ORBITADOR: A tool to analyze the stability of periodical dynamical systems

Abstract

Tool Presentation: We present ORBITADOR, a tool for stability analysis of dynamical systems. ORBITADOR uses a method that generates a bounded invariant set of a differential system with a given set of initial conditions around a point x0 to prove the existence of a limit cycle. 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 with adding a small bounded uncertainty. This uncertainty allows using an approximate value T of the exact period. We successfully applied ORBITADOR to several classical examples of periodical systems.

Qualitative Reasoning and Cyber-Physical Systems: Abstraction, Modeling, and Optimized Simulation

“…We will also improve the capacities of our qualitative models by adding new analyses on the different equations. For example, using works on semi-qualitative reasoning [12], [27], we think that with a better analysis of the flow equations F m,i , it will be possible to pilot more complex simulations integrating notions of uncertainty. If we can compute the zones of the state space where each of the F m,i is a contraction (i.e., where ∃k ∈ (0, 1) such that F m,i is k-Lipschitz), the management of uncertainty will become easier as we will know which areas of the state space can tolerate uncertainties (if a function f is a contraction, an initial uncertainty on f will only reduce with time).…”

Asymptotic error in Euler’s method with a constant step size

“…y k converges to 0. Using the software ORBITADOR [24], one calculates that L ď 0.5 and λ ă 0 on D (see Figure 1). Since }gpy k q} Ñ 0 (Figure 2), it follows by Theorem 2: δ µ0 pkhq Ñ 0 and |y k ´xpkhq| Ñ 0 as k Ñ 8 (see Figures 3 and 4).…”

“…Let us compute its minimum w ˚" pw 1 , w 2 q by gradient descent, with initial point w 0 " pw 0 1 , w 0 2 q " p9, 10q. Using the software ORBITADOR [24], we find L ă 7 and λ ď ´0.187 (see Figure 5). For h " 0.2 (which satisfies: h ă 2{L), it follows by Proposition 4 that }∇f pwpkhqq} Ñ 0 and C " L}∇f pwpkhqq} Ñ 0 respectively, as k Ñ 8…”

Qualitative Tendencies for Hybrid System Simulation

“…More advanced simulation methods such as the works on flow-pipe propagation [12] or closed expansivity study [13] tried to correct some drawbacks of the propagation of intervals. Finally, Tiwari opened another branch of qualitative modeling in [14], more adapted to systems represented by ODEs.…”