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This thesis is divided in three main parts. In the first part, we study equivariant forced systems of ODEs. We focus on periodic behaviour in equivariant forced systems and determine the H/K Theorem for forced systems of ODEs. This theorem classifies all possible symmetry pairs (H,K) of periodic solutions in such systems. We also analyze the possible periods to occur in these systems. In the second part, we study animal locomotion. Although many work has been done in this area, most is based on the mathematical assumption that locomotor CPGs should be modelled as autonomous systems of ODEs. We present a novel approach to this study by considering locomotor CPGs as forced systems of ODEs. By applying the results obtained in the first part, we show that a forced 4-cell system can exhibit the desired patterns of locomotion depending on the forcing to which it is subjected. The final part of this thesis is focused on the study of patterns in coupled cell networks. We divide our study into two different classes of networks: coupled cell networks with an internal symmetry group and general coupled cell networks. For the former type of network, we show that all isotropy subgroups of the internal symmetry group may occur as symmetry groups of some steady-state, and prove necessary conditions for a pair (H,K) to be the symmetry pair of some periodic solution in such networks. In the latter type of network, we analyze patterns of synchrony in such networks. We prove that rigid patterns of synchrony correspond to balanced equivalence relations. This result is known as the Rigid Synchrony Theorem. |
