Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay (Q964050)

We study the following system of two coupled relaxation oscillators of the van der Pol type with delay \[ \begin{cases} \varepsilon\ddot x_1-(1-x^2_1)\dot x_1+x_1=h_1(x_2(t-\tau)-x_1(t-\tau)),\\ \varepsilon\ddot x_2-(1-x^2_2)\dot x_2+x_2=h_2(x_1(t-\tau)-x_2(t-\tau)),\end{cases} \] where \(h_1\) and \(h_2\) are nonlinear functions. It is shown that this system can exhibit Hopf bifurcation as the time delay \(\tau\) passes certain critical values. The distribution of the eigenvalues of the linearized system is studied thoroughly in terms of the parameter \(\varepsilon\) and the linear parts of functions \(h_1\) and \(h_2\). The normal form theory for general retarded functional equations is applied to perform center manifold reduction and hence to obtain the explicit normal form Hopf bifurcation which can be used to determine the stability of the bifurcating periodic solutions and and the direction of Hopf bifurcation. Examples are given to illustrate the theoretical results.