Time-to-return functions in a two-dimensional Hamiltonian system

Time-to-return functions in a two-dimensional Hamiltonian system, André Zegeling (Guangxi Normal University, China)

Monday, 23. May 2022, 15:00

Abstract: In this talk, I will discuss the relation between the solutions of the boundary value problem $frac{d^2x(t)}{dt^2}+lambda f(x(t))=0$, with $x(0)=x(1)=Ainmathbb{R}$ and time-to-return functions $T_{n+frac{1}{2}}(h)$, with $ninmathbb{N}$ which can be regarded as generalizations of the period function $T(h)$ for a period annulus in an autonomous system. I will give a short historical overview of the methods used to study time-to-return functions. As an example to illustrate the concept, I will discuss the simplest possible case $f(u)=u(u+1)$. It is well-known that the period function $T(h)$ in this case is monotonically increasing. However, for the boundary value problem, other solution types exist for which the corresponding time-to-return functions $T_{n+frac{1}{2}}(h)$ are not monotonic. In some cases, situations arise with at least one local maximum and one local minimum (in the literature referred to as S-shaped bifurcations). For these cases it is an open problem, even for the quadratic Hamiltonian, to prove that no other local maxima or minima occur. At the end of the presentation, I will give a list of open problems for other types of autonomous differential equations.