A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems

Abstract

Abstract. We present a numerical method to prove certain statements about the global dynamics of infinitedimensional maps. The method combines set-oriented numerical tools for the computation of invariant sets and isolating neighborhoods, the Conley index theory, and analytic considerations. It not only allows for the detection of a certain dynamical behavior, but also for a precise computation of the corresponding invariant sets in phase space. As an example computation we show the existence of period points, connecting orbits, and chaotic dynamics in the Kot-Schaffer growth-dispersal model for plants.

Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square

“…The above theorem is the main tool for establishing strict topologically selfconsistent a priori bounds in section 2. Similar results for one-dimensional base domains can be found in [4,6,7].…”

“…The modes are the eigenfunctions of (5) linearized around u ≡ μ. In case of the unit square the modes are given by w ij = φ i,j given in (7). Kielhöfer shows in [13] that the continua for modes of the form w kk and w k0 + w 0k connect those two bifurcation points and are separated from each other (see Theorem 1.3).…”

“…However, for higherdimensional domains very little is known, and in fact, given current techniques there appears to be scant hope that the problem can be resolved by purely analytic techniques. Therefore, we have chosen to pursue this problem using newly developed methods that lead to rigorous numerical proofs of the existence of equilibria for partial differential equations [4,6,7,22]. As might be imagined, a proper description of our results is fairly technical and thus is presented in section 1.…”

Recent development in rigorous computational methods in dynamical systems

“…Since the graph G and the weight function w are both computed using interval arithmetic for the evaluation of f and f , no substantial change in the algorithms is necessary to obtain results valid for all f a for a in some (small) interval [a 1 , a 2 ]. Figure 8 shows the result of these calculations conducted with the software referred to in [15] to obtain explicit lower bounds for the expansion constant λ(a) in (11) for all a ∈ [1.5, 2], and thus proving Theorem 6.1 for those values of a for which the computed value of λ(a) is positive.…”

“…Therefore, we purposely omit some subjects and focus on the selected ones. Moreover, we would like to point out that in addition to the discussed results, there are also other important works in similar spirit which we do not mention here, for example, [4], [18], [11]. See also a survey article of K. Mischaikow [28] Aknowledgement.…”