Boundedness of solutions for a class of oscillating potentials without the twist condition
The authors prove that the second order differential equation \[ x''+x^{2n+1}f(x)+p(t)=0 \] with \(p(t+1)=p(t)\), \(f(x+T)=f(x)\) smooth and \(f(x)>0\), possesses Lagrangian stability by using a variant of Moser's twist theorem. The idea to prove the boundedness for all solutions is as follows. By means of several transformations, the system is transformed into a small perturbation of an integrable system. The Poincaré map of this system is close to a twist map. Using a variant of Moser's twist theorem, the authors prove the existence of large invariant curves diffeomorphic to circles and surrounding the origin in the \((x, \dot{x})\) plane. Every such curve is the base of a time-periodic invariant flow cylinder, which confines the solutions in its interior and therefore leads to boundedness of solutions.
scientific article; zbMATH DE number 5859375