Kang-Ling Liao 1, , Chih-Wen Shih 2, and Chi-Jer Yu 2,
Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, USA
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 300
The key of Marotto's theorem on chaos for multi-dimensional maps is the existence of snapback repeller. For practical application of the theory, locating a computable repelling neighborhood of the repelling fixed point has thus become the key issue. For some multi-dimensional maps $F$, basic information of $F$ is not sufficient to indicate the existence of snapback repeller for $F$. In this investigation, for a repeller $\bar{\bf z}$ of $F$, we start from estimating the repelling neighborhood of $\bar{\bf z}$ under $F^{k}$ for some $k ≥ 2$, by a theory built on the first or second derivative of $F^k$. By employing the Interval Arithmetic computation, we locate a snapback point ${\bf z}_0$ in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of $F$ along the orbit through ${\bf z}_0$. With this new approach, we are able to conclude the existence of snapback repellers under the valid definition, hence chaotic behaviors, in a discrete-time predator-prey model, a population model, and the FitzHugh nerve model.
Keywords: Snapback repeller, homoclinic orbit, chaos, Interval arithmetic.
Mathematics Subject Classification: Primary: 37C29, 37C70, 34C28.
Citation: Kang-Ling Liao, Chih-Wen Shih, Chi-Jer Yu. The snapback repellers for chaos in multi-dimensional maps. Journal of Computational Dynamics, 2018, 5 (1&2) : 81-92. doi: 10.3934/jcd.2018004
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Kang-Ling Liao Chih-Wen Shih Chi-Jer Yu