THE snapback repellers for chaos in multi-dimensional map

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 informa- tion of F is not sufficient to indicate the existence of snapback repeller for F. In this investigation, for a repeller z of F, we start from estimating the repelling neighborhood of 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 z ₀ in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of F along the orbit through 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.