Period-doubling cascades and mode interactions in coupled systems

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)