control system with dead zone and saturation constrains. The unperturbed system is a piecewise linear Hamiltonian system, which contains two parameters and exhibits quintuple well characteristic. The discontinuous unperturbed system, which is obtained by reducing the two parameters to zero, has saddle-like singularity and homoclinic-like orbit. Analytical expressions for the unperturbed homoclinic and heteroclinic orbits are derived by using Hamiltonian function for the piecewise linear system. The Melnikov analysis for nonsmooth planar systems is first described briefly, and the theorem for homoclinic bifurcations for the nonsmooth planar systems is also obtained and then employed to detect the homoclinic and heteroclinic tangency under the perturbation of a viscous damping and a periodic excitation. Finally, the chaotic attractors and the bifurcation diagrams are presented to show the bifurcations and chaotic dynamics of the piecewise linear system.

Realization of cascades on surfaces with finitely many moduli of topological conjugacy

A method for constructing cascades on surfaces is developed, which makes it possible to model structurally unstable discrete dynamical systems with finitely many orbits of heteroclinic tangency and preset moduli of topological conjugacy.

Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps**Dedicated to the memory of Leonid Pavlovich Shilnikov, a Master whose works strongly influenced the mathematical theory of dynamical systems.

We study the dynamics and bifurcations of two-dimensional reversible maps with non-transversal heteroclinic cycles containing symmetric saddle fixed points. We consider one-parameter families of reversible maps unfolding the initial heteroclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations and the birth of asymptotically stable, unstable and elliptic periodic orbits.

Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

We study periodic perturbations of planar quadratic vector fieldshaving infinite heteroclinic cycles, consisting of an invariantstraight line joining two saddle points at infinity and an arc oforbit also at infinity. The global study concerning the infinity ofthe perturbed system is performed by means of the Poincarécompactification in polar coordinates, from which we obtain a systemdefined on a set equivalent to a solid torus in $\mathbb{R}^3$, whoseboundary plays the role of the infinity. It is shown that forcertain type of periodic perturbation, there exist twodifferentiable curves in the parameter space for which the perturbedsystem presents heteroclinic tangencies and transversalintersections between the stable and unstable manifolds of twonormally hyperbolic lines of singularities at infinity. Thetransversality of the manifolds is proved using the Melnikov methodand implies, via the Birkhoff-Smale Theorem, in a complex dynamicalbehavior of the perturbed system solutions in a finite part of thephase space. Numerical simulations are performed for a particularexample in order to illustrate this behavior, which could be called'the chaos arising from infinity', because it depends on theglobal structure of the quadratic system, including the points atinfinity.

Renormalization in the Hénon family, II: the heteroclinic web

We study highly dissipative Henon maps $$F_{c,b}: (x,y) \mapsto (c-x^2-by, x)$$ with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of such maps (even away from W). We also prove that in the infinitely renormalizable case, the average Jacobian bF on the attracting Cantor set \({\mathcal{O}}_{F}\) is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside \({\mathcal{O}}_{F}\) if and only if there are no heteroclinic tangencies.

Necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency

We consider diffeomorphisms of orientable surfaces with the nonwandering set consisting of a finite number of hyperbolic fixed points and the wandering set containing a finite number of heteroclinic orbits of transversal and nontransversal intersection. We distinguish a meaningful class of diffeomorphisms and present a complete topological invariant for this class. The invariant is a scheme consisting of a set of numerical parameters and a set of geometric objects.

A classification of explosions in dimension one

AbstractA discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, anexplosionis a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.