Computability, noncomputability and undecidability of maximal intervals of IVPs

Abstract

Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solutions for the initial-value problemwhere E is an open subset of R m+1 , f is continuous in E and (t 0 , x 0 ) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f and (t 0 , x 0 ) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the "degree of noncomputability" by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.

Computational bounds on polynomial differential equations

“…This result is shown using methods which differ from those employed in [GZB07]. This result was already stated in [GBC07], but we now present its proof.…”

“…To address this kind of questions for IVPs, we use the computable analysis approach [PER89], [Ko91], [Wei00], which we presented in the end of Section 2. Using that approach, it was shown in [GZB07] that given an analytic IVP (17), defined with computable data, its corresponding maximal interval may be non-computable.…”

“…Motivated by the non-computability result obtained in [GZB07], this latter paper also addresses the following problem: while it is not possible to compute the maximal interval of (17) is it possible to compute some partial information about it? In particular, is it possible to decide if this maximal interval is bounded or not?…”

“…In [GZB07], it was shown that for the general class of analytic IVPs, the boundedness problem of the maximal interval is undecidable. Here we will deepen this result: we will show that the boundedness problem is still undecidable for PIVPs of degree greater or equal than 56 with parameters in Q[π].…”

Computing Domains of Attraction for Planar Dynamics

“…Before closing this section, we present some useful results from [14]. Recall that a function f : E → R m , E ⊆ R l , is said to be locally Lipschitz on E if it satisfies a Lipschitz condition on every compact set V ⊂ E. The following definition gives a computable analysis analog of this condition.…”