Replication of Chaos in Neural Networks, Economics and Physics

Abstract

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Existence of unpredictable solutions and chaos

“…Thus, if one shows the existence of an unpredictable solution of an equation, then the chaos exists. The present study as well as our previous results concerning replication of chaos [12] support the opinion of Holmes [20] that the theory of chaos has to be a part of the theory of differential equations. Since the main body of the results on chaotic motions has been formulated in terms of differential and difference equations, we may suggest that all these achievements have to be embedded and developed in the theory of dynamical systems or, more specifically, in the theory of differential equations or hybrid systems.…”

“…It was demonstrated in [13] that for such values of the parameter the map (12) possesses an unpredictable point in Λ. Let us consider system (13) with ζ 0 ∈ Λ.…”

“…For each natural number p , system (13) admits an unstable periodic solution with period 2p if p is odd and an unstable periodic solution with period p if p is even [5]. The reader is referred to [1][2][3][4][5][6][7][8][9][10][11][12] for more information about the dynamics of relay systems.…”

“…One can observe that the stabilized 2-periodic solutions seen in Figure 3 and Figure 4 are different, and this reveals the presence of periodic motions. (12). The value ε = 0.072 is used and the control is switched on at t = ζ25.…”

Poincaré chaos and unpredictable functions

“…Additionally, using the same dynamics we have introduced Poisson functions. All these make our duty of incorporating chaos investigation to theory of differential equations initiated in papers [9]- [14] and in the book [15] seems to proceed in the correct way.…”