A generalization of the Lefschetz fixed point theorem and detection of chaos

We consider the problem of existence of fixed points of a continuous map in (possibly) noninvariant subsets. A pair of subsets of induces a map given by if and elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: If is metrizable, and are compact ANRs, and is continuous, then has a fixed point in provided the Lefschetz number of is nonzero. Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic.

     

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.     

一类连续体上连续映射的周期点

设X是个阶有限的遗传可分解可链连续体, f:X→X是X上的连续自映射, On(x,f)={fi(x):0≤i≤n)是f的一个返回轨道, inf(On(x,f))相似文献   

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

Li–Yorke chaos and synchronous chaos in a globally nonlocal coupled map lattice

This paper investigates a globally nonlocal coupled map lattice. A rigorous proof to the existence of chaos in the scene of Li–Yorke in that system is presented in terms of the Marotto theorem. Analytical sufficient conditions under which the system is chaotic, and has synchronous behaviors are determined, respectively. The wider regions associated with chaos and synchronous behaviors are shown by simulations. Spatiotemporal chaos, synchronous chaos and some other synchronous behaviors such as fixed points, 2-cycles and 2²-cycles are also shown by simulations for some values of the parameters.     

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.     

Distributional chaos and factors

We show the existence of a dynamical system without any distributionally scrambled pair which is semiconjugated to a distributionally chaotic factor.     

Unimodal mappings and Li-Yorke chaos

Conditions for unimodal mappings to have domains with a Li-Yorke chaotic behavior of trajectories are found. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 679–689, May, 1998.     

一般三角帐篷映射混沌性与两种混沌互不蕴含性

将三角帐篷映射推广为一般的n-三角帐篷映射,并且借助于一般Bernoulli移位映射,Banks定理与Li-Yorke定理,首先证明：对于任意的正整数n,n-三角帐篷映射既是Devaney混沌的,也是Li-Yorke混沌的.然后,利用所得到的结果,通过实例展示：Devaney混沌与Li-Yorke混沌的互不蕴含性.     

A Note on “Multi-separation, Centrifugality and Centripetality Imply Chaos”

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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.     

The weak specification property and distributional chaos

In this paper, we first discuss almost periodic points in a compact dynamical system with the weak specification property. On the basis of this discussion, we draw two conclusions: (i) the weak specification property implies a dense Mycielski uniform distributionally scrambled set; (ii) the weak specification property and a fixed point imply a dense Mycielski uniform invariant distributionally scrambled set. These conclusions improve on some of the latest results concerning the specification property, and give a final positive answer to an open problem posed in [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), 31–43].     

A note on transitivity,sensitivity and chaos for graph maps

Two elementary proofs showing that (i) transitivity and sensitivity imply dense periodicity for maps on topological graphs and (ii) total transitivity and dense periodicity imply mixing for maps on spaces with an open subset homeomorphic with the open interval (0,1) are presented. As corollaries, one gets new and simple proofs that Auslander–Yorke chaos implies Devaney chaos, and weak mixing implies mixing for graph maps.     

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.     

Transitivity, mixing and chaos for a class of set-valued mappings

Consider the continuous map f : x → X and the continuous map f of K,(X) into itself induced by f, where X is a metric space and K(X) the space of all non-empty compact subsets of x endowed with the Hausdorff metric. According to the questions whether the chaoticity of f implies the chaoticity of f posed by Roman-Flores and when the chaoticity of f implies the chaoticity of f posed by Fedeli, we investigate the relations between f and f in the related dynamical properties such as transitivity, weakly mixing and mixing, etc. And by using the obtained results, we give the satisfied answers to Roman-Flores's question and Fedeli's question.     

Traveling waves,impulses and diffusion chaos in excitable media

In the present work it is shown, that the FitzHugh–Nagumo type system of partial differential equations with fixed parameters can have an infinite number of different stable wave solutions, traveling along the space axis with arbitrary speeds, and also traveling impulses and an infinite number of different states of spatiotemporal (diffusion) chaos. Those solutions are generated by cascades of bifurcations of cycles and singular attractors according to the FSM theory (Feigenbaum–Sharkovskii–Magnitskii) in the three-dimensional system of ordinary differential equations (ODEs), to which the FitzHugh–Nagumo type system of equations with self-similar change of variables can be reduced.     

Voltage transformer ferroresonance analysis using multiple scales method and chaos theory

In this article, the Multiple Scales Method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes subharmonic, quasi‐periodic, and also chaotic oscillations. In this article, the chaotic behavior and various ferroresonant oscillations modes of the voltage transformer is studied. This phenomenon consists of different types of bifurcations such as Period Doubling Bifurcation (PDB), Saddle Node Bifurcation (SNB), Hopf Bifurcation (HB) and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are illustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via Multiple Scales Method obtaining Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system. © 2013 Wiley Periodicals, Inc. Complexity 18: 34‐45, 2013