DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 13, Number 4, November 2005 pp. 1057–1067 NUMERICAL PERIODIC ORBITS OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS Nicola Guglielmi Dipartimento di Matematica Pura e Applicata Universit` a de L’Aquila I-67100 L’Aquila, Italy Christian Lubich Mathematisches Institut Universit¨at T¨ ubingen D-72076 T¨ ubingen, Germany Abstract. This paper deals with the long-time behaviour of numerical solu- tions of neutral delay differential equations that have stable hyperbolic periodic orbits. It is shown that Runge–Kutta discretizations of such equations have attractive invariant closed curves which approximate the periodic orbit with the full order of the method, in spite of the lack of a finite-time smoothing property of the flow. 1. Introduction. It is a basic question in the dynamics of numerical methods for evolution equations as to whether invariant sets of the equation have their counter- part in the discretization, and what order of approximation, if any, there is between the invariant sets of the continuous dynamical system and its discretization. These questions have been studied for many kinds of evolution equations, invariant ob- jects, and numerical methods; see, e.g., the reviews and numerous references in [13, 16]. In particular, for ordinary differential equations with a hyperbolic periodic orbit, it has been shown in [2, 3, 5] and [16, Sect. 6.6] that one-step methods have invariant closed curves which approximate the periodic orbit. For delay differential equations such results are shown in [6, 11], but no results in this direction have so far been given for neutral delay differential equations. Here, we analyze Runge-Kutta discretizations of systems of neutral delay differ- ential equations of the form x 0 (t)= f ( x(t),x(t - τ ) ) + Ax 0 (t - τ ), t ≥ 0, (1) with general (not necessarily consistent) initial functions. The particular form with a constant matrix A (of spectral radius < 1) is chosen for convenience, not as a necessity. We assume that the equation has a stable hyperbolic periodic orbit and ask for its approximation by an attractive invariant curve of the numerical method. The analysis of the analogous problem for delay differential equations in [11] used 2000 Mathematics Subject Classification. 65L05. Key words and phrases. Neutral delay differential equations, periodic orbit, numerical solution, Runge–Kutta methods, attractive invariant curves. 1057