Synchronous states in time-delay coupled periodic oscillators: a stability criterion (Q375575)

The authors consider two systems of two delay coupled oscillators represented by phase-locked loops with the measured phase differences \(\varphi^{(1,2)}(t)\) and \(\varphi^{(2,1)}(t)\). Firstly, for a node filter represented by a constant gain, the first order system NEWLINE\[NEWLINE \dot{\varphi}^{(i,j)}(t)=-\mu_{0}\varphi^{(i,j)}(t)-\mu_{1}[\sin(\varphi^{(i,j)}(t)+\omega_{M}\tau)-\sin(\varphi^{(j,i)}(t-\tau)+\omega_{M}\tau)],\;i\neq j, NEWLINE\]NEWLINE with positive constants \(\mu_{0}\) and \(\mu_{1}\) is obtained. Secondly, for a filter \(F(s)=\alpha_{0}/(s+\beta_{0})\), the second order system NEWLINE\[NEWLINE\ddot{\varphi}^{(i,j)}(t)=-\mu_{0}\dot{\varphi}^{(i,j)}(t)-\mu_{1}\varphi^{(i,j)}(t)-\mu_{2}[\sin(\varphi^{(i,j)}(t)+\omega_{M}\tau)-\sin(\varphi^{(j,i)}(t-\tau)+\omega_{M}\tau)],\;i\neq j,NEWLINE\]NEWLINE with positive constants \(\mu_{0}\), \(\mu_{1}\) and \(\mu_{2}\) is obtained.NEWLINENEWLINEFor both system a linear stability analysis of the trivial solution is carried out. For the first order system this shows that the trivial solution is always stable. For the second order system the trivial solution is stable for \(\tau=0\) and conditions for a first destabilizing Hopf-bifurcation are given. This is illustrated by numerical simulations. The authors also present simulations far from this bifurcation, where seemingly chaotic trajectories are presented.