Attractors as a bridge from topological properties to long-term behavior in dynamical systems

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-10-02 DOI:10.1016/j.topol.2025.109604

Aliasghar Sarizadeh

引用次数: 0

Abstract

This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions, including that the phase space of a topologically exact system is an attractor for its inverse, and vice versa, and that a system is topologically mixing if and only if its phase space is a physical attractor.

Through iterated function systems (IFSs), we illustrate classes of non-trivial topologically mixing and topologically exact IFSs. Additionally, we use IFSs to provide an example of topologically mixing system, generated by finite of homeomorphisms on a compact metric space, that is not topologically exact. These findings connect topological properties with attractor types, providing deeper insights into the long-term dynamics of such systems.

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吸引子是动力系统从拓扑性质到长期行为的桥梁

在关系的背景下，对吸引子、物理吸引子、固有吸引子、拓扑精确和拓扑混合等符号进行了改进和引入。我们建立了拓扑精确系统的相空间是其逆系统的吸引子，反之亦然的充分必要条件，以及当且仅当相空间是物理吸引子时系统是拓扑混合的。通过迭代函数系统（ifs），我们说明了非平凡的拓扑混合和拓扑精确的ifs类。此外，我们使用ifs提供了一个拓扑混合系统的例子，该系统是由紧致度量空间上的有限个同胚生成的，它不是拓扑精确的。这些发现将拓扑特性与吸引子类型联系起来，为此类系统的长期动态提供了更深入的见解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.