Sense Of Li-Yorke Research Articles

Snapback Repellers, Computational Chaos and Extreme Multistability in Discrete-Time Memristor Murali–Lakshmanan–Chua Circuit

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves.

Snap-back repellers and chaos in a class of discrete-time memristor circuits

In the last decade the flux-charge analysis method (FCAM) has been successfully used to show that continuous-time (CT) memristor circuits possess for structural reasons first integrals (invariants of motion) and their state space can be foliated in invariant manifolds. Consequently, they display an initial condition dependent dynamics, extreme multistability (coexistence of infinitely many attractors) and bifurcations without parameters. Recently, a new discretization scheme has been introduced for CT memristor circuits, guaranteeing that the first integrals are preserved exactly in the discretization. On this basis, FCAM has been extended to discrete-time (DT) memristor circuits showing that they also are characterized by invariant manifolds and they display extreme multistability and bifurcations without parameters. This manuscript considers the maps obtained via DT-FCAM for a circuit with a flux-controlled memristor and a capacitor and it provides a thorough and rigorous investigation of the presence of chaotic dynamics. In particular, parameter ranges are obtained where the maps have snap-back repellers at some fixed points, thus implying that they display chaos in the Marotto and also in the Li–Yorke sense. Bifurcation diagrams are provided where it is possible to analytically identify relevant points in correspondence with the appearance of snap-back repellers and the onset of chaos. The dependence of the bifurcation diagrams and snap-back repellers upon the circuit initial conditions and the related manifold is also studied.

Nonchaotic N-expansive homeomorphisms

In this paper, we give necessary conditions for an $N$ -expansive homeomorphism of a compact metric space to be nonchaotic in the Li–Yorke sense. As application we give a partial answer to a conjecture in [2].

Li-Yorke chaos in a class of controlled delay difference equations

A class of controlled delay difference equations is considered. Proof of the existence of chaos in the sense of Li-Yorke in the controlled system is given by applying the snap-back repeller theorem. Conditions for the existence of chaos are weak and lots of functions can be taken as controllers, which facilitates the chaotification of delay difference equations. One illustrative example is provided with computer simulations.

On the construction of one-dimensional discrete chaos theory based on the improved version of Marotto’s theorem

The purpose of this paper is to design one-dimensional chaotic maps with a non-zero intercept in the Y-axis and research the basic dynamic characteristics. First, by studying the common conditions that make one-dimensional discrete dynamical systems chaotic under the modulo operation, the general theory of the chaotic behavior of one-dimensional nonlinear functions with a non-zero origin is presented. It is proved that the discrete dynamic system derived from the proposed theory exhibits chaos phenomena in the sense of Li-Yorke, utilizing the improved version of Marotto’s theorem. Furthermore, the part of the discrete dynamical functions satisfying the proposed chaos theory is given. The function track, bifurcation diagram and Lyapunov exponential spectrum are also analyzed. Numerical simulations agree with the analytical framework for the suggested chaos theory. Finally, given the application of chaos in cryptographic communication, two significant corollaries of the original functions as general polynomials are put forward according to the theory of chaos structure.

Chaotic oscillations of one-dimensional coupled wave equations with mixed energy transports

In this paper, we consider the chaotic behavior of one-dimensional coupled wave equations with mixed partial derivative linear energy transport terms. The van der Pol-type symmetric nonlinearities are proposed at two boundary endpoints, which cause the energy of the coupled system to rise and fall within certain ranges. At the interconnected point of the two wave equations, the energy is injected into the system through a middle-point velocity feedback. We prove that when the parameters satisfy some conditions, the coupled wave equations have snapback repellers which can make the whole system chaos in the sense of Li-Yorke. Numerical simulations are presented to verify the theoretical results.

On Periodic Shadowing Property

In this paper, some properties of the periodic shadowing are presented. It is shown that a homeomorphism has the periodic shadowing property if and only if so does every lift of it to the universal covering space. Also, it is proved that continuous mappings on a compact metric space with the periodic shadowing and the average shadowing property also have the shadowing property and then are chaotic in the sense of Li-Yorke. Moreover, any distal homeomorphisms on a compact metric space with the periodic shadowing property do not have the asymptotic average shadowing property.

Estudio de una familia de funciones de periodo tres y su dinámica caótica

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><strong>Objetivo - </strong><span>construir sistemas dinámicos caóticos unidimensionales mediante el estudio de una familia de funciones con dominio y contradominio en el intervalo [0,1] la cual se define en términos de cuatro parámetros.</span></p><p><strong>Método - </strong><span>con base a los parámetros que definen a cada función que proponemos, se identificaron aquellas que tienen periodo tres, las cuales inducen un sistema caótico en el contexto de Li-Yorke. Los teoremas del punto fijo y de Sharkovskii fueron la herramienta fundamental de nuestro trabajo. </span></p><p><strong>Resultados - </strong><span>se obtuvo un conjunto de sistemas dinámicos caóticos, se describió un procedimiento sencillo para obtener sistemas dinámicos caóticos (adicionales a los obtenidos) y se sugiere como primera aplicación la obtención de números pseudoaleatorios.</span></p><p><strong>Limitaciones - </strong><span>los sistemas dinámicos construidos son caóticos en el sentido de Li-Yorke, -no necesariamente en el sentido de Devaney-. </span></p><p><strong>Principales hallazgos - </strong><span>las funciones estudiadas tienen una gráfica </span><span>en <em>forma de Zeta</em></span><span>, y para cada una de ellas se identifica a su respectiva </span><em>dual </em><span>(las gráficas que se obtienen presentan una relación de simetría), de esta manera se muestran las condiciones que deben verificar los parámetros -primal y dual- para obtener (y no obtener) período tres. </span></p></div></div></div>

Controlled shadowing property

In this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing and specication properties. The relation between the controlled shadowing and chaos in sense of Li-Yorke is studied. At the end we give some examples to investigate the controlled shadowing property.

Linear dynamics of semigroups generated by differential operators

Abstract During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C0-semigroups. We will also study in detail the dynamics of the translation C0-semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.

Devaney’s and Li-Yorke’s Chaos in Uniform Spaces

A definition of chaos in the sense of Li-Yorke is given for an action of a group on a uniform space, and it is shown that if a continuous action of an Abelian group G on a second countable Baire Hausdorff uniform space X without isolated points is chaotic in the sense of Devaney, then it is also chaotic in the sense of Li-Yorke.

Persistence of Li–Yorke chaos in systems with relay

It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is persistent even if a chaotic perturbation is applied. We consider chaos in a modified Li-Yorke sense such that infinitely many almost periodic motions take place in its basis. It is demonstrated that the system under investigation possesses countable infinity of chaotic sets of solutions. Coupled Duffing oscillators are used to show the effectiveness of our technique, and simulations that support the theoretical results are represented. Moreover, a chaos control procedure based on the Ott-Grebogi-Yorke algorithm is proposed to stabilize the unstable almost periodic motions embedded in the chaotic attractor.

Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

Conditions for maps to be topologically conjugate or semi-conjugate to subshifts of finite type and criteria of chaos

ABSTRACTIn this paper, some necessary and sufficient conditions are given for maps in Hausdorff spaces to be topologically conjugate or semi-conjugate to subshifts of finite type. These results not only extend some related existing results in metric spaces to Hausdorff spaces but also relax their conditions. In addition, it is shown that the strict A-coupled-expansion for a map in a Hausdorff space is a sufficient condition for it to be semi-conjugate to the subshift σA in Σ+m(A). Based on these results, two new criteria of chaos are established, in which the maps are shown to be chaotic either in the sense of both Devaney and Li-Yorke or in the sense of Li-Yorke.

Li-Yorke chaos generation by SICNNs with chaotic/almost periodic postsynaptic currents

We consider shunting inhibitory cellular neural networks with inputs and outputs that are chaotic in a modified Li-Yorke sense. The original Li-Yorke definition of chaos has been modified such that infinitely many periodic motions separated from the motions of the scrambled set are now replaced with almost periodic ones. Another principal novelty of the paper is that chaos is obtained as solutions of differential equations (neural networks) which are perturbed chaotically. To construct the original chaos, the special set of piecewise continuous postsynaptic currents is applied. It is shown that a control can be realized for the extended chaos in an effective way. This is the first time in the theory of neural networks that the Ott–Grebogi–Yorke control method is used to stabilize almost periodic motions. Our techniques can be performed to investigate chaotic dynamics in human brain activities, communication security, combinatorial optimization problems and control of legged robots. The results of this paper were announced in the 5th International Conference on Nonlinear Science and Complexity (August 4–9, 2014, Xi׳an, China) and in an International Conference on Nonlinear Dynamics and Complexity (May 11–15, 2015, La Manga, Spain).

POPULATION GROWTH MODELING WITH BOOM AND BUST PATTERNS: THE IMPULSIVE DIFFERENTIAL EQUATION FORMALISM

This note gives an overview on basic mathematical models describing the population dynamics of a single species whose vital dynamics has different time scales. We present five cases combining two time–scales with Malthusian growth in at least one scale. The dynamical behavior shows a progressive complexity, from "naive" to chaotic dynamics (in the Li–Yorke's sense). In addition, some open problems and new results are presented.

Chaos for Discrete Dynamical System

We prove that a dynamical system is chaotic in the sense of Martelli and Wiggins, when it is a transitive distributively chaotic in a sequence. Then, we give a sufficient condition for the dynamical system to be chaotic in the strong sense of Li-Yorke. We also prove that a dynamical system is distributively chaotic in a sequence, when it is chaotic in the strong sense of Li-Yorke.

Some properties of coupled-expanding maps in compact sets

In this paper, some properties of a strictly A A -coupled-expanding map in compact subsets of a metric space are studied, where A A is a transition matrix. It is shown that this map has a compact invariant set on which it is topologically semi-conjugate to the subshift for A A . If the subshift for A A has positive topological entropy, then the map is chaotic in the sense of Li-Yorke. Moreover, in the one-dimensional case, the map is at most two-to-one conjugate to the subshift for A A and chaotic in the sense of Devaney.

Anti-control and circuit implementation of discrete-time systems under limited regional conditions

In the Chen-Lai and Wang-Chen algorithm , the modular functions are both defined in (-∞,+∞). A modular function, however, in the implementation of electronic circuit, is more reasonable in line with the actual situation if it is defined in a finite region. We take for example the anti-control of a discrete time system, of which the modular function is sine function on the basis of a finite region. And in the sense of Li-Yorke, the chaotic sufficient condition and the rigorous theory proof are provided. As a result, ranges of specific circuit parameters can be determined by both the sufficient conditions resulting from the theorem, and a finite region defined by the device, or the constraint conditions of a dynamic range. Therefore, this provides a fundamental basis for the circuit design and its technology. Based on this method, the anti-control circuit of the discrete time system is designed, of which the modular function is sine function in a finite region. And the experimental results are given for confirming the feasibility of the method. The method presented in this paper can also be applied to the circuit implementation and the anti-control of a discrete time system, of which the modular function is other nonlinear function.

Coupled-expanding maps under small perturbations

This paper studies the $C^1$-perturbation problem of strictly$A$-coupled-expanding maps in finite-dimensional Euclidean spaces, where $A$ is an irreducible transition matrix with one row-sum no less than $2$.It is proved that under certain conditions strictly $A$-coupled-expanding maps are chaotic in the sense of Li-Yorke or Devaney under small $C^1$-perturbations. It is shown that strictly $A$-coupled-expanding maps are $C^1$ structurally stable in their chaotic invariant sets under certain stronger conditions. One illustrative example is provided with computer simulations.