Estimates Of Topological Entropy Research Articles

Topological Chaos for Differential Inclusions with Multivalued Impulses on Tori

A multivalued version of the Ivanov inequality for the lower estimate of topological entropy of admissible maps is applied to differential inclusions with multivalued impulses on tori via the associated Poincaré translation operators along their trajectories. The topological chaos in the sense of a positive topological entropy is established in terms of the asymptotic Nielsen numbers of the impulsive maps being greater than 1. This condition implies at the same time the existence of subharmonic periodic solutions with infinitely many variety of periods. Under a similar condition, the coexistence of subharmonic periodic solutions of all natural orders is also carried out.

The Dynamical Properties of Modified of Bogdanov Map

In the work, we study the general properties of modified of the Bogdanov map in the form Fa,m,k .
 
 We prove it has sensitive dependence on initial conditions and positive Lyapunov exponent with some conditions . So we give estimate of topological entropy of modified Bogdanov map .

Estimation of topological entropy in random dynamical systems

For Cr(r>1) random dynamical systems F on a compact smooth Riemannian manifold M, the fiber topological entropy is bounded above by an integral formula. Particularly, for the C∞ random dynamical systems, the integral formula coincides with the fiber topological entropy.

Topological entropy for impulsive differential equations

A positive topological entropy is examined for impulsive differential equations via the associated Poincare translation operators on compact subsets of Euclidean spaces and, in particular, on tori. We will show the conditions under which the impulsive mapping has the forcing property in the sense that its positive topological entropy implies the same for its composition with the Poincare translation operator along the trajectories of given systems. It allows us to speak about chaos for impulsive differential equations under consideration. In particular, on tori, there are practically no implicit restrictions for such a forcing property. Moreover, the asymptotic Nielsen number (which is in difference to topological entropy a homotopy invariant) can be used there effectively for the lower estimate of topological entropy. Several illustrative examples are supplied.

On a method of applied symbolic dynamics for investigation of dynamical systems

We consider a method of applied symbolic dynamics which may be used to obtain wide spectrum of characteristics of complex dynamical systems: approximation of invariant sets, estimation of topological entropy and approximation of invariant measures. The method has received wide acceptance in studying complex dynamical systems. The main idea of the method is to describe the system behaviour approximately by means of an oriented graph (called symbolic image). Such a graph is a representation of symbolic dynamical system which is more appropriately known as topological Markov chain. There exists a correspondence between trajectories of a given system and paths on the graph, which allows us to design algorithms for estimation of topological entropy, approximate invariant measures of a given system by using stationary flows on the graph and calculate corresponding metric entropies. These values characterize complex behaviour of dynamical systems such as the existence of trajectories with large periods and chaotic regimes. The results of experiments are given for systems with chaotic dynamics.

Survey on finding the periodic orbits in chaotic systems

The periodic orbits provide a skeleton for the organization of complex chaotic systems,for many important characteristics and dynamic properties of these systems can be determined by solving the periodic orbits,such as the accurate calculation of Lyapunov exponents,estimation of topological entropy and description of a chaotic invariant set. First,the paper reviewed the current commonly used four methods to find periodic orbits,which are NR algorithm,Broyden algorithm,SD algorithm and DL algorithm,and analyzed their characteristics and mutual relations.Second,the paper discussed the advantages,disadvantages and scope of each method with specific examples in detail.Finally,the paper pointed out that DL algorithm is more ideal among the four algorithms, and suggested the future research direction.

A new chaotic Hopfield neural network and its synthesis via parameter switchings

In this paper, we present a new chaotic attractor in Hopfield neural network. Numerical experiments show that the presented Hopfield neural network can display complex dynamics by changing the self-connection weight. Surprisingly, coexistence of a chaotic attractor and a limit cycle is found in this system, which means, the system can exhibit a chaotic attractor or a limit cycle according to different initial values, and this phenomenon is never reported before. We give a rigorous verification of existence of horseshoe chaos by virtue of topological horseshoes theory and estimates of topological entropy in the derived Poincaré maps. Finally, synthesis of the chaotic attractor is studied via parameter switching and a numerical example illustrates the effectiveness of this method.

Estimation of topological entropy via the direct Lyapunov method

This paper deals with the problem of estimation of the topological entropy for non-autonomous systems of differential equations via the second (direct) Lyapunov method. The main result of the paper is illustrated by examples concerning the Lorenz system and Duffing oscillator.

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.

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THREE-NEURON CELLULAR NEURAL NETWORKS

In this paper, we study a new class of simple three-neuron chaotic cellular neural networks with very simple connection matrices. To study the chaotic behavior in these cellular neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of the existence of horseshoe chaos using topological horseshoe theory and the estimate of topological entropy in derived Poincaré maps.

Horseshoe chaos in a class of simple Hopfield neural networks

A class of new simple Hopfield neural networks is revisited. To confirm the chaotic behavior in these Hopfield neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of existence of horseshoe chaos by virtue of topological horseshoes theory and estimates of topological entropy in the derived Poincaré maps.

Estimates of topological entropy of continuous maps with applications

We present a simple theory on topological entropy of the continuous maps defined on a compact metric space, and establish some inequalities of topological entropy. As an application of the results of this paper, we give a new simple proof of chaos in the so-calledN-buffer switched flow networks.

An estimate of topological entropy of N-buffer fluid networks

In this paper, we give an estimate of the topological entropy of a dynamical system defined on a simplex derived from the model of an N-buffer fluid network using an elementary theory of symbolic dynamics. Our treatment is easy to catch on even for readers with a less than good background of ergodic theory in dynamical systems.

Lyapunov’s direct method in estimates of topological entropy

An upper estimate for the topological entropy of a dynamical system defined by a system of ODE is obtained. The estimate involves the Lyapunov functions and Losinskii’s logarithmic norm. The proof uses the known fact that the topological entropy of a mapping acting in a compact space K can be estimated via the fractal dimension of K. Bibliography: 28 titles.