Algorithms for Constructing Isolating Sets of Phase Flows and Computer-Assisted Proofs with the Use of Interval Taylor Models

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

“…In this paper, we consider one algorithmic method for accelerating the CAP of the existence of periodic solutions. It can be noted that the estimates of solutions obtained with the help of low-order TMs (for example, the second order one from the example in [9]) at the initial interval of integration time can be sufficiently close to the real boundaries of the solution pencil; this allows us to show that the neighborhood of the cycle falls into itself in the topological approach. However, with further integration, these estimates begin to diverge exponentially.…”

“…In practice, the numerical errors do not allow the use of any arbitrarily small time intervals, and the intervals of the order of one tenth of the cycle period turn out to be practically suitable. Nevertheless, it turns out that even such a decrease in the lengths of the considered time intervals permits one to prove the existence of a periodic solution in the Van der Pol oscillator using only a second-order TM, as was shown in the example in [9].…”

“…The notation, definitions, assertions, and algorithms necessary for working with interval Taylor models are presented in [8,9]. In this section, we recall some of the most important of them, and also introduce several additional operations on TM and the corresponding algorithms that we will need further.…”

“…In our previous papers [8,9], CAP was considered as a tool for proving the existence of periodic trajectories in autonomous ODE systems. Both in our papers and in the papers of other authors using this approach to systems of ODEs (for example, [10]), validated estimates are constructed for solutions and families of solutions of these systems.…”

“…Unfortunately, in the transition to highdimensional systems, which ultimately are the subject of our consideration, such algorithms are unusable, because the complexity is too high. In [9], we tried to get around this difficulty by using the ideas applied in the topological approach to proving the existence of cycles (see, e.g., [10] and the references therein). When using this approach, it is not required to investigate the behavior of the solution over a long period of time; instead, it is sufficient to show that the entire neighborhood of the cycle falls into itself under the action of the phase flow in an arbitrarily short time.…”