Discretization and Morse–smale Dynamical Systems on Planar Discs

In a previous paper [12], we have shown that locally, in the vicinity of hyperbolic equilibria of autonomous ordinary differential equations, the time-h-map of the induced dynamical system is conjugate to the h-discretized system i.e. to the discrete dynamical system obtained via one-step discretization with stepsize h. The present paper is devoted to Morse-Smale dynamical systems defined on planar discs and having no periodic orbits. Using elementary extension techniques, we point out that local conjugacies about saddle points can be glued together: the time-h-map is globally conjugate to the h-discretized system. This is a discretization analogue of the famous Andronov-Pontryagin theorem [2], [18] on structural stability. For methods of order p, the conjugacy is O(hp)-near to the identity. The paper ends with some general remarks on similar problems. 0. Introduction and the Main Result Let | · | denote the Euclidean norm on R and consider the unit disc D = {x ∈ R ∣∣ |x| ≤ 1}. The boundary of D is denoted by ∂D. Assume that N is an open neighbourhood of D and that, for some positive integer p, the function f : N → R is of class C (with all derivatives bounded — the norm in C is defined by |f |p+1 = max{sup {f (x) | x ∈ N} | j = 0, 1, . . . , p+ 1}) and satisfies (i) x · f(x) < 0 whenever x ∈ ∂D; (ii) ẋ = f(x) has a finite number of equilibrium points in D, all hyperbolic; (iii) alphaand omegalimit sets of trajectories in D are equilibria; (iv) there are no saddle connections in D. Actually, conditions (i)–(iv) concern the local dynamical system Φ induced by the differential equation ẋ = f(x), x ∈ N , and not function f itself. Geometrically, (i) means that D is positively invariant for Φ and ∂D is transversally cutted by the trajectories. By (ii), the equilibria of ẋ = f(x) in D can be classified as Received June 28, 1993. 1980 Mathematics Subject Classification (1991 Revision). Primary 65L05; Secondary 58F09.