We present a computer assisted method for proving the existence of globally attracting fixed points of dissipative PDEs. An application to the viscous Burgers equation with periodic boundary conditions and a forcing function constant in time is presented as a case study. We establish the existence of a locally attracting fixed point by using rigorous numerics techniques. To prove that the fixed point is, in fact, globally attracting we introduce a technique relying on a construction of an absorbing set, capturing any sufficiently regular initial condition after a finite time. Then the absorbing set is rigorously integrated forward in time to verify that any sufficiently regular initial condition is in the basin of attraction of the fixed point.

We propose an efficient and generic algorithm for rigorous integration forward in time of partial differential equations written in the Fourier basis. By rigorous integration we mean a procedure which operates on sets and return sets which are guaranteed to contain the exact solution. The presented algorithm generates, in an efficient way, normalized derivatives which are used by the Lohner algorithm to produce a rigorous bound. The algorithm has been successfully tested on several partial differential equations (PDEs) including the Burgers equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation. The problem of rigorous integration in time of partial differential equations is a problem of large computational complexity and efficient algorithms are required to deal with PDEs on higher dimensional domains, like the Navier-Stokes equation. Technicalities regarding the various optimization techniques implemented in the software used in this paper will be reported elsewhere.

Abstract. We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF), which is derived from an overapproximation of the gradient of the solution-flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a well-defined region, thereby allowing LRT to compute a reachtube with a ball-overestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of well-established benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools.

We prove the existence of globally attracting solutions of the viscous Burgers equation with periodic boundary conditions on the interval for some particular choices of viscosity and nonautonomous forcing. The attracting solution is periodic if the forcing is periodic. The method is general and can be applied to other similar partial differential equations. The proof is computer assisted.Theorem 6.15. Let J = R, Z ⊂ J × X m , and T ⊂ Y m such that Z t is convex for t ∈ R. Assume that conditions (C1)-(C3) and (D) are satisfied on Z ⊕ T for a compatible norm | · |. Assume that Z is a trapping isolating segment for (64). GLOBALLY ATTRACTING SOLUTIONS FOR VISCOUS BURGERS EQUATION 809+/− i returns supremum/infimum of the real part and the imaginary part of (W 0 ) i , respectively.