On Points Focusing Entropy

Ewa Korczak-Kubiak,Anna Loranty,Ryszard Pawlak

doi:10.3390/e20020128

Abstract

In the paper, we consider local aspects of the entropy of nonautonomous dynamical systems. For this purpose, we introduce the notion of a (asymptotical) focal entropy point. The notion of entropy appeared as a result of practical needs concerning thermodynamics and the problem of information flow, and it is connected with the complexity of a system. The definition adopted in the paper specifies the notions that express the complexity of a system around certain points (the complexity of the system is the same as its complexity around these points), and moreover, the complexity of a system around such points does not depend on the behavior of the system in other parts of its domain. Any periodic system “acting” in the closed unit interval has an asymptotical focal entropy point, which justifies wide interest in these issues. In the paper, we examine the problems of the distortions of a system and the approximation of an autonomous system by a nonautonomous one, in the context of having a (asymptotical) focal entropy point. It is shown that even a slight modification of a system may lead to the arising of the respective focal entropy points.

Highlights

In many papers dealing with dynamical systems, their strong relation to difference equations is pointed out, which gives the possibilities of their wide applications in many fields of knowledge, including economics, biology, information flow or physics [2,3,4,5,6,7]
The definitions adopted in the paper specify the notions that express the complexity of a system around these points and the complexity of a system around such points does not depend on the behavior of the system in other parts of its domain
For a periodic nonautonomous dynamical system on X (NDS) consisting of continuous functions defined on the closed unit interval there exists an asymptotical focal entropy point

Summary

Introduction and Preliminaries

In many papers dealing with dynamical systems, their strong relation to difference equations is pointed out (see [1]), which gives the possibilities of their wide applications in many fields of knowledge, including economics, biology, information flow or physics [2,3,4,5,6,7]. In [13] was introduced a Bowen-like definition of entropy for an NDS consisting of continuous functions This definition was expanded for systems consisting of arbitrary functions in the paper [8]. Reasoning similar to that in the proofs of Lemma 4.3 and 4.5 [13] allows proving the following result concerning the entropy of an NDS consisting of not necessarily continuous functions. In [13] was introduced a new notion of asymptotical entropy, which, with respect to autonomous systems, coincides with the classical entropy. It is worth adding that the inequality from Lemma 2 is not true for entropy on subsets of the space, so the asymptotical entropy of a system on a set Y ⊂ X is defined as the following upper limit: h∗ ( f 1,∞ , Y ) = lim sup h( f n,∞ , Y ). We shall say that an NDS ( f 1,∞ ) of functions defined on M is irreducible at x0 if for n ∈ N, a function f 1n is irreducible at x0 , i.e., for any open neighborhood U of x0 , there exists a point y0 ∈ Int(M) ∩ U such that f 1n ( x0 ) 6= f 1n (y0 )

Focal Entropy Points of NDS

Disturbance and Approximation

Conclusions