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.
