In this paper, we have discussed the general concepts of chaos theory in dynamical systems and main characteristics of chaos. We have described manipulating chaos, different definitions of chaos and their interrelationship in this study. In this work we have found that one dimensional maps with complicated dynamical behaviour. We have solved some problems of tent map HENA RANI BISWAS et al. 92 and after solving those problems we observed that tent map is chaotic. We also discussed the bifurcation of one dimensional map and sketch bifurcation diagrams by using MATHEMATICA.

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In this paper, we study some chaos related properties of the forward shift map + on the generalized one-sided symbol space ∑ + , (≥ 2) ∈. We consider several notions of chaos available in the contemporary literature. In this paper, we prove that + is Devaney chaotic, Auslander-Yorke's chaotic and generically −chaotic. We prove that + is exact Devaney chaotic and as a consequence is mixing Devaney Chaotic and weak mixing Devaney Chaotic. We also provide examples to show that the forward shift map + on ∑ + is topologically conjugate to the map on the space or equivalently conjugate to the map () = (1) on the space /. Finally, we give a counterexample to prove that not all topologically transitive maps are totally transitive. We also give an example of a continuous function that is topologically transitive but not chaotic.

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