Li-Yorke and distributionally chaotic operators

We study Li-Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li-Yorke chaos in terms of the existence of irregular vectors. Sufficient “computable” criteria for distributional and Li-Yorke chaos are given, together with the existence of dense scrambled sets under some additional conditions. We also obtain certain spectral properties. Finally, we show that every infinite dimensional separable Banach space admits a distributionally chaotic operator which is also hypercyclic.     

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.      

A Devaney Chaotic System Which Is Neither Distributively nor Topologically Chaotic

Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.     

Distributional chaos for backward shifts

We provide sufficient conditions which give uniform distributional chaos for backward shift operators. We also compare distributional chaos with other well-studied notions of chaos for linear operators, like Devaney chaos and hypercyclicity, and show that Devaney chaos implies uniform distributional chaos for weighted backward shifts, but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic.     

Infinite-Dimensional Linear Dynamical Systems with Chaoticity

Summary. The authors present two results on infinite-dimensional linear dynamical systems with chaoticity. One is about the chaoticity of the backward shift map in the space of infinite sequences on a general Fréchet space. The other is about the chaoticity of a translation map in the space of real continuous functions. The chaos is shown in the senses of both Li-Yorke and Wiggins. Treating dimensions as freedoms, the two results imply that in the case of an infinite number of freedoms, a system may exhibit complexity even when the action is linear. Finally, the authors discuss physical applications of infinite-dimensional linear chaotic dynamical systems. Received January 27, 1997; second revision received August 8, 1997; final revision received January 12, 1998     

On λ-Power Distributional n-Chaos

For each real number λ∈ [0, 1], λ-power distributional chaos has been introduced and studied via Furstenberg families recently. The chaoticity gets stronger and stronger as λ varies from 1 to 0, where 1-power distributional chaos is exactly the usual distributional chaos. As a generalization of distributional n-chaos, λ-power distributional n-chaos is defined similarly. Lots of classic results on distributional chaos can be improved to be the versions of λ-power distributional n-chaos accordingly. A practical method for distinguishing 0-power distributional n-chaos is given. A transitive system is constructed to be 0-power distributionally n-chaotic but without any distributionally(n + 1)-scrambled tuples. For each λ∈ [0, 1], λ-power distributional n-chaos can still appear in minimal systems with zero topological entropy.     

完备度量空间中的混沌判定

本文研究完备度量空间上的离散动力系统的混沌标准,证明了如果完备度量空间X上的连续映射f具有正则非退化返回排斥子或连接不动点的正则非退化异宿环,则存在f的不变闭子集A,使得f限制在此不变闭子集上的子系统与两个符号的符号动力系统拓扑共轭,从而获得具有这类结构的连续映射f具有Devaney混沌、分布混沌、正拓扑熵及ω-混沌,...     

On synchronization of a forced delay dynamical system via the Galerkin approximation

A forced scalar delay dynamical system is analyzed from the perspective of bifurcation and synchronization. In general first order differential equations do not exhibit chaos, but introduction of a delay feedback makes the system infinite dimensional and shows chaoticity. In order to study the dynamics of such a system, Galerkin projection technique is used to obtain a finite dimensional set of ordinary differential equations from the delay differential equation. We compare the results of simulation with those obtained from direct numerical simulation of the delay equation to ascertain the accuracy of the truncation process in the Galerkin approximation. We have considered two cases, one with five and the other with eight shape functions. Next we study two types of synchronization by considering coupling of the above derived equations with a forced dynamical system without delay. Our analysis shows that it is possible to have synchronization between two such systems. It has been shown that the chaotic system with delay feedback can drive the system without delay to achieve synchronization and the opposite case is also equally valid. This is confirmed by the evaluation of the conditional Lyapunov exponents of the systems.     

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.     

Chaos in a topologically transitive system

The chaotic phenomena have been studied in a topologically transitive system and it has been shown that the erratic time dependence of orbits in such a topologically transitive system is more complicated than what described by the well-known technology "Li-Yorke chaos". The concept "sensitive dependency on initial conditions" has been generalized, and the chaotic phenomena has been discussed for transitive systems with the generalized sensitive dependency property.     

A note on distributional chaos with respect to a sequence

The aim of this note is to use methods developed by Kuratowski and Mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence. In particular, we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to Li and Yorke. Namely, positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence, which is not the case for distributional chaos as introduced by Schweizer and Smítal.     

Transient chaos in two coupled,dissipatively perturbed Hamiltonian Duffing oscillators

The dynamics of two coupled, dissipatively perturbed, near-integrable Hamiltonian, double-well Duffing oscillators has been studied. We give numerical and experimental (circuit implementation) evidence that in the case of small positive or negative damping there exist two different types of transient chaos. After the decay of the transient chaos in the neighborhood of chaotic saddle we observe the transient chaos in the neighborhood of unstable tori. We argue that our results are robust and they exist in the wide range of system parameters.     

Some simple global synchronization criterions for coupled time-varying chaotic systems

Based on the Lyapunov stabilization theory and matrix measure, this paper proposes some simple generic criterions of global chaos synchronization between two coupled time-varying chaotic systems from a unidirectional linear error feedback coupling approach. These simple criterions are applicable to some typical chaotic systems with different types of nonlinearity, such as the original Chua’s circuit and the Rössler chaotic system. The coupling parameters are determined according to the new criterion so as to ensure the coupled systems’ global chaos synchronization.     

一类一维混沌映射的拓扑条件

20世纪中期以来,人们在物理、天文、气象等领域中发现了大量的混沌现象.这些新发现引发了近几十年来对混沌现象的研究.由于它的困难程度和在解决实际问题中的巨大价值,对混沌现象的研究成为动力系统乃至数学中的一个长期的前沿和热点研究方向.混沌现象最本质的特征是初值敏感性,保证有初值敏感性的一个充分条件是系统具有正Lyapunov指数.因此研究系统是否具有正Lyapunov指数成为研究系统是否出现混沌的重要方法.从拓扑角度给出了一类一维映射出现混沌现象的充分条件.从拓扑的角度来研究,将加深对此类映射出现混沌的机理的认识.研究此类映射,最重要的是研究临界点、临界点轨道及它们的相互关系.我们采用临界点的逆像建立拓扑工具,使用这一拓扑工具分析临界点轨道与临界点的复杂关系,研究临界点逆轨道的运动形态、相应开集的拓扑特征,进而导出系统出现混沌的拓扑特征及它与Lyapunov指数之间的关系.     

Chaos in temporarily destabilized regular systems with the slow passage effect

We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.     

A search technique for global optimization in a chaotic environment

We describe a new algorithm which uses the trajectories of a discrete dynamical system to sample the domain of an unconstrained objective function in search of global minima. The algorithm is unusually adept at avoiding nonoptimal local minima and successfully converging to a global minimum. Trajectories generated by the algorithm for objective functions with many local minima exhibit chaotic behavior, in the sense that they are extremely sensitive to changes in initial conditions and system parameters. In this context, chaos seems to have a beneficial effect: failure to converge to a global minimum from a given initial point can often be rectified by making arbitrarily small changes in the system parameters.     

Bifurcation,chaos, and their control in a wheelset model

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.     

Chaos in Kolmogorov systems with proliferation - general criteria and applications

We provide general criteria for existence of chaotic dynamics in Kolmogorov systems with proliferation. As applications, we provide a new proof of chaoticity of a class of constant coefficient birth-and-death systems and also show that a finite difference discretization of a class of drift diffusion equations generates chaotic dynamics. We also present a general method of constructing chaotic Kolmogorov systems.     

Devaney’s chaos implies distributional chaos in a sequence

Let X be a complete metric space without isolated points, and let f:X→X be a continuous map. In this paper we prove that if f is transitive and has a periodic point of period p, then f is distributionally chaotic in a sequence. Particularly, chaos in the sense of Devaney is stronger than distributional chaos in a sequence.