Dynamics with Chaos and Fractals by Marat Akhmet & Mehmet Onur Fen & Ejaily Milad Alejaily

Author:Marat Akhmet & Mehmet Onur Fen & Ejaily Milad Alejaily
Language: eng
Format: epub
ISBN: 9783030358549
Publisher: Springer International Publishing

(8.1.1)

where A is a constant n × n real valued matrix, the function is rd-continuous and the function g(t, ζ) is defined through the equation g(t, ζ) = ζ k for t ∈ [θ 2k−1, θ 2k], such that is a sequence generated by the map

(8.1.2)

where ζ 0 ∈ Λ, F : Λ → Λ is a continuous function and Λ is a compact subset of In Eq. (8.1.1) the time scale is defined as in which is a strictly increasing sequence of real numbers such that as and ∑−∞(θ 2k − θ 2k−1) = ∞,

In the present chapter, we investigate the existence of chaos in the dynamics of Eq. (8.1.1). The system under discussion is a hybrid one, since it combines the continuous dynamics on the time scale with the discrete equation used in the right-hand side of the system. We theoretically prove that chaos exists in (8.1.1) provided that the map (8.1.2) is chaotic. For that purpose, we make use of the reduction technique to impulsive differential equations, which was presented by Akhmet and Turan [20]. As far as we know, there is no paper on chaos in dynamics on time scales. The reason is that the dynamics is essentially non-autonomous and it is difficult to verify the ingredients of chaos for unspecified time scales. That is why we utilize the time scale introduced in the papers [20, 21] and the method of reduction of the dynamics to impulsive differential equations [20].

The rest of this chapter is organized as follows. In Sect. 8.2, some preliminary results as well as basic concepts about DETS are mentioned. Section 8.3 is devoted to the bounded solutions of (8.1.1). In Sect. 8.4, we give the description of the chaos of equation (8.1.1) and prove its presence rigorously. An example concerning Duffing equations on a time scale is presented in Sect. 8.5 to support the theoretical results. Finally, some concluding remarks are given in Sect. 8.6.

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