S-shaped bifurcations in a two-dimensional Hamiltonian system

We study the solutions to following Dirichlet boundary problem: d 2x(t) dt2 + λ f(x(t)) = 0, where x ∈ R, t R+, with conditions: x(0) x(1) A R. Especially we focus on varying parameters and in case phase plane representation of equation contains a saddle loop filled period annulus surrounding center. introduce concept mixed which take values above below A, generalizing well-studied positive solutions. This leads generalization so-called function for annulus. derive expansions these functions formulas derivatives generalized functions. The main result is that under generic conditions f(x) S-shaped bifurcations occur. As consequence there exists an open interval sufficiently small can be found such three same type exist. show how concepts relate simplest possible x(x 1) despite its simple form difficult problems remain.