Topological Horseshoe Research Articles

A Note on Hidden Transient Chaos in the Lorenz System

Abstract In this paper, the classic Lorenz system is revisited. Some dynamical behaviors are shown with the Rayleigh number ρ $\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.

Entropy, Chaos, and Weak Horseshoe for Infinite‐Dimensional Random Dynamical Systems

AbstractIn this paper, we study the complicated dynamics of infinite‐dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we prove if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li‐Yorke. The complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system.© 2017 Wiley Periodicals, Inc.

Simple hyperchaotic memory system with large topological entropy

The memory elements have been the research hot spot for a long time. However, there is a litter works on the linear memory element. This paper presents a study of a new memory system containing a linear memory element. The study shows that the system not only has two kinds of route to hyperchaos, but also exists many kinds of coexisting attractors in the phase space. Moreover, the system can generate more complex hyperchaotic behaviors. To prove it, we find a new kind of topological horseshoes with two-directional expansions that consist of three disconnected compact sets. This new kind of horseshoes suggests that the topological entropy of the hyperchaotic attractor is larger than other hyperchaotic attractors reported before. For detailed study of the hyperchaotic invariant set, we also demonstrate a method to extract the orbits from the hyperchaotic horseshoes.

The dynamic analysis of a chaotic system

A three-dimensional chaotic system which contains different parameters is proposed in this article. By applying numerical and analytical methods, the basic properties of the system, such as dynamical behaviors, dissipation, sensitivity, power spectrum, Poincaré map, and Lyapunov exponents, are investigated. In addition, the existence of both Hopf bifurcation and topological horseshoe is presented. The obtained results clearly show that this is a new chaotic system.

A New 3D Autonomous Continuous System with Two Isolated Chaotic Attractors and Its Topological Horseshoes

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.

Topological horseshoe and its uniform hyperbolicity in the HP model.

A famous food-chain model proposed by Hastings and Powell is numerically restudied. The existence and uniform hyperbolicity of chaotic invariant sets are demonstrated by means of the topological horseshoe theory and the Conley-Moser conditions, indicating that, for a fixed cross section, the second return Poincaré map of the model possesses a closed uniformly hyperbolic chaotic invariant set, on which it is topologically conjugate to the 2-shift map.

A simple chaotic system with non-hyperbolic equilibria

This paper constructs a three-dimensional autonomous chaotic system with simple algebraic structure of only five terms. Interestingly, this system is found to possess two non-hyperbolic equilibria, thus it is not under the class of Šil’nikov sense chaos for owning neither heteroclinic orbit nor homoclinic orbit. Dynamical properties have been revealed with the help of bifurcation route, Poincaré map, frequency spectrum, amplitude modulation and topological horseshoe. We hope that our work can motivate a stimulus for the further study on non-Šil’nikov chaotic system with non-hyperbolic equilibrium.

Amplitude-Phase Modulation, Topological Horseshoe and Scaling Attractor of a Dynamical System**Supported by Hunan Provincial Natural Science Foundation of China under Grant No. 2016JJ4036, University Natural Science Foundation of Jiangsu Province under Grant No. 14KJB120007 and the National Natural Science Foundation of China under Grant Nos. 11504176 and 11602084

A three-dimensional autonomous chaotic system is discussed in this paper. Some basic dynamical properties of the system, including phase portrait, Poincaré map, power spectrum, Kaplan–Yorke dimension, Lyapunov exponent spectra, signal amplitude and topological horseshoe are studied theoretically and numerically. The main finding by analysis is that the signal amplitude can be modulated via controlling the coefficients of the linear term, cross-product term and squared term simultaneously or respectively, and the phase of x3 can be modulated by the product of the coefficients of the linear term and cross-product term. Furthermore, scaling chaotic attractors of this system are achieved by modified projective synchronization with an optimization-based linear coupling method, which is safer for secure communications than the existed synchronization scheme since the scaling factors can be regarded as the security encoding key.

Topological horseshoe analysis for a three-dimensional anti-control system and its application

This paper designs a new three-dimensional dynamical system to be chaotic by the anti-control method. Then, by the theory of topological horseshoe, we prove that there exists a invariant compact set in the Poincare mapping section of the new anti-control system such that it is semi-conjugate to a 2-shift dynamics with computer-assisted, so the new anti-controlled system is chaotic. Furthermore, an image encryption scheme is designed based on the chaotic system. First, the image pixel positions are scrambled by the pseudo-random sequences generated from the chaotic system. Second, the pixel values are encrypted for multiple rounds by the chaotic stream cipher, so it achieves the double chaotic encryption. Finally, the statistical analyses are given to show the feasibility and security of the proposed encryption scheme.

On the analysis of local bifurcation and topological horseshoe of a new 4D hyper-chaotic system

In this paper, a new four-dimensional (4D) smooth quadratic autonomous system with complex hyper-chaotic dynamics is presented and analyzed. The Lyapunov exponent (LE) spectrum, bifurcation diagram and various phase portraits of the system are provided. The stability, Hopf bifurcation and pitchfork bifurcation of equilibrium point are discussed by using the center manifold theorem and bifurcation theory. Numerical simulation results are consistent with the theoretical analysis. Besides, by combining the topological horseshoe theory with a computer-assisted method of Poincaré maps and utilizing the algorithm for finding horseshoes in 3D hyper-chaotic maps, a horseshoe with two-directional expansions in the 4D hyper-chaotic system is successfully found, which rigorously proves the existence of hyper-chaos in theory.

A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos

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.

A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization

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.

Local bifurcation analysis and topological horseshoe of a 4D hyper-chaotic system

In this paper, a 4D autonomous system with complex hyper-chaotic dynamics is introduced. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. Basic dynamical properties are also analyzed. In order to clarify the evolution of the complex dynamic behaviors of the system, the local bifurcation is studied and a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincare–Andronov–Hopf bifurcation theorem. Numerical simulation results are consistent with the theoretical analysis. Besides, we present a rigorous study on the hyper-chaotic system by combining the topological horseshoe theory with a computer-assisted approach of Poincare maps. Utilizing the algorithm for finding horseshoes in 3D hyper-chaotic maps, a horseshoe with two-directional expansion in the 4D hyper-chaotic system has been found, which rigorously proves the existence of hyper-chaos in theory.

Horseshoe chaos and topological entropy estimation in a palaeoclimate dynamical model

In this paper, a three-dimensional palaeoclimate dynamical model by Saltzman is revisited. We find that chaotic behavior displays in this three-dimensional dynamical system in numerical studies. To verify the chaoticity, we resort to Poincaré section and Poincaré map technique to present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoe theory, and give an estimation of the topological entropy.

Strange attractors generated by a fractional order switching system and its topological horseshoe

Chaos generation in a new fractional order unstable dissipative system with only two equilibrium points is reported. Based on the integer version of an unstable dissipative system (UDS) and using the same system’s parameters, chaos behavior is observed with an order less than three, i.e., 2.85. The fractional order can be decreased as low as 2.4 varying the eigenvalues of the fractional UDS in accordance with a switching law that fulfills the asymptotic stability theorem for fractional systems. The largest Lyapunov exponent is computed from the numerical time series in order to prove the chaotic regime. Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order switching system avoiding the possible numerical bias of Lyapunov exponents. Finally, an electronic circuit is designed to synthesize this fractional order chaotic system.

Dynamic analysis of a chaotic system

A three-dimensional (3D) autonomous chaotic system is reported in this paper. Some basic properties are analyzed carefully by means of phase portraits, bifurcation diagrams, Lyapunov exponent spectrums and equilibrium points. The remarkable particularity of the system is that it possesses the dynamical properties as constant Lyapunov exponent spectrum and adjustable signal amplitude versus parameters b, d, and f. Further, based on the topological horseshoe theory, the horseshoe chaos in this system is investigated.

Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement

For a three-dimensional autonomous four-wing chaotic attractor, this paper rigorously verifies its chaotic properties by using topological horseshoe theory and numerical calculations. Firstly, an appropriate Poincare section of the chaotic attractor is selected by numerical analysis. Accordingly, a certain first return Poincare map is defined in the Poincare section. Thereafter, by utilizing numerical calculations and topological horseshoe theory, a one-dimensional tensile topological horseshoe in the Poincare section is discovered, which revealed that the four-wing attractor has a positive topological entropy, and verifies the existence of chaos in this four-wing attractor. Finally, by using a FPGA chip, the four-wing chaotic attractor was physically implemented, which is more suitable for engineering applications.

Some new insights into a known Chen‐like system

In the paper entitled ‘A novel chaotic system and its topological horseshoe’ in [Nonlinear Analysis: Modelling and Control 18 (1) (2013) 66–77], proposed the 3D chaotic system, , and discussed some of its dynamics according to theoretical and numerical analysis of its parameters . The present work is devoted to giving some new insights into the system for b≥0. Combining theoretical analysis and numerical simulations, some new results are formulated. On the one hand, after some known errors, mainly the distribution of its equilibrium point which is pointed out, correct results are formulated. On the other hand, some of its more rich dynamical properties hiding and not found previously, such as the stability, fold bifurcation, pitchfork bifurcation, degenerated pitchfork bifurcation, and Hopf bifurcation of its isolated equilibria, the dynamics of non‐isolated equilibria, the singularly degenerate heteroclinic cycle, the heteroclinic orbit, and the dynamics at infinity are clearly revealed. Using these results, one can easily explain those interesting phenomena for invariant Lyapunov exponent spectrum and amplitude control that are presented in the known literature. What is more important, we probably demonstrate a new route to chaos. Copyright © 2015 John Wiley & Sons, Ltd.

A New Chaotic System with Positive Topological Entropy

This paper introduces a new simple system with a butterfly chaotic attractor. This system has rich and complex dynamics. With some typical parameters, its Lyapunov dimension is greater than other known three dimensional chaotic systems. It exhibits chaotic behavior over a large range of parameters, and the divergence of flow of this system is not a constant. The dynamics of this new system are analyzed via Lyapunov exponent spectrum, bifurcation diagrams, phase portraits and the Poincaré map. The compound structures of this new system are also analyzed. By means of topological horseshoe theory and numerical computation, the Poincaré map defined for the system is proved to be semi-conjugate to 3-shift map, and thus the system has positive topological entropy.

A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium

A new hyper-chaotic system is presented in this paper by adding a smooth flux-controlled memristor and a cross-product item into a three-dimensional autonomous chaotic system. It is exciting that this new memristive system can show a four-wing hyper-chaotic attractor with a line equilibrium. The dynamical behaviors of the proposed system are analyzed by Lyapunov exponents, bifurcation diagram and Poincare maps. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of hyperchaos in the system is verified theoretically. Finally, an electronic circuit is designed to implement the hyper-chaotic memristive system.