Pattern generation in diffusive networks: How do those brainless centipedes walk?

In this paper we study the existence and stability of linear invariant manifolds in a network of diffusively coupled identical dynamical systems. Symmetry under permutation of different units of the network is helpful to construct explicit formulae for linear invariant manifolds of the network, in order to classify them, and to examine their stability through Lyapunovs direct method. A particular attention is drawn to the situation when all the subsystems without interconnections are globally asymptotically stable and the oscillatory behavior is forced via diffusive coupling.