Snap-back Repeller Research Articles

Chaotic Dynamics of a Two-Dimensional Spatiotemporal Discrete System with Delayed Perturbations

This paper is mainly concerned with the existence of chaos in a two-dimensional spatiotemporal discrete system with delayed perturbations. By applying the snap-back repeller theory, two criteria of chaos in the sense of both Li–Yorke and Devaney are established. Moreover, the lower bounds of the parameters that make the system chaotic are obtained. Finally, an example and its computer simulations of the dynamical behavior, spatiotemporal graphs, and the trends of the largest Lyapunov exponents with respect to the initial values and parameters are given to demonstrate the correctness of the theoretical results.

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.

The Existence of Li–Yorke Chaos in a Discrete-Time Glycolytic Oscillator Model

This paper investigates an autonomous discrete-time glycolytic oscillator model with a unique positive equilibrium point which exhibits chaos in the sense of Li–Yorke in a certain region of the parameters. We use Marotto’s theorem to prove the existence of chaos by finding a snap-back repeller. The illustration of the results is presented by using numerical simulations.

Snapback repellers and homoclinic orbits in a logistic map system

Snapback repellers and homoclinic orbits in a logistic map system

Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

Complex dynamics of the simplest neuron model: Singular chaotic Shilnikov attractor as specific oscillatory neuron activity

We study complex dynamics of the Chialvo model that is the simplest neuron-type model in form of a four-parameter family of two-dimensional noninvertible maps (endomorphisms). Main elements of bifurcation diagram in the plane of two parameters have been constructed in which regions corresponding to both quasi-periodic and chaotic oscillations are selected. We also indicate special regions corresponding to singular discrete chaotic Shilnikov attractors that we consider as a new type of the so-called snap-back repellers (over an unstable focus). The study of time series was carried out in which there were classified patterns of specific oscillatory activities in the cases when homoclinic orbits to the unstable focus exists and, when such orbits were not yet formed but a strange attractor already exists. New dynamical characteristics are proposed, with the help of which it is possible to assess the level of distinctness of atypical oscillatory activity.

Existence of chaos in controlled first-order partial difference equations with general delay controllers

This paper is concerned with chaos in a class of controlled first-order partial difference equations with general delay controllers. Three criteria of chaos are established by applying snap-back repeller theory. All the equations are proved to be chaotic in the sense of both Devaney and Li-Yorke when they satisfy some weak conditions. Three examples are provided with computer simulations.

Anti-control of chaos in coupled delay difference equations based on snap-back repellers

<p style='text-indent:20px;'>In this paper, we focus on anti-control of chaos for a class of coupled delay difference equations (CDDEs). By applying the snap-back repeller theory, the CDDE is proved to be chaotic in the sense of both Devaney and Li-Yorke when the parameters of the CDDE satisfy some very weak conditions, which makes the anti-control of chaos for CDDE become easy. At the end, two illustrative examples are provided with computer simulations.</p>

Nonisotropic chaos induced by snap-back repellers and heteroclinic cycles of 3-D hyperbolic PDEs

Nonisotropic chaos induced by snap-back repellers and heteroclinic cycles of 3-D hyperbolic PDEs

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.

Chaos criteria and chaotification schemes on a class of first-order partial difference equations.

This article is involved in chaos criteria and chaotification schemes on one kind of first-order partial difference equations having non-periodic boundary conditions. Firstly, four chaos criteria are achieved by constructing heteroclinic cycles connecting repellers or snap-back repellers. Secondly, three chaotification schemes are obtained by using these two kinds of repellers. For illustrating the usefulness of these theoretical results, four simulation examples are presented.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

Chaotic Dynamical Behavior of Coupled One-Dimensional Wave Equations

In this paper, we consider the chaotic oscillation of coupled one-dimensional wave equations. The symmetric nonlinearities of van der Pol type are proposed at the two boundary endpoints, which can cause the energy of the system to rise and fall within certain bounds. At the interconnected point of the wave equations, the energy is injected into the system through an anti-damping velocity feedback. We prove the existence of the snapback repeller when the parameters enter a certain regime, which causes the system to be chaotic. Numerical simulations are presented to illustrate our theoretical results.

Bifurcations in a one-parameter family of Lotka-Volterra 2D transformations

A particular system of two-dimensional Lotka-Volterra maps, Ta:(x′,y′)=(x(a−x−y),xy), unfolding a map originally proposed by Sharkovsky for a=4, is considered. We show the routes to chaos leading to the dynamics of map T4. For map T4 we show that even if the stable set of the origin O includes a set dense in an invariant area, the only homoclinic points of O belong to the x−axis, as well as the cycles leading to heteroclinic connections, while many internal cycles are snap-back repellers. We also show that a particular 6-cycle known analytically for map T4 exists, and is known explicitly in closed form, for any a∈(3,4] appearing at a supercritical Neimark-Sacker bifurcation of the positive fixed point. Moreover, we show the existence of infinitely many k−cycles on the x−axis (for any k>3), which are topological attractors of map Ta for a∈(3.96,4) and saddle cycles transversely attracting at a=4.

Bifurcation and chaotic behavior in the discrete BVP oscillator

In this work, a new class of discrete Bonhoeffer–van der Pol (BVP) system with an odd function is proposed and investigated. At first, the necessary and sufficient conditions on the existence and stability of the fixed points for this system are given. We then show the system passes through various bifurcations of codimension one, including pitchfork bifurcation, saddle–node bifurcation, flip bifurcation and Neimark–Sacker bifurcation under some certain parameter conditions. The center manifold theorem and bifurcation theory are the main tools in the analysis of the local bifurcations. Furthermore, we prove rigorously there exists Marotto’s chaos in this discrete BVP system, which means the fixed point eventually evolves into a snap-back repeller. Finally, numerical simulation evidences are provided not only to further demonstrate our theoretical analysis, but also to exhibit the complex dynamical phenomena, such as the period-9, -17, -18 orbits, attracting invariant cycles, quasi-periodic orbits, ten-coexisting chaotic attractors, etc. These phenomena illustrate relatively rich dynamical behaviors of the discrete BVP oscillator.

Symbolic Methods for Analysing Bifurcations and Chaos of Two Five-Parameter Families of Planar Quadratic Maps

In this work, we analyze the dynamical behaviors of two five-parameter families of planar quadratic maps by utilizing strategies of symbolic computation. We are going to use computer algebra methods to clarify how to detect the stability of equilibrium points to analyze chaos and also the bifurcation of planar maps. Based on strategies for solving the systems in types of semi-algebraic and by utilizing an algorithmic approach, we obtain respectively for the two maps, sufficient conditions on the parameters to have a prescribed number of (stable) equilibrium points; necessary conditions on the parameters to undergo a certain type of bifurcation or to have chaotic behavior induced by snapback repeller.

Chaos analysis for a class of hyperbolic equations with nonlinear boundary conditions

ABSTRACT A system governed by a one-dimensional hyperbolic equation with a mixing transport term and both ends being general nonlinear boundary conditions is considered in this paper. By using the snap-back repeller theory, we rigorously prove that the system is chaotic in the sense of both Devaney and Li-Yorke when the system parameters satisfy certain conditions. Finally, numerical simulations are further presented to illustrate the theoretical results.

Bifurcation Analysis of the γ-Ricker Population Model Using the Lambert W Function

In this work, we present the dynamical study and the bifurcation structures of the [Formula: see text]-Ricker population model. Resorting to the Lambert [Formula: see text] function, the analytical solutions of the positive fixed point equation for the [Formula: see text]-Ricker population model are explicitly presented and conditions for the existence and stability of these fixed points are established. The main focus of this work is the definition and characterization of the Allee effect bifurcation for the [Formula: see text]-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the [Formula: see text]-Ricker population model is associated with the asymptotic behavior of the Lambert [Formula: see text] function in a neighborhood of zero. The theoretical results describe the global and local bifurcations of the [Formula: see text]-Ricker population model, using the Lambert [Formula: see text] function in the presence and absence of the Allee effect. The Allee effect, snapback repeller and big bang bifurcations are investigated in the parameters space considered. Numerical studies are included.

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.