Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

Abstract

Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integrating ODEs over large time intervals, the interval residual included in the TMs begins to grow exponentially and becomes the dominant part of the estimate of the solution pencil, making it practically unusable. To eliminate this deficiency, the creators of the TM—K. Makino and M. Berz—proposed the idea of so-called “shrink wrapping.” We formalize the original algorithm within the framework of the TM definitions we have adopted and propose our own version of the “shrink wrapping,” more accurately adapted to the problem of the computer-aided proof of the existence of periodic trajectories.