Topological Horseshoe Research Articles

Homoclinic/heteroclinic recurrent orbits and horseshoe

In this paper, we consider the existence of homoclinic/heteroclinic recurrent (periodic, almost periodic, almost automorphic, Birkhoff recurrent, Poisson stable) orbits of the system with the following form z˙=g(z)+μh(t,z,μ). We prove that, under certain conditions on h, if μ is sufficiently small, then the equation admits homoclinic/heteroclinic orbits which have the same character of recurrence as the function h. Our approach is based on the Melnikov method. In particular, we construct topological horseshoes.

New Topological Horseshoe in the Chen System

In this paper, we explore the rich dynamics in the Chen system using Runge–Kutta method and Deep Neural Networks (DNNs). Compared with the Lorenz system, we find that the first return map [Formula: see text] of the Chen system exhibits a more complex structure of continuous regions in the Poincaré section. Moreover, the existence of six crossing blocks with respect to the second return map implies that there is a closed invariant set [Formula: see text] in the Poincaré section such that [Formula: see text] is semi-conjugate to a 6-shift map, thus demonstrating remarkably chaos in the Chen system.

Constructing a New Multi-Scroll Chaotic System and Its Circuit Design

Multi-scroll chaotic systems have complex dynamic behaviors, and the multi-scroll chaotic system design and analysis of their dynamic characteristics is an open research issue. This study explores a new multi-scroll chaotic system derived from an asymptotically stable linear system and designed with a uniformly bounded controller. The main contributions of this paper are given as follows: (1) The controlled system can cause chaotic behavior with an appropriate control position and parameters values, and a new multi-scroll chaotic system is proposed using a bounded sine function controller. Meanwhile, the dynamical characteristics of the controlled system are analyzed through the stability of the equilibrium point, a bifurcation diagram, and Lyapunov exponent spectrum. (2) According to the Poincaré section, the existence of a topological horseshoe is proven using the rigorous computer-aided proof in the controlled system. (3) Numerical results of the multi-scroll chaotic system are shown using Matlab R2020b, and the circuit design is also given to verify the multi-scroll chaotic attractors.

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.

Invariant tori, topological horseshoes, and their coexistence in piecewise smooth hybrid systems

Invariant tori, topological horseshoes, and their coexistence in piecewise smooth hybrid systems

Learning Topological Horseshoe via Deep Neural Networks

Deep Neural Networks (DNNs) have been successfully applied to investigations of numerical dynamics of finite-dimensional nonlinear systems such as ODEs instead of finding numerical solutions to ODEs via the traditional Runge–Kutta method and its variants. To show the advantages of DNNs, in this paper, we demonstrate that the DNNs are more efficient in finding topological horseshoes in chaotic dynamical systems.

Uniform hyperbolicity of a class of scattering maps

In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a numerically sound method for computing quantum resonance states, did not exist. To address this challenge, we recently introduced a new class of quantum maps (Mertig and Shudo 2018 Phys. Rev. E 97 042216). For these quantum maps, we showed that, quantum resonance states can numerically be computed using theoretically grounded methods such as complex scaling or weak absorbing potentials (Mertig and Shudo 2018 Phys. Rev. E 97 042216). However, proving uniform hyperbolicty for this class of quantum maps was not straight forward. Going beyond that work this article generalises the class of scattering maps and provides mathematical proofs for their uniform hyperbolicity. In particular, we show that the suggested class of two-dimensional symplectic scattering maps satisfies the topological horseshoe condition and uniform hyperbolicity. In order to prove these properties, we follow the procedure developed in the paper by Devaney and Nitecki (1979 Commun. Math. Phys. 67 137–46). Specifically, uniform hyperbolicity is shown by identifying a proper region in which the non-wandering set satisfies a sufficient condition to have the so-called sector bundle or cone field. Since no quantum map is known where both a proof of uniform hyperbolicity and a methodologically sound method for numerically computing quantum resonance states exist simultaneously, the present result should be valuable to further test fractal Weyl laws and related topics such as chaotic eigenfunction hypothesis.

Computer‐assisted analysis of chaos in a three‐species food chain model

AbstractIn this paper, a three‐species food chain model with Holling type IV and Beddington–DeAngelis functional responses is formulated. Numerical simulations show that this system can generate chaos for some parameter values. But the mechanism behind chaos is still unclear only through numerical simulations. Then, using the topological horseshoe theories and Conley–Moser conditions, we present a computer‐assisted analysis to show the chaoticity of this system in the topological sense, that is, it has positive topological entropy. We prove that the Poincaré map of this model possesses a closed uniformly hyperbolic chaotic invariant set, and it is topologically conjugate to a 2‐shift map. At last, we consider the impact of fear on this three‐species model. It is an important factor in controlling chaos in biological models, which has been validated in other models.

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.

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.

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.

Topological horseshoes for surface homeomorphisms

In this work, we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy, and we obtain several applications. For homeomorphisms of the open annulus A with zero topological entropy, we show that rotation numbers exist for all points with nonempty ω-limit sets, and that if A is a generalized region of instability, then it admits a single rotation number. We also offer a new proof of a recent result of Passeggi, Potrie, and Sambarino, showing that zero entropy dissipative homeomorphisms of the annulus having a circloid as an attractor have a single rotation number. Our work also studies homeomorphisms of the sphere without horseshoes. For these maps, we present a structure theorem in terms of fixed point free invariant subannuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most two distinct periodic orbits and that, in many cases, the restriction of the homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti et al.

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.

Tri-valued memristor-based hyper-chaotic system with hidden and coexistent attractors

Recently, the nonlinear dynamics of memristor has attracted much attention. In this paper, a novel four-dimensional hyper-chaotic system (4D-HCS) based on a tri-valued memristor is found. Theoretical analysis shows that the 4D-HCS has complex hyper-chaotic dynamics such as hidden attractors and coexistent attractors. We also experimentally analyze the dynamics behaviors of the 4D-HCS in aspects of the phase diagram, bifurcation diagram, Lyapunov exponential spectrum, power spectrum and the correlation coefficient. To rigorously verify the chaotic behavior, we analyze the topological horseshoe of the system and calculate the topological entropy. In addition, the comparison with binary-valued memristor-based chaotic system shows that a chaotic system with a tri-valued memristor can generate hyper-chaos and coexistent attractors, while the one with a binary-valued memristor cannot. This finding suggests that applying three- or multi-value memristors in chaotic circuits can produce more complex dynamic properties than binary-valued memristors. To show the easy implementation of the 4D-HCS, we implement the 4D-HCS in an analogue circuit-based hardware platform, and the implementation results are consistent with the theoretical analysis. Finally, using the 4D-HCS, we design a pseudorandom number generator to explore its potential application in cryptography.

On integrability of the Nosé–Hoover oscillator and generalized Nosé–Hoover oscillator

The Nosé–Hoover system is a basic primitive model for the molecular dynamics simulations, which describes the equilibrium characterized by canonical distributions at a constant temperature. Its simplest form, called the Nosé–Hoover oscillator (NH), is a three-dimensional quadratic polynomial system that admits both regular invariant tori and chaotic trajectories. Recently, a similar system, called the generalized Nosé–Hoover oscillator (GNH), was constructed to show the coexistence of invariant tori and topological horseshoe for time-reversible systems in [Formula: see text]. This paper aims to study the integrability of both NH and GNH models. We show that (i) in the case of [Formula: see text], both NH and GNH models are integrable by quadratures and the general solutions are given; (ii) in the case of [Formula: see text] the non-existence of either global analytic first integrals or Darboux first integrals of two models are discussed and a complete characterization of Darboux polynomials and exponential factors are given; (iii) both NH and GNH models are not rationally integrable in an extended Liouville sense except for several parameter values by an extended Morales–Ramis theory; (iv) the reduced systems of NH and GNH models at the Poincaré compactification balls are integrable, which yields complete descriptions of dynamics at infinity for NH and GNH models. Interestingly, the topological structure of NH and GNH models at infinity strongly depends on the sign of [Formula: see text]. These results may help us better understand the complex and rich dynamics of nonlinear time-reversible systems.

Quasi-graphs, zero entropy and measures with discrete spectrum

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.

Chaos explosion and topological horseshoe in three-dimensional impacting hybrid systems with a single impact surface

For a class of three-dimensional impacting hybrid systems comprising a linear system of ordinary differential equations and a reset map, and having a single impact surface, this paper studies the phenomena of impacting homoclinic bifurcation leading to periodic orbits, horseshoes and chaos explosions. More precisely, it is proved that the homoclinic bifurcation can result in the impacting periodic sinks or impacting periodic saddle orbits when the impact surface is a plane and the reset map satisfies some basic conditions, but it is not easy to find horseshoes in this case. Furthermore, when the single impact surface is not a plane and the reset map has a certain rotational property, it is proved that a topological horseshoe will appear suddenly when bifurcation parameter passes through some threshold, thus, a kind of chaos explosion takes place. These main results are illustrated by several examples, in which it can be seen that the newborn chaotic invariant sets might be chaotic attractors from the perspective of numerical simulation.

Chaos Generation With Impulse Control: Application to Non-Chaotic Systems and Circuit Design

Chaos has been successfully applied in many fields to improve the performance of engineering systems, such as communication, vibration compact, and mixing. Generating chaos from originally non-chaotic systems is a relevant topic because of potential applications. In this work, the impulse control is shown to generate chaos from non-chaotic system. Using non-chaotic Chen system as an example, we prove by analytical and numerical methods that chaos is indeed generated. The features of the chaos generated by impulse control are analysed using Lyapunov exponents, bifurcation diagram, power spectrum, Poincare mapping and Kaplan-Yorke dimension. Furthermore, we demonstrate the chaotic attractor generation by impulse control using a circuit experiment. The last but not minor point is that the existence of topological horseshoe is given by rigorous computer-aided proof.

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'.

Symbolic dynamics in a binary asteroid system

We highlight the existence of a topological horseshoe arising from an a–priori stable model of the binary asteroid dynamics. The inspection is numerical and uses correctly aligned windows, as described in a recent paper by A. Gierzkiewicz and P. Zgliczyński, combined with a recent analysis of an associated secular problem.