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

POSTED ON: January 1, 2015


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(θ i )g(θ i-1 )fori=1,2,N,

θ 0 (t)

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



are constants,

is taken to be


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. 

) 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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