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Symmetry and phaselocking in chains of weakly coupled oscillators. (English) Zbl 0596.92011
Weakly coupled chains of oscillators with nearest-neighbor interactions are analyzed for phaselocked solutions. It is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound. It is also shown that qualitative behavior of these solutions depends on whether the coupling is ”diffusive” or ”synaptic”, terms defined in the paper.
The methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two-point (continuum) boundary value problem that is easier to analyze; the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law. An application to the neurophysiology of motor behaviour is discussed briefly.

92Cxx Physiological, cellular and medical topics
92B05 General biology and biomathematics
35Q99 Partial differential equations of mathematical physics and other areas of application
35B25 Singular perturbations in context of PDEs
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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