Slowly coupled oscillators: Phase dynamics and synchronization. (English) Zbl 1058.92002

Summary: We extend results of P. Frankel and T. Kiemel [SIAM J. Appl. Math 53, 1436–1446 (1993; Zbl 0783.34026)] to a network of slowly coupled oscillators. First, we use I. G. Malkin’s theorem [Lyapunov methods in the theory of nonlinear oscillations. (Russian) (1949; Zbl 0041.52704); Some problems in nonlinear oscillations theory. (1956; Zbl 0070.08703)] to derive a canonical phase model that describes synchronization properties of a slowly coupled network. Then, we illustrate the result using slowly coupled oscillators (1) near Andronov-Hopf bifurcations, (2) near saddle-node on invariant circle bifurcations, and (3) near relaxation oscillations. We compare and contrast synchronization properties of slowly and weakly coupled oscillators.


92B05 General biology and biomathematics
34C23 Bifurcation theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
92B20 Neural networks for/in biological studies, artificial life and related topics
34C25 Periodic solutions to ordinary differential equations
37N25 Dynamical systems in biology
92-04 Software, source code, etc. for problems pertaining to biology


Matlab; XPPAUT
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