First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. (English) Zbl 0702.34039

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.
Reviewer: H.Hofer


34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70H05 Hamilton’s equations
35J60 Nonlinear elliptic equations
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