Generation of bounded invariants via stroboscopic set-valued maps: Application to the stability analysis of parametric time-periodic systems

TL;DR

This paper introduces a method to generate bounded invariants for differential systems using stroboscopic set-valued maps, enabling stability analysis of parametric time-periodic systems, exemplified on Van der Pol's system.

Contribution

The paper presents a novel approach to construct bounded invariants as tubes around approximate solutions, applicable to parametric systems for stability analysis.

Findings

Invariant tubes are effectively constructed around approximate solutions.

The method guarantees solutions do not converge to equilibrium points.

Application to Van der Pol's system demonstrates practical utility.

Abstract

A method is given for generating a bounded invariant of a differential system with a given set of initial conditions around a point $x_0$. This invariant has the form of a tube centered on the Euler approximate solution starting at $x_0$, which has for radius an upper bound on the distance between the approximate solution and the exact ones. The method consists in finding a real $T>0$ such that the "snapshot" of the tube at time $t=(i+1)T$ is included in the snapshot at $t=iT$, for some integer $i$. In the phase space, the invariant is therefore in the shape of a torus. A simple additional condition is also given to ensure that the solutions of the system can never converge to a point of equilibrium. In dimension 2, this ensures that all solutions converge towards a limit cycle. The method is extended in case the dynamic system contains a parameter $p$, thus allowing the stability analysis of the system for a range of values of $p$. This is illustrated on classical Van der Pol's system.