Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens' clocks, is studied using the codimension-three Elphick-Huygens equivariant normal form presented here. The focus is on the effect that the Huygens symmetry assumption has on the dynamic behavior of the system. Periodic solutions include the classical in-phase and anti-phase normal modes that are forced by the symmetry, as well as pairs of mixed mode phase-locked periodic solutions. The escapement paradox is explained. A theorem based on topological degree theory establishes the existence of quasi-periodic solutions in an invariant 3-torus that resembles a 2-torus slightly thickened to a solid toroidal shell, with the two principal radii of the 2-torus slowly modulated in time-that is, a toroidal breather. Secondary bifurcations from the in-phase and anti-phase normal modes are explored, of codimension one and two, and it is shown that an Arnold tongue plays a fundamental role in the determination of whether secondary bifurcation gives birth to phase-locked periodic solutions or to quasi-periodic solutions. Detailed numerical analysis, using MATLAB, extends the local bifurcation analysis to a more global picture that includes coexistence of multiple stable solutions and a "swallowtail" bifurcation of periodic solutions.

In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.

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Re-entrant spiral waves are observed in many different situations in nature, perhaps most importantly in excitable electrophysiological tissue where they are believed to be responsible for pathological conditions such as cardiac arrhythmias, epileptic seizures and hallucinations. Mathematically, spiral waves occur as solutions to systems of reaction-diffusion partial differential equations (RDPDEs) which are frequently used as models for electrophysiological phenomena. Because of the invariance of these RDPDEs with respect to the Euclidean group SE(2) of planar translations and rotations, much progress has been made in understanding the dynamics and bifurcations of spiral waves using the theory of group-equivariant dynamical systems. In reality however, Euclidean symmetry is at best an approximation. Inhomogeneities and anisotropy in the medium of propagation of the waves break the Euclidean symmetry, and can lead to such phenomena as anchoring and drifting. In this paper, we study the effects on quasiperiodic meandering spiral waves of a small perturbation which breaks the continuous SE(2) symmetry, but preserves the symmetry of a regular square lattice.

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