Permanency and bifurcations of bounded solutions near homocilincs with symetric eigenvalues

we consider a system with homoclinic orbit, We decompose the corresponding variational equation on the space of solutions and provide sufficient conditions for the permanency of homoclinic in the space of $C^1$ vector fields. We also provide new sufficient conditions for the persistence and multiple bifurcations of the bounded solutions nearby. our results can be verified numerically and do not meet the limitations of classic methods (like Melnikon integrals and Poincare map)