The snapback repellers for chaos in multi-dimensional maps

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.