Distributional chaos in random dynamical systems

In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e. a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.     

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

In 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.      

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.     

Limitations of frequency domain approximation for detecting chaos in fractional order systems

In this paper, we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases. For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in this paper we show that this method is not always reliable for detecting chaos. This approximation can numerically demonstrate chaos in the non-chaotic fractional order systems, or eliminate chaotic behavior from a chaotic fractional order system.     

Omega-limit sets and invariant chaos in dimension one

Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31–43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent.     

Minimum entropy control of chaos via online particle swarm optimization method

One of the recently developed approaches for control of chaos is the minimum entropy (ME) control technique. In this method an entropy function based on the Shannon definition, is defined for a chaotic system. The control action is designed such that the entropy as a cost function is minimized which results in more regular pattern of motion for the system trajectories. In this paper an online optimization technique using particle swarm optimization (PSO) method is developed to calculate the control action based on ME strategy. The method is examined on some standard chaotic maps with error feedback and delayed feedback forms. Considering the fact that the optimization is online, simulation results show very good effectiveness of the presented technique in controlling chaos.     

On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals

In this paper, we have examined effects of forcing a periodic Colpitts oscillator with periodic and chaotic signals for different values of coupling factors. The forcing signal is generated in a master bias-tuned Colpitts oscillator having identical structure as that of the slave periodic oscillator. Numerically solving the system equations, it is observed that the slave oscillator goes to chaotic state through a period-doubling route for increasing strengths of the forcing periodic signal. For forcing with chaotic signal, the transition to chaos is observed but the route to chaos is not clearly detectable due to random variations of the forcing signal strength. The chaos produced in the slave Colpitts oscillator for a chaotic forcing is found to be in a phase-synchronized state with the forced chaos for some values of the coupling factor. We also perform a hardware experiment in the radio frequency range with prototype Colpitts oscillator circuits and the experimental observations are in agreement with the numerical simulation results.     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization

In this paper, we investigate the ultimate bound and positively invariant set for a new chaotic system via the generalized Lyapunov function theory. For this system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set. In addition, the two-dimensional bound with respect to x-z and y-z are established. Finally, the result is applied to the study of completely chaos synchronization, an exact threshold is given with the system parameters. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

非线性弹性梁中的混沌带现象

研究了非线性弹性梁的混沌运动,梁受到轴向载荷的作用。非线性弹性梁的本构方程可用三次多项式表示。计及材料非线性和几何非线性,建立了系统的非线性控制方程。利用非线性Galerkin法,得到微分动力系统。采用Melnikov方法对系统进行分析后发现,当载荷P₀和f满足一定条件时,系统将发生混沌运动,且混沌运动的区域呈现带状。还详尽分析了从次谐分岔到混沌的路径,确定了混沌发生的临界条件。     

Bifurcation analysis and linear control of the Newton–Leipnik system

In this paper, we study a sort of chaotic system—Newton–Leipnik system which possesses two strange attractors. The static and dynamic bifurcations of the system are studied. The chaos controlling is performed by a simpler linear controller, and numerical simulation of the control is supplied. At the same time, Lyapunov exponents of the system show that the result of the chaos controlling is right.     

Structurally stable but chaotic limit set of E-infinity Cantorian space–time

In the present work, first we give some definitions and theorems on hyperbolic maps, structurally stability and deterministic chaos. The limit set of the Kleinian transformation acting on the E-infinity Cantorian space–time turned out to be a set of periodic continued fractions as shown in [Chaos, Solitons & Fractals, 21 (2004) 9]. That set has a hyperbolic structure and is structurally stable. Subsequently, we show that the appearance of transversal homoclinic points induces a chaotic behavior in that set.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Chaos for Subshifts of Finite Type

This paper deals with chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperiodic matrix, there is a finitely chaotic set with full Hausdorff dimension. Moreover, for any subshift of finite type determined by a matrix, we point out that the cases including positive topological entropy, distributional chaos, chaos and Devaney chaos are mutually equivalent.     

Estimation of chaotic parameter regimes via generalized competitive mode approach

A procedure, we call it generalized competitive mode (GCM), is proposed to estimate the parameter regimes of chaos in nonlinear systems by implementing a mathematical version of mode competition. The idea is that for a system to be chaotic there must exist at least two GCMs in the system. The Lorenz system and a thin plate in flow-induced vibrations system are analyzed to find chaotic regimes by this procedure.     

Generalization on the chaos control of 4-D chaotic systems using recursive backstepping nonlinear controller

Using recursive backstepping nonlinear technique generalized controllers are designed for chaos control of two 4-D chaotic systems, namely: the Lorenz–Stenflo (LS) system that models low-frequency short-wavelength gravity waves and a new 4-D chaotic system containing three cross products that was recently introduced by Qi et al. Using numerical simulation the designed controllers are shown to be effective in stabilizing the two systems at the origin as well as controlling them to track desired trajectories.     

Co-Existence of Chaos and Stable Periodic Orbits in a Simple Discrete Neural Network

We show that a simple discrete network of two identical neurons can demonstrate chaotic behavior near the origin. This is complementary to the results in Wu and Zhang (Disc. Contin. Dynam. Syst. Series B, 4 (2004), 853-865), where it was shown that the same system can have a large capacity of stable periodic orbits in a region away from the origin.     

Golden mean relevance for chaos inhibition in a system of two coupled modified van der Pol oscillators

In this work, we present a novel evidence of the importance of the golden mean criticality of a system of oscillators in agreement with El Naschie’s E-infinity theory. We focus on chaos inhibition in a system of two coupled modified van der Pol oscillators. Depending on the coupling between the two oscillators, the system shows chaotic behavior for different ranges of the coupling parameter. Chaos suppression, as a transition from irregular behavior to a periodical one, is induced by perturbing the system with a harmonic signal with amplitude considerably lower than the value which causes entrainment. The frequency of the perturbation is related to the main frequencies in the spectrum of the freely running system (without perturbation) by the golden mean. We demonstrate that this effect is also obtained for a perturbation with frequency such that the ratio of half the frequency of the first main component in the freely running chaotic spectrum over the frequency of the perturbation is very close (five digits coincidence) to the golden mean. This result is shown to hold for arbitrary values of the coupling parameter in the various ranges of chaotic dynamics of the free running system.     

Robust chaos in piecewise nonsmooth map of the plane

In this paper we give sufficient conditions for the occurrence of robust chaotic attractors in piecewise nonsmooth map of the plane. The application of these results is illustrated by two 2D discontinuous maps. We have reported some analytical results on the existence of robust chaos in a general piecewise nonsmooth map of the plane via the search for super chaotic attractors. Some elementary examples are also given and discussed.     

Chaotic synchronization of a fractional-order system based on washout filter control

In this paper, chaos in a fractional-order neural network system with varying time delays is presented, and chaotic synchronization system with varying time delays is constructed. The stability of constructed synchronization system is analyzed by Laplace transformation theory. In addition, the bifurcation graph of the chaotic system is illustrated. The study results show that the chaos in such fractional-order neural networks with varying time delay can be synchronized, and Washout filter control can be used to reduce the range of coupled parameter.