Prof. Mintchev Co-Authors "Stability of a Family of Travelling Wave Solutions in a Feedforward Chain of Phase Oscillators"

January 01, 2015

Abstract

The paper concerns a chain of identical phase oscillators, each coupled to only its nearest neighbour on one side. The governing equations are

θ˙i=ω+ϵZ(θi)g(θi-1)fori=1,2,N,

where
θ0(t)

is some prescribed function of time. Each
θi[0,2π)

;
ω

and
ϵ

are constants,
Z

is taken to be
Z(θ)=1-cosθ

and
g

is a particular “pulse” function. (The results do not depend on the exact form of these, nor the values of parameters.) The model can be regarded as describing a feedforward network of theta neurons. The authors are interested in waves that travel with an approximately uniform profile and speed. They prove (under certain hypotheses) for a doubly-infinite chain (i.e. 
i=-,,

) that a family of such waves, each with constant speed and profile, does exist. (The family is parametrised by the wave’s temporal period.) They also prove that such a wave is stable to a large class of specified perturbations. The authors then give the results of careful numerical experiments which suggest that the hypotheses needed above are true.

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