Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos

Abstract

Many systems exhibit behaviour which can be described by three coupled oscillators. Provided the coupling is not too strong, such systems can be modelled by maps of the two-torus to itself. In this paper we describe and explain those aspects of the bifurcation diagram for two-parameter families of torus maps ƒ that involve change of mode-locking type. We introduce the concepts of partial and full mode-locking. For the coupled oscillators, these notions correspond to the presence of one or two rational relations between the frequencies, respectively. Numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations. In order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time-1 map of a flow on the torus. Then we analyse those condimension-1 and -2 bifurcations for flows on the torus which change the set of frequency ratios. We find a large variety of bifurcation diagrams, in particular many involving homotopically non-trivial saddle connections. Next we ask how the picture changes when the time-1 maps of a family of flows are perturbed to a general family of diffeomorphisms. We find toroidal chaos in the neighbourhood of all these bifurcations, meaning that there exist orbits which perform a pseudo-random sequence of rotations in different directions around the torus. The corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk. Finally, we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps, which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation.