Poincare chaos and unpredictable functions

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AKHMET M., Fen M. O.

COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, vol.48, pp.85-94, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48
  • Publication Date: 2017
  • Doi Number: 10.1016/j.cnsns.2016.12.015
  • Journal Name: COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.85-94
  • Keywords: Poincare chaos, Unpredictable function, Unpredictable solutions, Unpredictable sequence, QUASI-MINIMAL SETS, DYNAMICAL SYNTHESIS
  • Middle East Technical University Affiliated: Yes

Abstract

The results of this study are continuation of the research of Poincare chaos initiated in the papers (M. Akhmet and M.O. Fen, Commun Nonlinear Sci Numer Simulat 40 (2016) 1-5; M. Akhmet and M.O. Fen, Turk J Math, doi:10.3906/mat-1603-51, in press). We focus on the construction of an unpredictable function, continuous on the real axis. As auxiliary results, unpredictable orbits for the symbolic dynamics and the logistic map are obtained. By shaping the unpredictable function as well as Poisson function we have performed the first step in the development of the theory of unpredictable solutions for differential and discrete equations. The results are preliminary ones for deep analysis of chaos existence in differential and hybrid systems. Illustrative examples concerning unpredictable solutions of differential equations are provided.(C) 2016 Elsevier B.V. All rights reserved.