Expansiveness, Lyapunov exponents and entropy for set valued maps

In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. We define Lyapunov exponent for set valued maps and prove that positiveness of its Lyapunov exponent implies positiveness for the topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points for set valued maps defined on Peano spaces: either it is empty or the whole space.Comment: 24 pages, 1 figur

A reinterpretation of set differential equations as differential equations in a Banach space

Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example, which can be treated in support function representation, but not in the Hukuhara setting

Stability of Solutions of Fuzzy Differential Equations

In this paper, We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and comparison principle for Lyapunov - like functions, we give some sufficient criterias for the stability and asymptotic stability of solutions of fuzzy differential equations.Comment: 10 pages, 5 Theorems, interesting result in Exponential Stabilit

A Note on the Asymptotic Stability of Fuzzy Differential Equations

We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and the comparison principle for Lyapunov-like functions, we give sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equations.Вивчено стійкість розв'язків нечітких диференціальних рівнянь за допомогою другого методу Ляпунова. За допомогою масштабних рівнянь та принципу порівняння для рівнянь типу Ляпунова встановлено достатні умови стабільності та асимптотичної стабільності розв'язків нечітких диференціальних рівнянь

First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment

In this paper, the solution procedure of a first order linear non homogeneous ordinary differential equation in fuzzy environment is described. It is discussed for three different cases. They are i) Ordinary Differential Equation with initial value as a fuzzy number, ii) Ordinary Differential Equation with coefficient as a fuzzy number and iii) Ordinary Differential Equation with initial value and coefficient are fuzzy numbers. Here fuzzy numbers are taken as Generalized Triangular Fuzzy Numbers (GTFNs). An elementary application of population dynamics model is illustrated with numerical example. Keywords: Fuzzy Ordinary Differential Equation (FODE), Generalized Triangular fuzzy number (GTFN), strong solution

Existence Results for Functional Dynamic Equations with Delay

Time scale, arbitrary nonempty closed subset of the real numbers (with the topology and ordering inherited from the real numbers) is an efficient and general framework to study different types of problems to discover the commonalities and highlight the essential differences. Sometimes, we may need to choose an appropriate time scale to establish parallels to known results. We present a few recent results from existence theory of funcational dynamic equations including a few (counter) examples. In particular, we discuss first order functional dynamic equations with delay xDelta(t)=f(t,xt) on a time scale. Here, xt is in Crd([-tau,0],Rn) and is given by xt(s)=x(t+s), -tau \u3c s\u3c 0. We consider an appropriate timescale on which such delay equations can be studied meaningfully. We establish an existence result for the solutions of such problems. We also present a few examples

Full averaging of fuzzy impulsive differential inclusions

In this paper the substantiation of the method of full averaging for fuzzy impulsive differential inclusions is studied. We extend the similar results for impulsive differential inclusions with Hukuhara derivative (Skripnik, 2007), for fuzzy impulsive differential equations (Plotnikov and Skripnik, 2009), and for fuzzy differential inclusions (Skripnik, 2009)