The Research On Some Properties Of Chaos To Multiple Mappings

The discovery of chaos phenomenon and the initiation of chaos theory are one of the greatest scientific discoveries in the 20th century.The study on chaos has become a major project in nonlinear science,which has gotten a rapid development and rich achievements.The study on chaos theory is to find the phenomena with inner rules which seemed complex and random emerging during the evolution of the system.It covers many fields and is widely used in various practical problems.However,the classical sense of chaos only considers the role of a mapping.Such problems like the ill-posed or inverse problems in differential equations,the static n-tuple mans-game problem in game theory,the toll highway problem in economics,etc.,it is impossible to be determined only by one mapping but by multiple mappings.The study on such issues has stimulated the booming of chaotic theory of set-valued dynamical systems.It is a special new definition of the set-valued map involved in this paper,which is called the multiple mappings.Let?X,d?be a compact metric space,its metric be d,and F={f₁,f₂,...,f_n} be a multiple mappings consisting of n-tuple of continuous maps from X to itself.i.e.That is a semigroup generated by n-tuple of continuous maps from X to itself.For any point x?X,F?x?= {f₁?x?,f₂?x?,...,f_n?x?} is a non-empty compact subset of X,and we investigated the chaotic dynamic properties of the point of X under F from the view of set-valued.F is the multiple mappings from X to its set-valued space k?X?F is a continuous map,and the metric d on X naturally induces a Hausdorff metrics dH on the set-valued space k?X?,and thus generates a new dynamical system.The?X,F?is called multiple mappings system,and several Hausdorff metric chaos are defined in the system?X,F?.In this paper,therefore,we studied some properties of several Hausdorff metric chaos for the system?X,F?,including?1?two topological conjugacy dynamical systems to multiple mappings have simultaneously Hausdorff metric Li-Yorke chaos and distributional chaos.?2?Hausdorff metric Li-Yorke ?-chaos is equivalent to Hausdorff metric distributional ?-chaos in a sequence.?3?by giving two examples,we show that there is non-mutual implication between that dynamical system?X,F ={f₁,f₂,...,f_n}?is Hausdorff metric Li-Yorke chaos and that fi?i=1,2,...,n?are Li-Yorke chaos.?4?The system?X,F?is Hausdorff metric distributional chaos if and only if?X,F^N?is Hausdorff metric distributional chaos.