Topological Horseshoe Research Articles

Horseshoe and entropy in a fractional-order unified system

This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincaré section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.

Topological horseshoe in nonlinear Bloch system

This paper demonstrates rigorous chaotic dynamics in nonlinear Bloch system by virtue of topological horseshoe and numerical method. It considers a properly chosen cross section and the corresponding Poincaré map, and shows the existence of horseshoe in the Poincaré map. In this way, a rigorous verification of chaos in the nonlinear Bloch system is presented.

Topological horseshoe analysis and the circuit implementation for a four-wing chaotic attractor

The paper first analyzes a newly reported three-dimensional four-wing chaotic attractor, and observes all kinds of attractors, including periodic and chaotic, by numerical simulation. Then, the chaotic characteristic of the system is proved by investigating the existence of a topological horseshoe in it, based on the topological horseshoe theory. At last, an electronic circuit is designed to implement the chaotic system. The results of circuit experiment coincided well with those of numerical simulation.

Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model

Complex nonlinear economic dynamics in a Cournot duopoly model proposed by M. Kopel is studied in detail in this work. By utilizing the topological horseshoe theory proposed by Yang XS, the authors detect the topological horseshoe chaotic dynamics in the Cournot duopoly model for the first time, and also give the rigorous computer-assisted verification for the existence of horseshoe. In the process of the proof, the topological entropy of the Cournot duopoly model is estimated to be bigger than zero, which implies that this economic system definitely exhibits chaos. In particular, the authors observe two different types of economic intermittencies, including the Pomeau–Manneville Type-I intermittency arising near a saddle-node bifurcation, and the crisis-induced attractor widening intermittency caused by the interior crisis, which lead to the appearance of intermittency chaos. The authors also observe the transient chaos phenomenon which leads to the destruction of chaotic attractors. All these intermittency phenomena will help us to understand the similar dynamics observed in the practical stock market and the foreign exchange market. Besides, the Nash-equilibrium profits and the chaotic long-run average profits are analyzed. It is numerically demonstrated that both firms can have higher profits than the Nash-equilibrium profits, that is to say, both of the duopolists could be beneficial from a chaotic market. The controlled Cournot duopoly model can make one firm get more profit and reduce the profit of the other firm, and control the system to converge to an equilibrious state, where the two duopolists share the market equally.

Some novel double-scroll chaotic attractors in Hopfield networks

In this paper, a 3-neuron chaotic Hopfield network is presented. Numerical simulations show that the network displays rich dynamics by changing the self-connection weight. The dynamic of the network is studied by using the Lyapunov exponents spectrum, bifurcation diagram, power spectrum and topological horseshoe theory. It is shown that the proposed network exhibits some novel double-scroll chaotic attractors.

A SIMPLE METHOD FOR FINDING TOPOLOGICAL HORSESHOES

This paper presents an efficient method for finding horseshoes in dynamical systems by using several simple results on topological horseshoes. In this method, a series of points from an attractor of a map (or a Poincaré map) are firstly computed. By dealing with the series, we can not only find the approximate location of each short unstable periodic orbit (UPO), but also learn the dynamics of almost every small neighborhood of the attractor under the map or the reverse map, which is very helpful for finding a horseshoe. The method is illustrated with the Hénon map and two other examples. Since it can be implemented with a computer software, it becomes easy to study the existence of chaos and topological entropy by virtue of topological horseshoe.

Adaptive Set-Oriented Computation of Topological Horseshoe Factors in Area and Volume Preserving Maps

We describe an automatic chaos verification scheme, based on set-oriented numerical methods, which is especially well suited to the study of area- and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, in contrast to existing methods, does not require the computation of chain recurrent sets. We provide several example computations in dimensions two and three.

Chaotic dynamics in a vertically driven planar pendulum

Using elementary phase-plane analysis, combined with results from the theory of topological horseshoes and linked twist maps, we prove the presence of chaos-like dynamics for a vertically driven planar pendulum and other, more general, related equations.

TOPOLOGICAL HORSESHOES AND COMPUTER ASSISTED VERIFICATION OF CHAOTIC DYNAMICS

In this tutorial paper, we present a history of Smale horseshoe and an overview of the progress of topological horseshoe theory. Then we offer a pedagogical exposition of elements of topological horseshoe theory with a lot of examples. Finally we demonstrate some typical applications of topological horseshoe theory to practical dynamical systems.

Computer-assisted verification of chaos in the model of nuclear spin generator

In this paper, chaos in a nuclear spin generator is revisited. To confirm the numerically demonstrated chaotic behavior in the nuclear spin generator, we resort to Poincaré map technique and present a rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoes theory.

Rapid fluctuation for topological dynamical systems

In this paper, we introduce a new notion called rapid fluctuation to characterize the complexity of a general topological dynamical system. As a continuation of the former work [Huang, Chen, Ma, J. Math. Anal. Appl., 2006, 323: 228–252], here we prove that a Lipschitz dynamical system defined on a compact metric space has a rapid fluctuation if it has either a quasi shift invariant set or a topological horseshoe. As an application, the rapid fluctuation of a discrete predator-prey model is considered.

Horseshoe chaos and topological entropy estimate in a simple power system

This paper presents rigorous arguments on existence of chaos and an estimate of topological entropy in a simple power system by means of topological horseshoe theory and computer computations.

Horseshoe in a modified Van der Pol–Duffing circuit

In this paper, we study a modified Van der Pol–Duffing circuit, and present a rigorous verification for existence of chaos in this system. The arguments are given in the manner of computer-assisted proof by using topological horseshoe theory.

A computer-assisted proof for the existence of horseshoe in a novel chaotic system

The dynamics of a novel chaotic system are studied, and a rigorous computer-assisted proof for existence of horseshoe in this system is given. A Poincaré section is properly chosen to obtain the Poincaré map, which is proved to be semi-conjugate to the 4-shift map by utilizing topological horseshoe theory. This implies the entropy of the system is no less than log 4, and the system definitely exhibits chaos.

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THE SMOOTH CHUA'S EQUATION

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

Dynamics in numerics: On a discrete predator-prey model

In this paper, we consider the dynamics of a discrete predator-prey model which is a result of discretization of the corresponding continuous Lotka-Volterra predator-prey model. Among the topis are equilibria and their stability, existence and stability of a period-2 orbit, as well the chaotic behavior of F. The chaos here is in the sense of topological horseshoe and is obtained for certain range of parameter values by applying a recent result from [5]. Our results are in contrast to the recent ones in [6] which claimed that if the predator-prey interaction is replaced by cooperative or competitive interaction, the discretization preserves the property of convergence to the equilibrium, regardless of the step size.

HORSESHOE IN RTD-BASED CELLULAR NEURAL NETWORKS

In this paper, the dynamics of discrete-time RTD-based cellular neural networks is studied. By virtue of topological horseshoe theory, it can be shown that the dynamics of the map derived from the discretization of RTD-based cellular neural networks is semi-conjugate to the dynamics of three-shift map.

Complex dynamics in a nerve fiber model with periodic coefficients

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo-type equation v x x − g v + n ( x ) F ( v ) = 0 , previously considered by Grindrod and Sleeman [P. Grindrod, B.D. Sleeman, A model of a myelinated nerve axon: threshold behaviour and propagation, J. Math. Biol. 23 (1985) 119–135. [6]] and by Chen and Bell [P.-L. Chen, J. Bell, Spine-density dependence of the qualitative behavior of a model of a nerve fiber with excitable spines, J. Math. Anal. Appl. 187 (1994) 384–410.] in the study of nerve fiber models. In some recent works we discussed the case of nonexistence of nontrivial solutions as well as the case in which many positive periodic solutions may arise, the different situations depending upon threshold parameters related to the weight function n ( x ) . Here we show that for a step function n ( x ) (or for small perturbations of it) it is possible to obtain infinitely many periodic solutions and chaotic dynamics, due to the presence of a topological horseshoe (according to Kennedy and Yorke).

A topological horseshoe in the hyperchaotic Rössler attractor

This Letter reports a horseshoe with two-directional expansion in the 4D hyperchaotic Rössler system. In order to show that it is indeed a horseshoe with two-dimensional expansion, some simple results on topological horseshoe which are applicable to 3D hyperchaotic maps are presented. In this way, a computer-assisted verification of hyperchaoticity is presented.

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THREE-NEURON CELLULAR NEURAL NETWORKS

In this paper, we study a new class of simple three-neuron chaotic cellular neural networks with very simple connection matrices. To study the chaotic behavior in these cellular neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of the existence of horseshoe chaos using topological horseshoe theory and the estimate of topological entropy in derived Poincaré maps.