Snapback repellers and homoclinic orbits for multi-dimensional maps

Marotto extended Li–Yorkeʼs theorem on chaos from one-dimension to multi-dimension through introducing the notion of snapback repeller in 1978. Due to a technical flaw, he redefined snapback repeller in 2005 to validate this theorem. This presentation provides two methodologies to facilitate the application of Marottoʼs theorem. The first one is to estimate the radius of repelling neighborhood for a repelling fixed point. This estimation is of essential and practical significance as combined with numerical computations of snapback points. Secondly, we propose a sequential graphic-iteration scheme to construct homoclinic orbit for a repeller. This construction allows us to track the homoclinic orbit. Applications of the present methodologies with numerical computation to a chaotic neural network and a predator–prey model are demonstrated.