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

Izhikevich, Eugene M.; Hoppensteadt, Frank C.

SIAM J. Appl. Math. 63, No. 6, 1935-1953 (2003).

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.

MSC:

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

Keywords:

phase model; Andronov-Hopf; saddle-node on invariant circle; Class 1 excitability; relaxation oscillators; Malkin theorem; MATLAB

Citations:

Zbl 0783.34026; Zbl 0041.52704; Zbl 0070.08703

Software:

Matlab; XPPAUT