Smale Horseshoe Research Articles

Invariant manifolds of the Bonhoeffer–van der Pol oscillator

The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer–van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (which are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.

Hopf bifurcation and Si’lnikov chaos of Genesio system

Genesio system, which is a three-dimensional system with only one quadratic nonlinear term, is considered. It has two equilibrium points for some parameters. The Hopf bifurcation is discussed, and the existence of the homoclinic orbit for this system has been proven by using the undetermined coefficient method. As a result, the Si’lnikov criterion guarantees that the Genesio system has Smale horseshoe chaos.

SHIL'NIKOV HETEROCLINIC ORBITS IN A CHAOTIC SYSTEM

In this paper, a chaotic system which exhibits a chaotic attractor with only three equilibria for some parameters is considered. The existence of heteroclinic orbits of the Shil'nikov type in a chaotic system has been proved using the undetermined coefficient method. As a result, the Shil'nikov criterion guarantees that the system has Smale horseshoes. Moreover, the geometric structures of the attractor are determined by these heteroclinic orbits.

Topological characterization of deterministic chaos: enforcing orientation preservation

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

Embeddings of low-dimensional strange attractors: Topological invariants and degrees of freedom

When a low-dimensional chaotic attractor is embedded in a three-dimensional space its topological properties are embedding-dependent. We show that there are just three topological properties that depend on the embedding: Parity, global torsion, and knot type. We discuss how they can change with the embedding. Finally, we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings. These results apply only to chaotic attractors of genus one, which covers the majority of cases in which experimental data have been subjected to topological analysis. This means that the conclusions drawn from previous analyses, for example that the mechanism generating chaotic behavior is a Smale horseshoe mechanism, a reverse horseshoe, a gateau roulé, an S -template branched manifold, etc., are not artifacts of the embedding chosen for the analysis.

DYNAMICAL SYSTEMS ON THREE MANIFOLDS PART I: KNOTS, LINKS AND CHAOS

In this paper, we give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tube neighborhood of the braid. The second one, generating chaotic systems, is realized by modeling the Smale horseshoe.

A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System

This paper presents a three-dimensional topological horseshoe in the hyperchaotic generalized Hénon map. It looks like a planar Smale horseshoe with an additional vertical expansion, so we call it 3D Smale horseshoe. In this way, a computer assisted verification of existence of hyperchaos is provided by means of interval analysis.

Smale horseshoe

Smale horseshoe

К вопросу о классификации линейных и нелинейных подков Смейла

К вопросу о классификации линейных и нелинейных подков Смейла

A hyperelliptic realization of the horseshoe and baker maps

We present a generalization of the functional equation for the Weierstrassk -function for hyperelliptic surfaces of infinite genus arising from iteration of the horseshoe and baker maps. The ramified cover of these infinite genus surfaces over the complex plane are associated to a quadratic differential of finite norm with simple poles accumulating to infinity. We study the geometry of its critical trajectories emanating from these poles and their rate of accumulation.

S˘i’lnikov-type orbits of Lorenz-family systems

This paper studies the Lorenz-family system which is known to establish a topological connection among the Lorenz, Chen and L u ¨ systems in the parametric space. The existence of S˘i’lnikov heterclinic orbits is proved using an undetermined coefficient method. As a consequence, the S˘i’lnikov criterion along with some technical conditions guarantees that the Lorenz-family system has both Smale horseshoes and horseshoe type of chaos. It is this heteroclinic orbit that determines the geometric structure of the corresponding chaotic attractor.

Alternative determinism principle for topological analysis of chaos

The topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits has proven to be a powerful method, however knot theory can only be applied to three-dimensional systems. Still, the core principles upon which this approach is built--determinism and continuity--apply in any dimension. We propose an alternative framework in which these principles are enforced on triangulated surfaces rather than curves, and we show that in dimension 3 our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

Reductions and simplifications of orbital sums in a Hamiltonian repeller

We solve the “center of mass puzzle”, namely the empirical observation that in the presence of a complete Smale horseshoe, the sum of the coordinates of orbital points for low periodic orbits of the Hamiltonian Hénon map reduces to simple rational numbers every so often. We show that such reductions occur systematically in phase-space and arise from specific factorizations of the equations defining orbital coordinates and their sums. The factorizations manifest sets of correlated orbital clusterings and imply an algebraic way of ordering orbits.

Interaction between charged particle and strained superlattice and chaotic behaviours of the system

The effect on a particle motion of a deflecting channel is equated to modulation by a weak-periodic potential of similar shape as the channel. The motion equation of a particle is reduced to the nonlinear differential equation with a weak-periodic modulation by sine-squared potential. The critical conditions for the revelation of Smale horseshoe are analyzed using the Melnikov method. It is predicted that the chaotic behaviours may exit in the interaction of a particles between a crystal. Theory is provided to describe PME effects for the semicoductor superlattice.

Si’lnikov homoclinic orbits in a new chaotic system

Abstract In the present paper, a new chaotic system is considered, which is a three-dimensional quadratic system and exhibits two 1-scroll chaotic attractors simultaneously with only three equilibria for some parameters. The existence of Si’lnikov homoclinic orbits in this system has been proven by using the undetermined coefficient method. As a result, the Si’lnikov criterion guarantees that the system has Smale horseshoes. Moreover, the geometric structures of attractors are determined by these homoclinic orbits.

Active control with delay of horseshoes chaos using piezoelectric absorber on a buckled beam under parametric excitation

We consider the control of an undamped buckled beam, subjected to parametric excitations. Control with delay is applied to suppress chaotic vibrations. Using the Melnikov function approach, we obtain the threshold condition for the inhibition of Smale horseshoes chaos. Emphasis is laid on the effect of time delay.

Smale horseshoes and chaos in discretized perturbed NLS systems (II)—Smale horseshoes

The existence of Smale horseshoes for a certain discretized perturbed nonlinear Schroedinger (NLS ) equations was established by using n-dimensional versions of the Conley-Moser conditions. As a result, the discretized perturbed NLS system is shown to possess an invariant set ? on which the dynamics is topologically conjugate to a shift on four symbols.

Smale horseshoes and chaos in discretized perturbed NLS systems(I)—Poincaré map

The existence of Smale horseshoes for a certain discretized perturbed nonlinear Schroedinger (NLS) equations was established by using n-dimensional versions of the Conley-Moser conditions. As a result, the discretized perturbed NLS system is shown to possess an invariant set ∧ on which the dynamics is topologically conjugate to a shift on four symbols.

Destroying ergodicity in geodesic flows on surfaces

On surfaces, metrics that contain focusing cap outside of which the curvature is non-positive give rise to ergodic, and indeed Bernouli, geodesic flow. There exist C∞ small perturbations of these metrics that destroy the closed geodesic in the focusing cap, thereby producing what we call partially focusing systems. We prove that arbitrarily close to the ergodic focusing system are non-ergodic partially focusing systems. The proof involves perturbing near a homoclinic connection so as to produce a one-parameter family of horseshoe maps which leads to the existence of elliptic periodic orbits.Dedicated to the memory of Vladimir F Lazutkin.

Smale horseshoe via the anti-integrability

Abstract The embedding of the Bernoulli shift into the horseshoe map can be viewed as it is inherited from the anti-integrable limit where the horseshoe has infinite contraction in one direction and infinite expansion in the other. At the limit the map is virtually a subshift of finite type with four symbols or equivalently a full shift with two symbols. We present an algebraic explanation for the equivalence of the two shifts.