Symmetry and phaselocking in chains of weakly coupled oscillators. (English) Zbl 0596.92011

Kopell, N.; Ermentrout, G. B.

Commun. Pure Appl. Math. 39, 623-660 (1986).

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.

MSC:

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

Keywords:

diffusive coupling; synaptic coupling; asymptotic stability; numerical stability; Weakly coupled chains of oscillators; nearest-neighbor interactions; phaselocked solutions; symmetry properties; scaling behavior; singularly perturbed two-point (continuum) boundary value problem; continuum equation; neurophysiology of motor behaviour

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