Monotonicity of the period function and chaotic dynamics in a class of singular ODEs
This paper deals with a periodically perturbed singular Duffing equation of the type \[ u'' - \frac{1}{u^\alpha} = e(t), \] where \(\alpha \geq 1\) and \(e: \mathbb{R} \to \mathbb{R}\) is a locally integrable and periodic function. More precisely, the authors prove the existence of chaotic dynamics (in the coin-tossing sense), as well as the presence of infinitely many subharmonic solutions, when the forcing term \(e(t)\) is a stepwise function with large period. It is also pointed out that this result is stable under perturbations, meaning that the result still holds if the forcing term \(e(t)\) is close to a stepwise function as above, and a forcing term of the type \(cu'\), with \(c\) small, is introduced in the equation. The proof relies on the theory of topological horseshoes, in the framework of a so-called ``Linked Twist Maps'' geometry, along a line of research extensively developed by both the authors in the last years. As an auxiliary result for the proof, the authors also prove, using the celebrated Chicone's theorem, the strict monotonocity of the period map in the autonomous case, a fact which can be considered of independent interest. The paper contains several numerical simulations illustrating the complex behavior of the Poincaré map. Moreover, the final section provides an interesting discussion about the dynamics of the equation from various points of view.
scientific article; zbMATH DE number 7442658
scientific article