Chaotic features of the generalized shift map and the complemented shift map

Abstract

Now a days, the discussion of chaotic dynamics has become increasingly popular. A particular class of dynamical systems is defined as chaotic dynamical systems. The idea of dynamical systems has gone through phases and has even been given different names. Chaos is gradually becoming a part of our daily life. Devaney's definition of chaos is considered a general and strong definition of chaos. It is based on the strength of topological transitivity in the discovery of chaos. In this research, we reviewed the Proximity theorem 3.3.1 and its proof and using this theorem we have found strong chaotic features of the shift map 𝜎, on Σ����. We have also proved the chaotic features of the generalized shift map. We have found the essential chaotic properties of the complemented shift map and confirmed that 𝜎���� is chaotic on Σ����. Presently, we are using the shift map as a chaotic model of a dynamical system. This research has established that the generalized shift map and the complemented shift map are chaotic. So, we can use the above two maps as constituting the new model for chaotic dynamical systems.