Research On Chaoticity And Sensitivity Of Dynamical Systems

Chaos theory has been developed for nearly forty years.Many scholars have discussed the relationship between various chaotic concepts and their related dynamical behaviors(such as sensitivity,topological transitivity,etc.)in nonlinear systems,and achieved fruitful results.This thesis focuses on discussing the related chaotic properties of symbolic dynamical sys-tems,g-fuzzification and non-autonomous discrete dynamical systems,especially the dynamical characteristics of sensitivity,specification property and transitivity and obtain the following re-sults:1.The symbolic dynamic system is used to illustrate how complex a class of extreme mapping can be.Based on this idea,another important concept to measure the complexity of the system-topological entropy is investigated,and examples of zero topological entropy and positive topological entropy are constructed.In addition,it is proved that there is a dynamic system with zero topological entropy,including a dense,invariant,extremal,transitive,uncount-able distributionally scrambled set consisting of proper quasi-weakly almost periodic points.The above conclusions further discuss the orbit7 s structure theory by Zhou Zuoling and He Weihong(1995)in Science in China Ser A.2.Specification property and mixing of Zadeh's extension system is studied.Firstly,it is discussed the relationship between sensitivity,syndetic sensitivity,transitivity,syndetic tran-sitivity,d(or d)-shadowing property and(strong)specification property of the original system.Secondly,it is proved that when a compact system has shadowing property,the system is mixing if and only if it is full and has(almost)specification-property if and only if for any ??(0,1],its Zadeh's extension system is mixing if and only if for any ??(0,1],its Zadeh's extension system is full and has(almost)specification-property.This conclusion further discusses the is-sue of an original system and its induced fuzzification system proposed by Kupka et al.(2011)in Fuzzy Sets&Systems and Journal of the London Mathematical Society.Finally,we ex-tend the conclusion of g-fuzzification to a product system,it is proved that the product system is multi-sensitive(respectively,F-sensitive)if and only if its factor system is multi-sensitive(respectively,F-sensitive).3.In a non-autonomous discrete system which converges uniformly,it is first proved that multi-sensitivity and ergodic sensitivity are preserved under any iterative operation.Then the existence conditions are given with sensitivity of several strong forms.The relationship be-tween the chaoticity(for example,Martelli chaos,Kato chaos,Ruelle-Takens chaos)of a non-autonomous product system and its factor system is discussed afterwards.The above questions are obtained from several perspectives of studying the chaotic definition proposed by Li Jian and Ye Xiangdong(2016)in Acta Mathematica Sinica.Analogous to autonomous dynamical systems,it is discussed multi-sensitivity and F-sensitivity of a non-autonomous product fuzzi-fication system.Moreover,multi-transitivity and a-transitivity are preserved under any iteration.Finally,it is studied several transitivity relations between the original non-autonomous system and its non-autonomous fuzzification system.In Chaos Solitons&Fractals,Sanchez(2017)positively stated that some properties of autonomous dynamical systems are not preserved in non-autonomous dynamical systems.But the above results indicate that some dynamical prop-erties of autonomous systems are preserved in non-autonomous dynamical systems.