Subharmonic solutions bifurcated from homoclinic orbits for weakly coupled singular systems

The problem of bifurcation from homoclinic solution towards periodic solution was considered for weekly coupled singular systems. By using functional analytic approach based on the Lyapunov–Schmidt reduction, we obtained some functions H:R^d-1×R→R^d$H:R^{d-1}\times R\to R^d$. The simple roots of the equations, H(α,β)=0$H(\alpha ,\beta )=0$, correspond to the existence of subharmonic solutions. And if the vector field is 2-period, then for any integer m  , the weakly coupled singular system has 2m$2m$-period solution.