Period‐doubling cascades and mode interactions in coupled systems

Abstract Suppose that an iterated map exhibits a period‐doubling cascade. If two such maps are coupled, then the synchronised state will exhibit the same period‐doubling cascade but there is also the additional possibility of symmetry‐breaking bifurcations to non‐synchronised states. By introducing a second parameter, a symmetry‐breaking bifurcation and a period‐doubling bifurcation can be made to occur at the same point, resulting in a mode interaction. As the second parameter is varied from the value at the mode interaction, a second symmetry‐breaking bifurcation may occur from the period 2 solutions, which will then be involved in another mode interaction at the next period‐doubling bifurcation point. In this way, a complete cascade of mode interactions can occur. A local analysis of such a mode interaction is considered. The global consequences together with a classification of different cases are then analysed. Renormalisation theory is used to determine the universal behaviour and parameter scalings of such a system. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)