## 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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