×

Twist character of the least amplitude periodic solution of the forced pendulum. (English) Zbl 1189.37064

Summary: We will derive some twist criteria for the periodic solution of a periodic scalar Newtonian equation using the third order approximation. As an application to the forced pendulum \(\ddot x+\omega^2 \sin x=p(t)\), we will find an explicit bound \(P(\omega)\) for the \(L^1\) norm, \(\| p\| _1\), of the periodic forcing \(p(t)\) using the frequency \(\omega\) as a parameter such that the least amplitude periodic solution of the forced pendulum is of twist type when \(\| p\| _1 < P(\omega)\). The bound \(P(\omega)\) has the order of \(O(\omega^{1/2})\) when \(\omega\) is bounded away from resonance of orders \(\leq 4\) and \(\omega \to +\infty\).

MSC:

37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
34D20 Stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI