Computing Domains of Attraction for Planar Dynamics

In this note we investigate the problem of computing the domain of attraction of a flow on R 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-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? Since there are many open questions about general classes of dynamical system, we restrict ourselves to a well-studied case, the continuous-time DS defined on the plane R by x′ = f(x), (1) where f : R → R and t is the independent variable. Some techniques introduced on this paper are based on [5]. They are essentially adaption and enhancements, that allow to correct some results of [5]. 1 Differential equations Here we give a summary of results concerning the ODE (1). For more details, the reader is referred to [6], [7], [8]. Definition 1. Let φ(t, x0) denote the solution of (1) corresponding to the initial condition x(0) = x0. The function φ(·, x0) : R → R is called a solution curve, trajectory, or orbit of (1) through the point x0. 1. A point y is called an equilibrium point of (1) if f(y) = 0. An equilibrium point y0 is called hyperbolic if none of the eigenvalues of the gradient matrix Df(y0) of f at y0 has zero real part. 2. An equilibrium point x0 of (1) is called stable if for any ε > 0, there exists a δ > 0 such that |φ(t, x̃)− x0| < ε for all t ≥ 0, provided |x̃− x0| < δ. Furthermore, x0 is called asymptotically stable if it is stable and there exists a (fixed) δ0 > 0 such that limt→∞ φ(t, x̃) = x0 for all x̃ satisfying |x̃− x0| < δ0. Given a trajectory Γx0 = φ(·, x0), we define the positive half-trajectory as Γ x0 = {φ(t, x0)|t ≥ 0}. When the context is clear, we often drop the subscript x0 and write simply Γ and Γ. It is not difficult to see that if x0 is an equilibrium point of (1), then φ(t, x0) = x0 for all t ≥ 0. It is also known that if all eigenvalues of Df(x0) of an hyperbolic equilibrium point x0 are negative, then x0 is asymptotically stable; in this case x0 is called a sink. While many results about the long term dynamics of (1) focus on fixed points, especially hyperbolic ones, since this is the easiest case to tackle, fixed points are not the sole objects to which trajectories converge as we now will see. Definition 2. 1. A point p ∈ R is an ω-limit point of the trajectory φ(·, x) of the system (1) if there is a sequence tn →∞ such that limn→∞ φ(tn, x) = p. Definition 3. The set of all ω-limit points of the trajectory Γ is called the ωlimit set of Γ ; written as ω(Γ ) or ω(x) if Γ = φ(·, x). Definition 4. A cycle or periodic orbit of (1) is any closed solution curve of (1) which is not an equilibrium point. A cycle Υ is stable if for each ε > 0 there is a neighborhood U of Υ such that for all x ∈ U , d(Γ x , Υ ) < ε. A cycle Υ is asymptotically stable (we also say that Υ is a limit cycle) if for all points x0 in some neighborhood U of Υ one has limt→∞ d(φ(t, x0), Υ ) = 0.