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

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.