Codimension-one and -two bifurcation analysis of a two-dimensional coupled logistic map

Abstract

This paper devotes to a detailed bifurcation analysis of a two-dimensional non-invertible map, obtained using a symmetric coupling between one-dimensional logistic maps. The critical normal form coefficients method is employed to detect bifurcations and to explore further critical conditions without explicit reduction to the center manifold. The results show that the two-dimensional map undergoes codimension-one (codim-1) bifurcations such as transcritical, pitchfork, period-doubling, Neimark–Sacker, and codim-2 bifurcations including transcritical-flip, pitchfork-flip, strong resonances 1:2, 1:3, 1:4. For each bifurcation, the critical normal form coefficients are calculated to check the non-degeneracy conditions and predict the bifurcation scenarios around the bifurcation points. To validate the theoretical results, all bifurcation curves of fixed points are plotted with the aid of the numerical continuation method. Weak resonances are also specified by the isoclines on the bi-parameter plane. The results will help in understanding the occurrence and the structure of bifurcation cascades observed in many coupled discrete systems.