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

POSTED ON: January 1, 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

${\stackrel{˙}{\theta }}_{i}=\omega +ϵZ\left({\theta }_{i}\right)g\left({\theta }_{i-1}\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}i=1,2\cdots ,N,$

where
${\theta }_{0}\left(t\right)$

is some prescribed function of time. Each
${\theta }_{i}\in \left[0,2\pi \right)$

;
$\omega$

and
$ϵ$

are constants,
$Z$

is taken to be
$Z\left(\theta \right)=1-cos\theta$

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.
$i=-\infty ,\cdots ,\infty$

) 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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