Journal
CHAOS SOLITONS & FRACTALS
Volume 118, Issue -, Pages 1-17Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.chaos.2018.11.013
Keywords
Space and time discretization; Neimark-Sacker bifurcation; Flip bifurcation; Chaos; Turing instability; Pattern
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Funding
- National Natural Science Foundation of China [10971009, 10771196, 11771033]
- National Scholarship Fund [201303070222, 201706020203, 201706020094]
- undamental Research Funds for the Central Universities
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This paper is concerned with the spatiotemporal behaviors of a Gierer-Meinhardt system in discrete time and space form. Through the linear stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Based on the bifurcation theory, as well as center manifold theorem, we derive the critical parameter values of the flip, Neimark-Sacker and Turing bifurcation respectively. Besides, the specific parameter expression to form patterns are also determined. In order to identify chaos among regular behaviors, we calculate the Maximum Lyapunov exponents. The results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamics, such as period doubling cascade, invariant cycles, periodic windows, chaotic behaviors, and some striking Turing patterns, e.g. circle, mosaic, spiral, spatiotemporal chaotic patterns and so on, which can be produced by flip-Turing instability, Neimark-Sacker-Turing instability and chaos. (C) 2018 Elsevier Ltd. All rights reserved.
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