Computing Domains of Attraction for Planar Dynamics

Abstract

Abstract. In this note we investigate the problem of computing the domain of attraction of a flow on R 2 for a given attractor. We consider an operator that takes two inputs, the description of the flow and a cover of the attractors, and outputs the domain of attraction for the given attractor. We show that: (i) if we consider only (structurally) stable systems, the operator is (strictly semi-)computable; (ii) if we allow all systems defined by C 1 -functions, the operator is not (semi-)computable. We also address the problem of computing limit cycles on these systems.Many problems about dynamical systems (DSs) are concerned with their long term behavior. For example, given some trajectory, where will it end up? Which are the invariant sets of a DS? Which are its attractors?Recently, with the advent of increasingly powerful digital computers, numerous new ideas and concepts related to these question have appeared (e.g. sensitive dependence on initial conditions, chaos, strange attractors, Mandelbrot set). However it is interesting to note that the computer, while being an invaluable tool to get some intuition about a DS, is rarely used to prove results. Usually the formal analysis of DSs is done analytically (but often relies on information provided by numerical simulations), using heavy mathematics with little reliance on the computer. A notable exception is the proof that the Lorenz strange attractor exists and is robust under small perturbations [1], [2].One of the reasons for this phenomenon is that computers introduce truncation errors which, in conjunction with other properties such as sensitive dependency on initial conditions, is likely to produce simulated trajectories that cannot be proved accurate. Of course, there are many results exhibiting that these simulations are valuable; the foremost of such results is perhaps the Shadowing Lemma [3], [4]. However the accuracy of a particular simulation, especially when we are interested in global properties, can usually be put into question.In this paper we deal with a particular type of the problems mentioned above: is it possible to conceive a computer program that, given an input that describes a dynamical system (DS) as well as an attractor of this DS, computes (rigourously) the basin of attraction of the given attractor?

Computability and Dynamical Systems

“…For example, in [25], we show that the operator F is strictly semi-computable if we consider only structurally stable systems; on the other hand, F fails to be semi-computable if all C 1 systems are permitted, where F is the operator that takes two inputs, the description of the flow and a cover of an attractor, and outputs the domain of attraction for the given attractor. In [25] we also demonstrate how to decide whether or not there are limit cycles, and furthermore how to compute hyperbolic ones when given a compact set without an equilibrium point (equilibrium points are computable from f ). As a consequence, all kinds of hyperbolic attractors in the plane can be computed, though their domains of attraction cannot.…”

“…As a matter of fact, our results about computability of domains of attraction presented in [25] are based on some of these techniques and provide a good example of how control theory may be of use in dynamical systems.…”

“…If some robustness to errors is allowed (in a weaker form than that required by structural stability), then usually the reachability problem is decidable as was mentioned in [28], but previously seen in other classes [38], [39], [23]. This fact was used in [25]. The idea is to cover some region with a grid of points (more precisely, small squares) and follow the individual evolution of each point to get an estimate of the domain of attraction.…”

Computability in planar dynamical systems

“…We will establish results about important structures (equilibrium points, periodic orbits, basins of attraction) for both the general case of (1), and the restricted case of structurally stable system. This paper is an improved journal version, with new results, of the conference paper [GZ09].…”