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Bifurcation for second-order Hamiltonian systems with periodic boundary conditions. (English) Zbl 1144.37025

Summary: Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a function \(u\), and prove that the set of bifurcation points for the solutions of the system is not \(\sigma \)-compact. Then, we deal with a linear system depending on a real parameter \(\lambda >0\) and on a function \(u\), and prove that there exists \(\lambda ^{\ast }\) such that the set of the functions \(u\), such that the system admits nontrivial solutions, contains an accumulation point.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
47J30 Variational methods involving nonlinear operators
37C60 Nonautonomous smooth dynamical systems
34C25 Periodic solutions to ordinary differential equations
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References:

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