The authors prove the existence of homoclinic solutions for a second order vectorial differential system: (1) \(\ddot U+F(U)=0\) (when F is nonlinear, diagonal, and the system is Hamiltonian), or, more generally: \((2)\quad M.\ddot U+G(U)=0.\) In (2), M is a symmetric definite positive matrix; \(G=\text{grad} H\); \(G^{-1}(0)=H^{-1}(0)=\{0\}\). Besides a condition on the linearization of (2) near zero, they assume that \(U^ T_{(t)}M.U(t)\) is increasing for \(\| U(t)\|\) large enough. They derive their result (existence of a nontrivial solution connecting 0 at \(+\infty\) and -\(\infty)\) from the study of the deformation of a convenient set by the solution flow, using the notion of Wazewski sets.
The paper is related to a previous one by Hofer and Toland where the same result was obtained under notably distinct assumptions by means of the Brouwer degree. The assumptions cover the case of FitzHugh-Nagumo type systems (case of bundle of parallel unmyelinated nerve axons) and a special form of the Hodgkin-Huxley model. The authors refer to a forthcoming paper in J. Math. Appl. Med. Biol. for more details.