Topological Horseshoe Theory Research Articles

Dynamic Analysis and FPGA Implementation of a New Linear Memristor-Based Hyperchaotic System with Strong Complexity

Chaotic or hyperchaotic systems have a significant role in engineering applications such as cryptography and secure communication, serving as primary signal generators. To ensure stronger complexity, memristors with sufficient nonlinearity are commonly incorporated into the system, suffering a limitation on the physical implementation. In this paper, we propose a new four-dimensional (4D) hyperchaotic system based on the linear memristor which is the most straightforward to implement physically. Through numerical studies, we initially demonstrate that the proposed system exhibits robust hyperchaotic behaviors under typical parameter conditions. Subsequently, we theoretically prove the existence of solid hyperchaos by combining the topological horseshoe theory with computer-assisted research. Finally, we present the realization of the proposed hyperchaotic system using an FPGA platform. This proposed system possesses two key properties. Firstly, this work suggests that the simplest memristor can also induce strong nonlinear behaviors, offering a new perspective for constructing memristive systems. Secondly, compared to existing systems, our system not only has the largest Kaplan-Yorke dimension, but also has clear advantages in areas related to engineering applications, such as the parameter range and signal bandwidth, indicating promising potential in engineering applications.

Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments

Abstract This article deals with the existence, multiplicity, minimal complexity, and global structure of the subharmonic solutions to a class of planar Hamiltonian systems with periodic coefficients, being the classical predator-prey model of V. Volterra its most paradigmatic example. By means of a topological approach based on techniques from global bifurcation theory, the first part of the paper ascertains their nature, multiplicity and minimal complexity, as well as their global minimal structure, in terms of the configuration of the function coefficients in the setting of the model. The second part of the paper introduces a dynamical system approach based on the theory of topological horseshoes that permits to detect, besides subharmonic solutions, “chaotic-type” solutions. As a byproduct of our analysis, the simplest predator-prey prototype models in periodic environments can provoke chaotic dynamics. This cannot occur in cooperative and quasi-cooperative dynamics, as a consequence of the ordering imposed by the maximum principle.

Chaos of the Six-Dimensional Non-Autonomous System for the Circular Mesh Antenna

In the process of aerospace service, circular mesh antennas generate large nonlinear vibrations under an alternating thermal load. In this paper, the Smale horseshoe and Shilnikov-type multi-pulse chaotic motions of the six-dimensional non-autonomous system for circular mesh antennas are first investigated. The Poincare map is generalized and applied to the six-dimensional non-autonomous system to analyze the existence of Smale horseshoe chaos. Based on the topological horseshoe theory, the three-dimensional solid torus structure is mapped into a logarithmic spiral structure, and the original structure appears to expand in two directions and contract in one direction. There exists chaos in the sense of a Smale horseshoe. The nonlinear equations of the circular mesh antenna under the conditions of the unperturbed and perturbed situations are analyzed, respectively. For the perturbation analysis of the six-dimensional non-autonomous system, the energy difference function is calculated. The transverse zero point of the energy difference function satisfies the non-degenerate conditions, which indicates that the system exists Shilnikov-type multi-pulse chaotic motions. In summary, the researches have verified the existence of chaotic motion in the six-dimensional non-autonomous system for the circular mesh antenna.

A topological approach to the problem of chaotic tides

We study the existence of subharmonic solutions and solutions with complicated dynamics in a Duffing-type equation which arises from the study of forced tides. This is a model which involves an almost enclosed basin, connected to the sea by a narrow strait. Such a basin turns out to be a nonlinear Helmholtz resonator. The proofs are based on developments of the theory of topological horseshoes in connection with some geometric features associated to linked twist maps.

Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight.

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'.

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

In this paper we prove the existence of multiple periodic (harmonic and subharmonic) solutions for a class of planar Hamiltonian systems which includes the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincare-Birkhoff fixed point theorem as well as some refinements on the side of the theory of topological horseshoes. A Duffing-type equation and an exponential nonlinearity case are studied as applications.

Analysis, Stabilization, and DSP-Based Implementation of a Chaotic System with Nonhyperbolic Equilibrium

This paper reports an autonomous dynamical system, and it finds that one nonhyperbolic zero equilibrium and two hyperbolic nonzero equilibria coexist in this system. Thus, it is difficult to demonstrate the existence of chaos by Šil’nikov theorem. Consequently, the topological horseshoe theory is adopted to rigorously prove the chaotic behaviors of the system in the phase space of Poincaré map. Then, a single control scheme is designed to stabilize the dynamical system to its zero-equilibrium point. Besides, to verify the theoretical analyses physically, the attractor and stabilization scheme are further realized via DSP-based technique.

Topological horseshoe in a single-scroll Chen system with time delay

Abstract The proof that an attractor is chaotic is not trivial. A single-scroll attractor in the Chen system with time delay is investigated through both theory and simulation. The Chen system with time delay is infinite dimensional parameterized by the time delay τ. The detailed procedure operations for finding topological horseshoe in the delay differential equation are different from that one in ordinary differential equation. We show the existence of chaos by using both the topological horseshoe theory and its corollary, and the Smale horseshoe construction if the geometry of the attractor on the 2D plane section satisfies certain conditions. This paper presents both methods for establishing the presence of the horseshoe in a Chen system with time delay. In the first method, we select two quadrilaterals in the 2D transversal section, and calculate the relationship of the quadrilaterals under the map. In the second method, we select quadrilaterals in the neighborhood of a short unstable periodic orbit in the section and obtain the approximate location of the quadrilaterals under the map, yielding the Smale horseshoe. By using the above two methods, the geometrical expansion of the quadrilaterals under the map satisfies the Topological Horseshoe Corollary and also the Smale horseshoe construction, thus showing that the time-delayed single-scroll attractor is indeed chaotic.

Chaos in periodically forced reversible vector fields

We discuss the appearance of chaos in time-periodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension~$1$ reversible vector fields and discuss the ways a time-dependent periodic forcing term of pulse form may be added to them to yield topological chaotic behaviour. Chaos here means that the resulting dynamics is semi-conjugate to a shift in a finite alphabet. The results rely on the classification of reversible vector fields and on the theory of topological horseshoes. This work is part of a project of studying periodic forcing of symmetric vector fields.

Global exponential stabilization for chaotic brushless DC motor with simpler controllers

In this paper, computer assisted proofs in dynamics (CAPD) group and conventional dynamic analysis methods are applied to find the parameters that make the brushless DC motor (BLDCM) system chaotic. For the purpose to make the BLDCM system exponentially stable, four simple controllers with just one state variable are proposed, which means that it only needs one sensor instead of two or more in practice. Numerical simulations show that the motor with the new controllers can converge efficiently. Finally, the topological horseshoe theory is introduced to verify the existence of chaos in BLDCM system.

Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

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.

Dynamical analysis and FPGA implementation of a large range chaotic system with coexisting attractors

Abstract Based on the generalized 3-D Lu chaotic system, this paper presents a new 3-D autonomous continuous chaotic system with a large range parameter b ∈ [ 7500 ] . The system has coexisting chaotic attractors depending on the distances from unstable equilibrium points to a given initial point. The topological horseshoes and entropy are obtained by the aid of the topological horseshoes theory and numerical computations. The results show that the new system is chaotic. Moreover, the system’s chaotic characteristics are verified by Multisim simulation and FPGA implementation.

Ultimate boundary estimations and topological horseshoe analysis of a new 4D hyper-chaotic system

In this paper, we first estimate the boundedness of a new proposed 4-dimensional (4D) hyper-chaotic system with complex dynamical behaviors. For this system, the ultimate bound set Ω1 and globally exponentially attractive set Ω2 are derived based on the optimization method, Lyapunov stability theory and comparison principle. Numerical simulations are presented to show the effectiveness of the method and the boundary regions. Then, to prove the existence of hyper-chaos, the hyper-chaotic dynamics of the 4D nonlinear system is investigated by means of topological horseshoe theory and numerical computation. Based on the algorithm for finding horseshoes in three-dimensional hyper-chaotic maps, we finally find a horseshoe with two-directional expansions in the 4D hyper-chaotic system, which can rigorously prove the existence of the hyper-chaos in theory.

The Coexistence of Invariant Tori and Topological Horseshoe in a Generalized Nosé–Hoover Oscillator

This paper is devoted to the study of dynamical complexity of a generalized Nosé–Hoover oscillator which is a three-dimensional quadratic polynomial system. Precisely, a lot of moderately conservative regions are found, each of which is filled with different sequences of nested tori with various knot types and is embedded in the “chaotic region”. This shows that the generalized Nosé–Hoover oscillator may possess so-called “fat fractal” structure in phase space. In addition, horseshoe chaos can be demonstrated by applying the topological horseshoe theory to a Poincaré map defined in a proper cross-section, which further shows the coexistence of infinitely stable periodic trajectories and infinite saddle periodic trajectories.

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.

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.

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.

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.