First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems (Q914929)

We study the existence of nontrivial homoclinic orbits of the Hamiltonian system \[ (*)\quad \dot x=JA(x)+JR'(t,x)x\in {\mathbb{R}}^{2n}, \] where \(J\) is the matrix \(\begin{pmatrix} 0&I\\ -I&0 \end{pmatrix}\), \(\epsilon^{JA}\) has no eigenvalues on the unit circle, \(R\) is \(T\)-periodic in \(t\) and superquadratic at infinity. In order to find homoclinic orbit of (*) we associate to it a family of first order elliptic problems and study the set of solutions. We prove, using the Sard-Smale theory for Fredholm operators, that this set cannot be compact. On the other hand, if (*) has no homoclinic orbits then the set of solutions of our family of operators would be compact, thus proving our assertion.