Coupled oscillators with internal symmetries
We investigate how the addition of an internal Z2 symmetry can affect the patterns seen in systems of coupled oscillators. To do this we first consider the theoretical implications, and discover that we must investigate bifurcations in the presence of S3, Z2 | S3 (the wreath product) and Z2 X S3 symmetries. We do this for both steady-state and Hopf bifurcations. We also consider the general case of steady-state bifurcations with Sn, Z2 | Sn and Z2 X Sn symmetries. We then try to answer the question of what it means for an oscillator to have an internal symmetry, and then how the form of coupling chosen between three oscillators with internal symmetries affects the global symmetry of the system. To this end we also introduce the notion of skew-equivariance, a generalisation of the notion of equivariance. It turns out that the addition of an internal Z2 symmetry to a network of three coupled oscillators can have a substantial effect on the patterns of oscillation observed, which as well as showing theoretically we also show by some numerical experiments. Finally we apply our results in two applications. The first is towards a model of how insects walk, their gaits, this problem being one of the main motivations for this work. The results found here show that the addition of an internal Z2 symmetry into the oscillators used to model locomotion, which can easily be justified by thinking of each leg as a pendulum, can be used to produce a much better model than those used in Wood  and . The second application is considering a network of three clusters of three oscillators, where applying skew-equivariance to the coupling produces a new set of solutions to those calculated by Dangelmayer et al. , where they consider D3 x D3 symmetry, by producing global symmetries of Z2 | S3 and Z2 X S3.