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Journal articles on the topic 'Topological horseshoes'

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Author: Grafiati

Published: 9 March 2023

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1

Kennedy, Judy, and James A. Yorke. "Topological horseshoes." Transactions of the American Mathematical Society 353, no. 6 (February 15, 2001): 2513–30. http://dx.doi.org/10.1090/s0002-9947-01-02586-7.

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2

LI, QINGDU, and XIAO-SONG YANG. "A SIMPLE METHOD FOR FINDING TOPOLOGICAL HORSESHOES." International Journal of Bifurcation and Chaos 20, no. 02 (February 2010): 467–78. http://dx.doi.org/10.1142/s0218127410025545.

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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.

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3

HUAN, SONGMEI, QINGDU LI, and XIAO-SONG YANG. "HORSESHOES IN A CHAOTIC SYSTEM WITH ONLY ONE STABLE EQUILIBRIUM." International Journal of Bifurcation and Chaos 23, no. 01 (January 2013): 1350002. http://dx.doi.org/10.1142/s0218127413500028.

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To confirm the numerically demonstrated chaotic behavior in a chaotic system with only one stable equilibrium reported by Wang and Chen, we resort to Poincaré map technique and present a rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoes theory.

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4

YUAN, QUAN, and XIAO-SONG YANG. "COMPUTER ASSISTED VERIFICATION OF CHAOS IN THE SMOOTH CHUA'S EQUATION." International Journal of Bifurcation and Chaos 18, no. 08 (August 2008): 2391–96. http://dx.doi.org/10.1142/s0218127408021762.

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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.

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5

LI, QINGDU, and XIAO-SONG YANG. "TWO KINDS OF HORSESHOES IN A HYPERCHAOTIC NEURAL NETWORK." International Journal of Bifurcation and Chaos 22, no. 08 (August 2012): 1250200. http://dx.doi.org/10.1142/s0218127412502008.

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This paper studies the hyperchaotic dynamics in a four-dimensional Hopfield neural network by virtue of topological horseshoe. Two different horseshoes (chaotic invariant sets) are found in this network with the same parameters. Numerical studies show that the first one expands only one-dimensionally and the second one expands two-dimensionally. Computer simulation also shows that there exists a heteroclinic connection from the second horseshoe to the first one, which indicates that the chaotic set of this system can have a very complicated structure composed of different kinds of expansions.

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6

Yang, Xiao-Song. "Topological horseshoes in continuous maps." Chaos, Solitons & Fractals 33, no. 1 (July 2007): 225–33. http://dx.doi.org/10.1016/j.chaos.2005.12.030.

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7

YANG, XIAO-SONG. "TOPOLOGICAL HORSESHOES AND COMPUTER ASSISTED VERIFICATION OF CHAOTIC DYNAMICS." International Journal of Bifurcation and Chaos 19, no. 04 (April 2009): 1127–45. http://dx.doi.org/10.1142/s0218127409023548.

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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.

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8

Wójcik, Klaudiusz, and Piotr Zgliczyński. "Topological horseshoes and delay differential equations." Discrete & Continuous Dynamical Systems - A 12, no. 5 (2005): 827–52. http://dx.doi.org/10.3934/dcds.2005.12.827.

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9

GONCHENKO, SERGEY, MING-CHIA LI, and MIKHAIL MALKIN. "GENERALIZED HÉNON MAPS AND SMALE HORSESHOES OF NEW TYPES." International Journal of Bifurcation and Chaos 18, no. 10 (October 2008): 3029–52. http://dx.doi.org/10.1142/s0218127408022238.

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We study hyperbolic dynamics and bifurcations for generalized Hénon maps in the form [Formula: see text] (with b, α small and γ > 4). Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to the local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary.

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10

Yuan, Quan, Fang-Yan Yang, and Lei Wang. "A Note on Hidden Transient Chaos in the Lorenz System." International Journal of Nonlinear Sciences and Numerical Simulation 18, no. 5 (July 26, 2017): 427–34. http://dx.doi.org/10.1515/ijnsns-2016-0168.

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AbstractIn this paper, the classic Lorenz system is revisited. Some dynamical behaviors are shown with the Rayleigh number $\rho $ somewhat smaller than the critical value 24.06 by studying the basins characterization of attraction of attractors and tracing the one-dimensional unstable manifold of the origin, indicating some interesting clues for detecting the existence of hidden transient chaos. In addition, horseshoes chaos is verified in the famous system for some parameters corresponding to the hidden transient chaos by the topological horseshoe theory.

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11

Kočan, Zdeněk, Veronika Kurková, and Michal Málek. "Horseshoes, Entropy, Homoclinic Trajectories, and Lyapunov Stability." International Journal of Bifurcation and Chaos 24, no. 02 (February 2014): 1450016. http://dx.doi.org/10.1142/s0218127414500163.

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We consider six properties of continuous maps, such as the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, or Lyapunov instability on the set of periodic points. The relations between the considered properties are provided in the case of graph maps, dendrite maps and maps on compact metric spaces. For example, by [Llibre & Misiurewicz, 1993] in the case of graph maps, the existence of an arc horseshoe implies the positivity of topological entropy, but we construct a continuous map on a Peano continuum with an arc horseshoe and zero topological entropy. We also formulate one open problem.

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12

Zhou, Ping, and Meihua Ke. "A New 3D Autonomous Continuous System with Two Isolated Chaotic Attractors and Its Topological Horseshoes." Complexity 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/4037682.

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Based on the 3D autonomous continuous Lü chaotic system, a new 3D autonomous continuous chaotic system is proposed in this paper, and there are coexisting chaotic attractors in the 3D autonomous continuous chaotic system. Moreover, there are no overlaps between the coexisting chaotic attractors; that is, there are two isolated chaotic attractors (in this paper, named “positive attractor” and “negative attractor,” resp.). The “positive attractor” and “negative attractor” depend on the distance between the initial points (initial conditions) and the unstable equilibrium points. Furthermore, by means of topological horseshoes theory and numerical computation, the topological horseshoes in this 3D autonomous continuous system is found, and the topological entropy is obtained. These results indicate that the chaotic attractor emerges in the new 3D autonomous continuous system.

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13

Cian, Giuseppe. "Some remarks on topological horseshoes and applications." Nonlinear Analysis: Real World Applications 16 (April 2014): 74–89. http://dx.doi.org/10.1016/j.nonrwa.2013.09.007.

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14

Nunes, Pollyanna Vicente, and Fábio Armando Tal. "Transitivity and the existence of horseshoes on the 2-torus." Nonlinearity 36, no. 1 (December 8, 2022): 199–230. http://dx.doi.org/10.1088/1361-6544/aca252.

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Abstract We study the relationship between transitivity and topological chaos for homeomorphisms of the two torus. We show that if a transitive homeomorphism of T 2 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 T 2 is homotopic to a Dehn twist, then either it is aperiodic or it has a topological horseshoe.

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15

Li, Ming-Chia, and Mikhail Malkin. "Topological horseshoes for perturbations of singular difference equations." Nonlinearity 19, no. 4 (February 14, 2006): 795–811. http://dx.doi.org/10.1088/0951-7715/19/4/002.

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16

Zhang, Xu. "Chaotic Polynomial Maps." International Journal of Bifurcation and Chaos 26, no. 08 (July 2016): 1650131. http://dx.doi.org/10.1142/s0218127416501315.

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This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li–Yorke or Devaney. This type of maps includes both the Logistic map and the Hénon map. For some diffeomorphisms with the expansion dimension equal to one or two in three-dimensional spaces, the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets on which the systems are topologically conjugate to the two-sided fullshift on finite alphabet are obtained; for some expanding maps, the chaotic region is analyzed by using the coupled-expansion theory and the Brouwer degree theory. For three types of higher-dimensional polynomial maps with degree two, the conditions under which there are Smale horseshoes and uniformly hyperbolic invariant sets are given, and the topological conjugacy between the maps on the invariant sets and the two-sided fullshift on finite alphabet is obtained. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.

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17

KOČAN, ZDENĚK, VERONIKA KORNECKÁ-KURKOVÁ, and MICHAL MÁLEK. "Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites." Ergodic Theory and Dynamical Systems 31, no. 1 (February 2, 2010): 165–75. http://dx.doi.org/10.1017/s0143385709001011.

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AbstractIt is known that the positiveness of topological entropy, the existence of a horseshoe and the existence of a homoclinic trajectory are mutually equivalent, for interval maps. The aim of the paper is to investigate the relations between the properties for continuous maps of trees, graphs and dendrites. We consider three different definitions of a horseshoe and two different definitions of a homoclinic trajectory. All the properties are mutually equivalent for tree maps, whereas not for maps on graphs and dendrites. For example, positive topological entropy and the existence of a homoclinic trajectory are independent and neither of them implies the existence of any horseshoe in the case of dendrites. Unfortunately, there is still an open problem, and we formulate it at the end of the paper.

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18

Li, Jian, Piotr Oprocha, and Guohua Zhang. "Quasi-graphs, zero entropy and measures with discrete spectrum." Nonlinearity 35, no. 3 (February 11, 2022): 1360–79. http://dx.doi.org/10.1088/1361-6544/ac4b3a.

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Abstract In this paper, we study dynamics of maps on quasi-graphs and characterise their invariant measures. In particular, we prove that every invariant measure of a quasi-graph map with zero topological entropy has discrete spectrum. Additionally, we obtain an analog of Llibre–Misiurewicz’s result relating positive topological entropy with existence of topological horseshoes. We also study dynamics on dendrites and show that if a continuous map on a dendrite whose set of all endpoints is closed and has only finitely many accumulation points, has zero topological entropy, then every invariant measure supported on an orbit closure has discrete spectrum.

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19

Sovrano, Elisa. "How to Construct Complex Dynamics? A Note on a Topological Approach." International Journal of Bifurcation and Chaos 30, no. 02 (February 2020): 2050034. http://dx.doi.org/10.1142/s0218127420500340.

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We investigate the presence of complex behaviors for the solutions of two different dynamical systems: one is of discrete type and the other is continuous. We give evidence of “chaos” in the framework of topological horseshoes and show how different problems can be analyzed by the same procedure.

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20

PEDERSON, STEVEN M. "ESSENTIAL ENTROPY-CARRYING HORSESHOES AS SET LIMITS." International Journal of Bifurcation and Chaos 22, no. 08 (August 2012): 1250195. http://dx.doi.org/10.1142/s0218127412501957.

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This paper studies the set limit of a sequence of invariant sets corresponding to a convergent sequence of piecewise monotone interval maps. To do this, the notion of essential entropy-carrying set is introduced. A piecewise monotone map f with an essential entropy-carrying horseshoe S(f) and a sequence of piecewise monotone maps [Formula: see text] converging to f is considered. It is proven that if each gi has an invariant set T(gi) with at least as much topological entropy as f, then the set limit of [Formula: see text] contains S(f).

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21

Chen, Yi-Chiuan, Shyan-Shiou Chen, and Juan-Ming Yuan. "Topological horseshoes in travelling waves of discretized nonlinear wave equations." Journal of Mathematical Physics 55, no. 4 (April 2014): 042701. http://dx.doi.org/10.1063/1.4870618.

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22

Pascoletti, Anna, and Fabio Zanolin. "A Crossing Lemma for Annular Regions and Invariant Sets with an Application to Planar Dynamical Systems." Journal of Mathematics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/267393.

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We present a topological result, namedcrossing lemma, dealing with the existence of a continuum which crosses a topological space between a pair of “opposite” sides. This topological lemma allows us to obtain some fixed point results. In the works of Pascoletti et al., 2008, and Pascoletti and Zanolin, 2010, we have widely exposed the crossing lemma for planar regions homeomorphic to a square, and we have also presented some possible applications to the theory of topological horseshoes and to the study of chaotic-like dynamics for planar maps. In this work, we move from the framework of the generalized rectangles to two other settings (annular regions and invariant sets), trying to obtain similar results. An application to a model of fluid mixing is given.

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23

BARTOŠ, ADAM, JOZEF BOBOK, PAVEL PYRIH, SAMUEL ROTH, and BENJAMIN VEJNAR. "Constant slope, entropy, and horseshoes for a map on a tame graph." Ergodic Theory and Dynamical Systems 40, no. 11 (April 22, 2019): 2970–94. http://dx.doi.org/10.1017/etds.2019.29.

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We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a map $g$ of constant slope. In particular, we show that in the case of a Markov map $f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope $e^{h_{\text{top}}(f)}$, where $h_{\text{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\text{top}}(f)$ is achievable through horseshoes of the map $f$.

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24

YANG, XIAO-SONG, and QINGDU LI. "HORSESHOES IN A NEW SWITCHING CIRCUIT VIA WIEN-BRIDGE OSCILLATOR." International Journal of Bifurcation and Chaos 15, no. 07 (July 2005): 2271–75. http://dx.doi.org/10.1142/s0218127405011631.

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In this paper we revisit a switching circuit designed by the authors and present a theoretical analysis on the existence of chaos in this circuit. For the ordinary differential equations describing this circuit, we give a computer-aided proof in terms of cross-section and Poincare map, by applying a modern theory of topological horseshoes theory to the obtained Poincare map, that this map is semiconjugate to the two-shift map. This implies that the corresponding differential equations exhibit chaos.

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25

Yang, Dawei, and Jinhua Zhang. "NON-HYPERBOLIC ERGODIC MEASURES AND HORSESHOES IN PARTIALLY HYPERBOLIC HOMOCLINIC CLASSES." Journal of the Institute of Mathematics of Jussieu 19, no. 5 (January 7, 2019): 1765–92. http://dx.doi.org/10.1017/s1474748018000579.

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We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$\ast$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures. As a consequence, we obtain the continuity of topological entropy.

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26

Kiriki, Shin, and Masaki Nakajima. "Blenders for a non-normally Henon-like family." Tamkang Journal of Mathematics 41, no. 2 (June 30, 2010): 149–66. http://dx.doi.org/10.5556/j.tkjm.41.2010.666.

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A blender is an indispensable concept presented by Bonatti and Diaz [3] to study high-dimensional $C^1$-robust transitive dynamics around heterodimensional cycles. In this paper, we present a certain Henon-like family of real quadratic diffeomorphisms on $\mathbb{R}^3$, which exhibits an phase transition from non-normal horseshoes to blenders. It can be observable from a rapidly jump of topological dimension for some projected stable segments in some characteristic region of $\mathbb{R}^3$.

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27

Gameiro, Marcio, Tomáš Gedeon, William Kalies, Hiroshi Kokubu, Konstantin Mischaikow, and Hiroe Oka. "Topological Horseshoes of Traveling Waves for a Fast–Slow Predator–Prey System." Journal of Dynamics and Differential Equations 19, no. 3 (May 18, 2006): 623–54. http://dx.doi.org/10.1007/s10884-006-9013-6.

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28

Pham, Viet-Thanh, Christos Volos, Sundarapandian Vaidyanathan, and Xiong Wang. "A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization." Advances in Mathematical Physics 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/4024836.

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Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control.

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29

Margheri, Alessandro, Carlota Rebelo, and Fabio Zanolin. "Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2191 (January 4, 2021): 20190385. http://dx.doi.org/10.1098/rsta.2019.0385.

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In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive–contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré–Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré–Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

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30

SLIJEPČEVIĆ, SINIŠA. "Variational construction of positive entropy invariant measures of Lagrangian systems and Arnold diffusion." Ergodic Theory and Dynamical Systems 40, no. 3 (September 25, 2018): 799–864. http://dx.doi.org/10.1017/etds.2018.59.

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We develop a variational method for constructing positive entropy invariant measures of Lagrangian systems without assuming transversal intersections of stable and unstable manifolds, and without restrictions to the size of non-integrable perturbations. We apply it to a family of $2\frac{1}{2}$ degrees of freedom a priori unstable Lagrangians, and show that if we assume that there is no topological obstruction to diffusion (precisely formulated in terms of topological non-degeneracy of minima of the Peierls barrier), then there exists a vast family of ‘horseshoes’, such as ‘shadowing’ ergodic positive entropy measures having precisely any closed set of invariant tori in its support. Furthermore, we give bounds on the topological entropy and the ‘drift acceleration’ in any part of a region of instability in terms of a certain extremal value of the Fréchet derivative of the action functional, generalizing the angle of splitting of separatrices. The method of construction is new, and relies on study of formally gradient dynamics of the action (coupled parabolic semilinear partial differential equations on unbounded domains). We apply recently developed techniques of precise control of the local evolution of energy (in this case the Lagrangian action), energy dissipation and flux.

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31

Zanini, Chiara, and Fabio Zanolin. "Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential." Complexity 2018 (December 2, 2018): 1–17. http://dx.doi.org/10.1155/2018/2101482.

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We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.

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32

Mitchener, W. Garrett. "Symmetric replicator dynamics with depletable resources." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 4 (April 2022): 043121. http://dx.doi.org/10.1063/5.0081182.

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The replicator equation is a standard model of evolutionary population game dynamics. In this article, we consider a modification of replicator dynamics, in which playing a particular strategy depletes an associated resource, and the payoff for that strategy is a function of the availability of the resource. Resources are assumed to replenish themselves, given time. Overuse of a resource causes it to crash. If the depletion rate is low enough, most trajectories converge to a stable equilibrium at which all initially present strategies are equally popular. As the depletion rate increases, these fixed points vanish in bifurcations. The phase space is periodic in each of the resource variables, and it is possible for trajectories to whirl around different numbers of times in these variables before converging to the stable equilibrium, resulting in a wide variety of topological types of orbits. Numerical solutions in a low-dimensional case show that in a cross section of the phase space, the topological types are separated by intricately folded separatrices. Once the depletion rate is high enough that the stable equilibrium in the interior of the phase space vanishes, the dynamics immediately become chaotic, without going through a period-doubling cascade; a numerical method reveals horseshoes in a Poincaré map. It appears that the multitude of topological types of orbits present before this final bifurcation generate this chaotic behavior. A periodic orbit of saddle type can be found using the symmetries of the dynamics, and its stable and unstable manifolds may generate a homoclinic tangle.

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33

Ma, You Jie, Shuang Song, and Xue Song Zhou. "Introduction of Topological Horseshoe Theory in Chaotic Research." Advanced Materials Research 811 (September 2013): 716–19. http://dx.doi.org/10.4028/www.scientific.net/amr.811.716.

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Over the past 10 years, nonlinear dynamics and chaotic theory attracted scholars and people got a deeper understanding of chaos. There are many methods for chaos research, and the method of using topological horseshoe is an important branch of those methods. So far, this is one of the core methods with mathematical rigor for chaos research. Based on simple thinking of geometric space, topological horseshoe build a bridge for numerical and theoretical studies of complex behavior of nonlinear systems so that people can carry out a series of studies for chaotic behavior. This paper introduces the basic content of topological horseshoe theory and the application to a simple power system.

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34

Wang, Chunmei, Chunhua Hu, Jingwei Han, and Shijian Cang. "A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos." Advances in Mathematical Physics 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/3142068.

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A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory.

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35

YUAN, QUAN, and XIAO-SONG YANG. "COMPUTER ASSISTED VERIFICATION OF CHAOS IN THREE-NEURON CELLULAR NEURAL NETWORKS." International Journal of Bifurcation and Chaos 17, no. 12 (December 2007): 4381–86. http://dx.doi.org/10.1142/s0218127407020026.

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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.

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36

Li, Qingdu, and Xiao-Song Yang. "A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System." Discrete Dynamics in Nature and Society 2007 (2007): 1–9. http://dx.doi.org/10.1155/2007/16239.

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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.

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37

Wang, Lei, XiaoSong Yang, WenJie Hu, and Quan Yuan. "Horseshoe Chaos in a Simple Memristive Circuit." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/546091.

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A simple memristive circuit model is revisited and the stability analysis is to be given. Furthermore, we resort to Poincaré section and Poincaré map technique and present rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoe theory.

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38

Li, Chunlai, Lei Wu, Hongmin Li, and Yaonan Tong. "A novel chaotic system and its topological horseshoe." Nonlinear Analysis: Modelling and Control 18, no. 1 (January 25, 2013): 66–77. http://dx.doi.org/10.15388/na.18.1.14032.

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Based on the construction pattern of Chen, Liu and Qi chaotic systems, a new threedimensional (3D) chaotic system is proposed by developing Lorenz chaotic system. It’s found that when parameter e varies, the Lyapunov exponent spectrum keeps invariable, and the signal amplitude can be controlled by adjusting e. Moreover, the horseshoe chaos in this system is investigated based on the topological horseshoe theory.

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39

Fan, Qing-Ju. "Topological horseshoe in nonlinear Bloch system." Chinese Physics B 19, no. 12 (December 2010): 120508. http://dx.doi.org/10.1088/1674-1056/19/12/120508.

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Zhang, Xu, and Guanrong Chen. "A simple topological model for two coupled neurons." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 7 (July 2022): 073124. http://dx.doi.org/10.1063/5.0097385.

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YANG, FANGYAN, QINGDU LI, and PING ZHOU. "HORSESHOE IN THE HYPERCHAOTIC MCK CIRCUIT." International Journal of Bifurcation and Chaos 17, no. 11 (November 2007): 4205–11. http://dx.doi.org/10.1142/s0218127407019743.

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The well-known Matsumoto–Chua–Kobayashi (MCK) circuit is of significance for studying hyperchaos, since it was the first experimental observation of hyperchaos from a real physical system. In this paper, we discuss the existence of hyperchaos in this circuit by virtue of topological horseshoe theory. The two disjoint compact subsets producing a horseshoe found in a specific 3D cross-section, both expand in two directions under the fourth Poincaré return map, this fact means that there exists hyperchaos in the circuit.

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LEFRANC, MARC, and PIERRE GLORIEUX. "TOPOLOGICAL ANALYSIS OF CHAOTIC SIGNALS FROM A CO2 LASER WITH MODULATED LOSSES." International Journal of Bifurcation and Chaos 03, no. 03 (June 1993): 643–50. http://dx.doi.org/10.1142/s0218127493000544.

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Unstable periodic orbits have been extracted from chaotic time series coming from a CO 2 laser with modulated losses. Topological analysis of their organization reveals that chaos in this laser occurs through the formation of a Smale’s horseshoe.

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43

Li, Qingdu. "A topological horseshoe in the hyperchaotic Rössler attractor." Physics Letters A 372, no. 17 (April 2008): 2989–94. http://dx.doi.org/10.1016/j.physleta.2007.11.071.

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Takeuchi, Noriaki, Tachiki Nagai, and Takashi Matsumoto. "Topological horseshoe in the R-L-diode circuit." Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 84, no. 3 (2000): 91–100. http://dx.doi.org/10.1002/1520-6440(200103)84:3<91::aid-ecjc10>3.0.co;2-y.

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CHEN, FENG-JUAN, JI-BIN LI, and FANG-YUE CHEN. "HORSESHOE IN RTD-BASED CELLULAR NEURAL NETWORKS." International Journal of Bifurcation and Chaos 18, no. 03 (March 2008): 689–94. http://dx.doi.org/10.1142/s0218127408020586.

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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.

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Boulant, G., M. Lefranc, S. Bielawski, and D. Derozier. "A Nonhorseshoe Template in a Chaotic Laser Model." International Journal of Bifurcation and Chaos 08, no. 05 (May 1998): 965–75. http://dx.doi.org/10.1142/s0218127498000772.

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We have performed a topological analysis of a chaotic regime of a modulated single-mode class-B laser model, at realistic parameter values. In contrast with previous numerical and experimental studies of this type of laser, we observe a topological structure which differs from the one described by the paradigmatic horseshoe template. In view of this result, class-B lasers appear to be good candidates for the first clear characterization of a nonhorseshoe template in an experimental system.

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BEDFORD, ERIC, and JOHN SMILLIE. "A symbolic characterization of the horseshoe locus in the Hénon family." Ergodic Theory and Dynamical Systems 37, no. 5 (March 8, 2016): 1389–412. http://dx.doi.org/10.1017/etds.2015.113.

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We consider the family of quadratic Hénon diffeomorphisms of the plane $\mathbb{R}^{2}$. A map will be said to be a ‘horseshoe’ if its restriction to the non-wandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for being a horseshoe based on an auxiliary coding which describes positions of points relative to the stable manifold of one of the fixed points. In addition we describe the topological conjugacy type of maps on the boundary of the horseshoe locus. We use complex techniques and we work with maps in a parameter region which is a two-dimensional analog of the familiar ‘$1/2$-wake’ for the quadratic family $p_{c}(z)=z^{2}$.

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LI, QINGDU, XIAO-SONG YANG, and SHU CHEN. "HYPERCHAOS IN A SPACECRAFT POWER SYSTEM." International Journal of Bifurcation and Chaos 21, no. 06 (June 2011): 1719–26. http://dx.doi.org/10.1142/s0218127411029380.

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This paper presents some rigorous arguments on a chaotic strange attractor in a spacecraft power system, which is a 3D system with hysteresis switching. By carefully picking a suitable cross-section with respect to the attractor, we find a topological horseshoe of the corresponding third-returned Poincaré map, thus giving a rigorous verification of the existence of chaos in this system. Numerical computation shows that the two Lyapunov exponents of the Poincaré map are both positive and that there exists a two-directional expansion in this horseshoe, suggesting that this attractor is hyperchaotic in nature.

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YANG, XIAO-SONG, and QINGDU LI. "CHAOS IN SIMPLE CELLULAR NEURAL NETWORKS WITH CONNECTION MATRICES SATISFYING DALE'S RULE." International Journal of Bifurcation and Chaos 17, no. 02 (February 2007): 583–87. http://dx.doi.org/10.1142/s0218127407017446.

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In this paper, it is shown that chaos can take place in simple three-dimensional cellular neural networks with connection matrices satisfying Dale's rule. In addition, a rigorous computer-assisted verification of chaoticity in these cellular neural networks is given by virtue of topological horseshoe theory.

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Xi, Lifeng. "Horseshoe effect and topological entropy of one-dimensional maps." Applied Mathematics 11, no. 2 (June 1996): 209–16. http://dx.doi.org/10.1007/bf02662014.

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