Global homoclinic bifurcation for damped systems
Recently, existence and other results have been obtained for the variational problem \((P^0)\) \(\ddot u= -V'(u, t)\), \(u(- \infty)= u(\infty)= 0\), \(\dot u(- \infty)= \dot u(\infty)= 0\), where \(u\in \mathbb{R}^p\), the potential \(V\) behaves roughly like \(V= -{1\over 2} |U|^2+ W(u, t)\) with \(W\) superquadratic and 1-periodic in \(t\). In the paper the author studies the existence of homoclinic orbits for a damped system, namely, solutions of \((P^\alpha)\) \(\ddot u= -V'(u, t)- \alpha\dot u\), \(u(- \infty)= u(\infty)= 0\), \(\dot u(- \infty)= \dot u(\infty)= 0\), where \(\alpha\geq 0\) is a damping parameter. It is shown that under certain hypotheses \((P^\alpha)\) has a branch of solutions emanating from a suitable solution of \((P^0)\). This branch is global in nature and is shown to be either unbounded or having a turning point. It is known that, in general, solutions of \((P^0)\) do not form a bounded set. In the paper it is shown by example that there are situations in which solutions of \((P^\alpha)\) are bounded. The results mentioned above depend essentially on the hypothesis that \(V\) is time- depended, since in the autonomous case \((P^\alpha)\) has no solutions for \(\alpha\neq 0\).
scientific article; zbMATH DE number 778952