Continuous Averaging in Dynamical Systems
نویسنده
چکیده
The method of continuous averaging can be regarded as a combination of the Lie method, where a change of coordinates is constructed as a shift along solutions of a differential equation and the Neishtadt method, well-known in perturbation theory for ODE in the presence of exponentially small effects. This method turns out to be very effective in the analysis of one-and multi-frequency averaging, exponentially small separatrix splitting and in the problem of an inclusion of an analytic diffeomorphism into an analytic flow. We discuss general features of the method as well as the applications. 1. The method There are several problems in the perturbation theory, of real-analytic ordinary differential equations (ODE), where standard methods do not lead to satisfactory results. We mention as examples the problem of an inclusion of a diffeomorphism into a flow in the analytic set up, and the problem of quantitative description of exponentially small effects in dynamical systems. In this cases one of possible approaches is an application of the continuous averaging method. The method appeared as an extension of the Neishtadt averaging procedure [14]. We begin with the description of the method. Let us transform the system ˙ z = u(z), (1.1) by using the change of variables z → Z(z, s). Here z is a point of the manifold M , u is a smooth vector field on M , s is a non-negative parameter, and the change is
منابع مشابه
Entropy operator for continuous dynamical systems of finite topological entropy
In this paper we introduce the concept of entropy operator for continuous systems of finite topological entropy. It is shown that it generates the Kolmogorov entropy as a special case. If $phi$ is invertible then the entropy operator is bounded with the topological entropy of $phi$ as its norm.
متن کاملOn Two-parameter Dynamical Systems and Applications
In this note some useful properties of strongly continuous two-parameter semigroups of operators are studied, an exponential formula for two-parameter semigroups of operators on Banach spaces is obtained and some applied examples of two-parameter dynamical systems are discussed
متن کاملOn the Notion of Fuzzy Shadowing Property
This paper is concerned with the study of fuzzy dynamical systems. Let (XM ) be a fuzzy metric space in the sense of George and Veeramani. A fuzzy discrete dynamical system is given by any fuzzy continuous self-map dened on X. We introduce the various fuzzy shad- owing and fuzzy topological transitivity on a fuzzy discrete dynamical systems. Some relations between this notions have been proved.
متن کاملNORMAL FORM SOLUTION OF REDUCED ORDER OSCILLATING SYSTEMS
This paper describes a preliminary investigation into the use of normal form theory for modelling large non-linear dynamical systems. Limit cycle oscillations are determined for simple two-degree-of-freedom double pendulum systems. The double pendulum system is reduced into its centre manifold before computing normal forms. Normal forms are obtained using a period averaging method which is appl...
متن کاملRandom Evolutions toward Applica- Tions
This article gives a short and elementary presentation of random evolutions toward applications in reliability and quality engineering. At first, the following two examples are presented: dynamical stochastic systems and increment processes both in Markov media. A dynamical system in continuous time is presented since nowadays they are widely used in dynamic reliability modeling. Limit theorems...
متن کامل