Bifurcation and dynamics of a normal form map

Abstract

This paper investigates the dynamics and stability properties of a so-called planar truncated normal form map. This kind of map is widely used in the applied context, especially in normal form coefficients of n-dimensional maps. We determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the system. We first analyze stability of the fixed points and the existence of local bifurcations. Our analysis shows the presence of a rich variety of local bifurcations, namely stable fixed points, periodic cycles, quasiperiodic cycles that are constraints to stable attractors called invariant closed curves, and chaos, where dynamics of the system change erratically. Our study is based on the numerical continuation method under variation of 1 and 2 parameters and computation of different bifurcation curves of the system and its iterations. For the all codimension 1 and 2 bifurcation points, we compute the corresponding normal form coefficients to reveal the criticality of the corresponding bifurcations as well as to identify different bifurcation curves that emerge around the corresponding bifurcation point. We further perform numerical simulations to characterize qualitatively different dynamical behaviors within each regime of parameter space.