Horseshoe chaos and topological entropy estimate in a simple power system

This paper presents rigorous arguments on existence of chaos and an estimate of topological entropy in a simple power system by means of topological horseshoe theory and computer computations.     

On essential entropy-carrying sets

In this paper we study properties of essential entropy-carrying sets of a continuous map on a compact metric space. If f:X→X$f:X\to X$ is continuous on a compact metric space X, then the intersection of all essential entropy-carrying sets of f may or may not be an essential entropy-carrying set of f  . When this intersection is an essential entropy-carrying set we denote it by E(f)$E(f)$, the least essential entropy-carrying set, otherwise we say that E(f)$E(f)$ does not exist. We present an example where E(f)$E(f)$ does not exist but also find a sufficient condition for E(f)$E(f)$ to exist. If f   is a piecewise monotone map, we show that E(f)$E(f)$ exists and is the finite union of the entropy-carrying sets in the Nitecki Decomposition of the nonwandering set of f intersected with the closure of the periodic points of f  . When E(f)$E(f)$ exists we study how it relates to other entropy-carrying sets of f including subsets of itself.     

The transversal homoclinic point of a saddle-node implies horseshoe

A method of identifying the existence of horseshoe for a two-dimension diffeomorphism is introduced and utilized to generalize the Birkhoff-Smale Theorem to the saddle-node case. Project supported by the National Natural Science Foundation of China (Grant No. 19531070).     

A commentary “on the periodic solutions of a forced second-order equation” by S. P. Hastings and J. B. McLeod

Summary We discuss the relationship of the shooting and perturbation methods used by Hastings and McLeod in the paper On the Periodic Solutions of a Forced Second-Order Equation to the geometrical techniques of nonlinear dynamical systems theory.     

On distributionally chaotic and null systems

By a topological dynamical system, we mean a pair (X,f), where X is a compactum and f is a continuous self-map on X. A system is said to be null if its topological sequence entropies are zero along all strictly increasing sequences of natural numbers. We show that there exists a null system which is distributionally chaotic. This system admits open distributionally scrambled sets, and its collection of all maximal distributionally scrambled sets has the same cardinality as the collection of all subsets of the phase space. Finally such system can even exist on continua.     

Hopf bifurcation and topological horseshoe of a novel finance chaotic system

This paper deals with the existence of both Hopf bifurcation and topological horseshoe for a novel finance chaotic system. First, through rigorous mathematical analysis, we show that a Hopf bifurcation occurs at systems’ three equilibriums S_0,1,2 and Hopf bifurcation at equilibrium S₀ is non-degenerate and supercritical. Second, the computer-assisted verifications for horseshoe chaos in the system are given. Simulation results are presented to support the analysis.     

树映射的链等价集与拓扑熵

本文讨论了树映射f的链等价集的性质,得到了f具有零拓扑熵的几个等价条件,并证明了:如果 f的一个链等价集是个无限集,那么这个链等价集的任何孤立点都是f的非周期的终于周期点.     

连续自映射拓扑压的两个注记

令T:X→ X是紧度量空间(X,d)上的连续映射.该文给出了T的拓扑压和T在非游荡集上的限制的拓扑压相等的不依赖于变分原理的一个直接证明.同时,还讨论了半共轭的两个系统的拓扑压之间的关系,证明了拓扑压在一致有限对一条件下是半共轭不变量.     

马蹄的存在蕴含横截同宿点的存在

对于微分同胚,横戴同宿点的存在蕴含Smale马蹄的存在,本证明了这一定理的逆定理成立,即Smale马蹄的存在也蕴含横同宿点的存在。     

树映射的不稳定流形,非游荡集与拓扑熵

设f是个端点数为n的树T上的连续自映射.本文得到了f的单侧不稳定流形与拓扑熵的关系,并证明了:（1）如果x∈i=0∞fi（Ω（f））-P（f）,那么,x的轨道是无限的;（2）如果f有一组可循环的不动点,那么h（f）≥In2（n-1）.     

Hyperchaotic set in continuous chaos–hyperchaos transition

Topological horseshoes with two-directional expansion imply invariant sets with two positive Lyapunov exponents (LE), which are recognized as a signature of hyperchaos. However, we find such horseshoes in two piecewise linear systems and one smooth system, which all exhibit chaotic attractors with one positive LE. The three concrete systems are the simple circuit by Tamaševičius et al., the Matsumoto–Chua–Kobayashi (MCK) circuit and the linearly controlled Lorenz system, respectively. Substantial numerical evidence from these systems suggests that a hyperchaotic set can be embedded in a chaotic attractor with one positive LE, and keeps existing while the attractor becomes hyperchaotic from chaotic. This paper presents such a new scenario of the continuous chaos–hyperchaos transition.     

THE IMPROVEMENT OF HORSESHOE CONDITION GIVEN BY JIANYIN ZHOU AND ITS APPLICATION

The horseshoe conditions[3] given by Jianyin Zhou is improved, then it is used to give asimpler proof of the conditions for generating horseshoe behavior produced by the equilibrum solutions of CNN Model.     

关于S~1上扩张映射的拓扑熵的改进

对文献 [1 ]中圆周上扩张映射的拓扑熵的结论给予了改进 ,得到了结论 :设 f∈ Cr(S1 ,S1 )是扩张映射 ,则 ent(f) =log|deg(f) |.