The snapback repellers for chaos in multi-dimensional maps

Abstract

Listen

Translate article

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 \begin{document}$F$\end{document} , basic information of \begin{document}$F$\end{document} is not sufficient to indicate the existence of snapback repeller for \begin{document}$F$\end{document} . In this investigation, for a repeller \begin{document}$\bar{\bf z}$\end{document} of \begin{document}$F$\end{document} , we start from estimating the repelling neighborhood of \begin{document}$\bar{\bf z}$\end{document} under \begin{document}$F^{k}$\end{document} for some \begin{document}$k ≥ 2$\end{document} , by a theory built on the first or second derivative of \begin{document}$F^k$\end{document} . By employing the Interval Arithmetic computation, we locate a snapback point \begin{document}${\bf z}_0$\end{document} in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of \begin{document}$F$\end{document} along the orbit through \begin{document}${\bf z}_0$\end{document} . 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.