Existence Of Horseshoes Research Articles

Transitivity and the existence of horseshoes on the 2-torus

We study the relationship between transitivity and topological chaos for homeomorphisms of the two torus. We show that if a transitive homeomorphism of is homotopic to the identity and has both a fixed point and a periodic point which is not fixed, then it has a topological horseshoe. We also show that if a transitive homeomorphisms of is homotopic to a Dehn twist, then either it is aperiodic or it has a topological horseshoe.

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.

Chaos in three-dimensional hybrid systems and design of chaos generators

The existence of horseshoes is proved in a class of 3-dim piecewise linear systems, in which a homoclinic orbit connecting the origin to itself is explicitly given. Based on these results, a mathematically rigorous methodology for design of chaos generators is proposed. Implementation of such chaos generators by circuit is easy and the chaotic attractor is robust under small perturbations.

New walking dynamics in the simplest passive bipedal walking model

This paper revisits the simplest passive walking model by Garcia et al. which displays chaos through period doubling from a stable period-1 gait. By carefully numerical studies, two new gaits with period-3 and -4 are found, whose stability is verified by estimates of eigenvalues of the corresponding Jacobian matrices. A surprising phenomenon uncovered here is that they both lead to higher periodic cycles and chaos via period doubling. To study the three different types of chaotic gaits rigorously, the existence of horseshoes is verified and estimates of the topological entropies are made by computer-assisted proofs in terms of topological horseshoe theory.

A simple computable criteria for the existence of horseshoes

In this note we present a simple computable criteria that assures the existence of hyperbolic horseshoes for certain diffeomorphisms of the torus. The main advantage of our method is that it is very easy to check numerically whether the criteria is satisfied or not.

Topological horseshoes in continuous maps

Abstract We present a framework on existence of horseshoes in continuous maps in terms of “crossing with respect to two pairs of fixed subsets”.

Horseshoes in chromosome’s attractors

Abstract In this paper, we study dynamics of a class of chromosome’s attractors. We show that these chromosome sequences are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincare maps derived from these chromosome’s attractors are semi-conjugate to the 2-shift map, and its entropy is no less than log 2. The chaotic behavior is robust in the following sense: chaos exists when one parameter varies from −5.5148 to −5.4988.

Horseshoe in a class of planar mappings

Bursting dynamics of mappings is investigated in this paper. We first present stability analysis of the mappings' equilibria with various parameters. Then for three mappingsP,P¯, andP^with different parameters, we study their powersP4,P¯6, andP^4. We show that the mappings thus obtained are chaotic by giving a rigorous verification of existence of horseshoes in these mappings. Precisely, we prove that the mappingP¯6is semiconjugate to the 3-shift mapping; the mappingsP4andP^4are semiconjugate to the 4-shift mapping.

Horseshoes in a class of simple circuits

Abstract In this paper we study dynamics of a class of simple single-scroll and two-scrolls type circuits, we show that these circuits are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincare map derived from the ordinary differential equations of the single-scroll circuit system is semi-conjugate to the 2-shift map; and the Poincare map of the two-scrolls circuit system is semi-conjugate to the 4-shift map.

Horseshoes in modified Chen’s attractors

Abstract In this paper we study dynamics of a class of modified Chen’s attractors, we show that these attractors are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincare maps derived from these modified Chen’s attractors are semi-conjugate to the 2-shift map.

Metric horseshoes

In this paper a framework for metric horseshoe theory is developed to provide a criterion on the existence of horseshoes in maps with intension to offer a potential approach to study chaos in concrete systems via computer computations and simulations.

Horseshoes in piecewise continuous maps

This paper presents a framework on existence of horseshoes in piecewise continuous maps, which is more applicable to practical problems.

The nonmonotonic twist maps and basic hyperbolic sets

The existence of horseshoes for a family of nonmonotonic twist maps is established. And it is proved that the Hausdorff dimension of the horseshoes goes to the dimension of phase space as the parameter goes to infinity.

GENERALIZING THE TWIST-AND-FLIP PARADIGM: II

In this letter we introduce a new concept, the control manifold, which simultaneously points us toward a general algorithm relating the existence of horseshoes in a Poincare map to the parameters of an ODE, while also providing a mechanism for understanding how to control nonlinear phenomena and therefore utilize these phenomena in a disciplined and predictable manner in the design of nonlinear circuits, systems, and neural networks.