Generation of bounded invariants via stroboscopic set-valued maps: Application to the stability analysis of parametric time-periodic systems
Abstract
A method is given for generating a bounded invariant of a differential system with a given set of initial conditions around a point x0. This invariant has the form of a tube centered on the Euler approximate solution starting at x0, 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.