We study properties of the dynamics underlying slow cluster oscillations in two systems of five globally coupled oscillators. These slow oscillations are due to the appearance of structurally stable heteroclinic connections between cluster states in the noise-free dynamics. In the presence of low levels of noise they give rise to long periods of residence near cluster states interspersed with sudden transitions between them. Moreover, these transitions may occur between cluster states of the same symmetry, or between cluster states with conjugate symmetries given by some rearrangement of the oscillators. We consider the system of coupled phase oscillators studied by Hansel et al. [Phys. Rev. E 48, 3470 (1993)] in which one can observe slow, noise-driven oscillations that occur between two families of two cluster periodic states; in the noise-free case there is a robust attracting heteroclinic cycle connecting these families. The two families consist of symmetric images of two inequivalent periodic orbits that have the same symmetry. For N = 5 oscillators, one of the periodic orbits has one unstable direction and the other has two unstable directions. Examining the behavior on the unstable manifold for the two unstable directions, we observe that the dimensionality of the manifold can give rise to switching between conjugate symmetry orbits. By applying small perturbations to the system we can easily steer it between a number of different marginally stable attractors. Finally, we show that similar behavior occurs in a system of phase-energy oscillators that are a natural extension of the phase model to two dimensional oscillators. We suggest that switching between conjugate symmetries is a very efficient method of encoding information into a globally coupled system of oscillators and may therefore be a good and simple model for the neural encoding of information.

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In the 1940s, the first generation of modern computers used vacuum tube oscillators as their principle components, however, with the development of the transistor, such oscillator based computers quickly became obsolete. As the demand for faster and lower power computers continues, transistors are themselves approaching their theoretical limit and emerging technologies must eventually supersede them. With the development of optical oscillators and Josephson junction technology, we are again presented with the possibility of using oscillators as the basic components of computers, and it is possible that the next generation of computers will be composed almost entirely of oscillatory devices. Here, we demonstrate how coupled threshold oscillators may be used to perform binary logic in a manner entirely consistent with modern computer architectures. We describe a variety of computational circuitry and demonstrate working oscillator models of both computation and memory.

We review some examples of dynamics displaying sequential switching for systems of coupled phase oscillators. As an illustration we discuss a simple family of coupled phase oscillators for which one can find robust heteroclinic networks between unstable cluster states. For N = 2k + 1 oscillators we show that there can be open regions in parameter space where the heteroclinic networks have the structure of an odd graph of order k; a class of graphs known from permutation theory. These networks lead to slow sequential switching between cluster states that is driven by noise and/or imperfections in the system. The dynamics observed is of relevance to modelling the emergent complex dynamical behaviour of coupled oscillator systems, e.g. for coupled chemical oscillators and neural networks.

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