**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 63, 1 (1994)

pp. 25-38

DISCRETIZATION AND MORSE-SMALE DYNAMICAL SYSTEMS ON PLANAR DISCS

B. M. GARAY

**Abstract**.
In a previous paper Ref. 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 Ref. 2, Ref. 18 on structural stability. For methods of order $p$, the conjugacy is $ O (h^p )$-near to the identity. The paper ends with some general remarks on similar problems.

**AMS subject classification**.

**Keywords**.
Discretization, planar Morse-Smale systems, structural stability

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