## 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

### MSC:

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

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