Chaotic orbits prediction and atypical bifurcations in a class of piece-wise linear noninvertible maps

Abstract
In this paper we apply one of the main results from the theory of noninvertible maps to predict the oscillations amplitude of a chaotic attractor, using only the lines where the piece-wise linear (PWL) map is not differentiable, their first iterates (called critical lines or curves), and their finite-rank iterates. This approach is also valid for smooth differentiable noninvertible maps, where the critical lines are defined as the first iterates of the set of points for which the Jacobian determinant of the map cancels. Moreover, a novel bifurcation encountered in the case of PWL map is presented: chaotic orbit arising from the collision of a typical bifurcation for smooth maps (the Neimark-Hopf, or center bifurcation) with the bifurcation typical only for piece-wise smooth and non differentiable maps (the border-collision bifurcation). Finally, it is shown that this novel bifurcation can give birth to two (or more) chaotic attractors, and the conditions allowing to predict the number of chaotic attractors which could coexist are proposed and discussed in the context of possible applications.