Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics

We prove the existence of a pair of positive T-periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic-like dynamics for the scalar second order ODEu″+aλ,μ(t)g(u)=0,u″+aλ,μ(t)g(u)=0, where g(x)g(x) is a positive function on R+R+, superlinear at zero and sublinear at infinity, and aλ,μ(t)aλ,μ(t) is a T  -periodic and sign indefinite weight of the form λa+(t)−μa−(t)λa+(t)−μa−(t), with λ,μ>0λ,μ>0 and large.