Stability switches and Hopf bifurcations in a pair of delay-coupled oscillators (Q883118)

The paper studies two delay coupled limit cycle oscillators \[ \dot z_1 = (a+i\omega -| z_1(t)| ^2)z_1(t) + k(z_2(t-\tau_2)-z_1(t)), \] \[ \dot z_2 = (a+i\omega -| z_2(t)| ^2)z_2(t) + k(z_1(t-\tau_1)-z_2(t)), \] where \(a,\omega >0\), \(z_1,z_2\) are complex variables, \(k>0\) is the coupling strength, \(\tau_1,\tau_2>0\). Main results provide stability analysis of the origin \(z_1,z_2=0\), conditions for the Hopf bifurcation, and analysis of the corresponding Hopf bifurcation using the center manifold reduction.