PHYSICAL REVIEW A VOLUME 43, NUMBER 10 15 MAY 1991 Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series G. Gouesbet Laboratoire d'Energetique des Systemes et Procedes, Institut National des Sciences Appliquees de Rouen, Boite Postale 08, 76131 Mont Saint Aignan CEDEX, France (Received 4 September 1990; revised manuscript received 4 December 1990) We emphasize that the ordinary differential equations of a continuous dynamical system, or at least of equivalent systems, can be reconstructed from numerical scalar time series. Methods are exemplified in the case of a strange, chaotic attractor generated by a mathematical model, namely, a Rossler band. Resultant validations rely (i) qualitatively on comparisons between original and reconstructed phase portraits, and (ii) quantitatively on comparisons between generalized dimen- sions D~ of original and reconstructed attractors. Some of the many lines of research offered by the presented results are discussed to stress potentialities of this kind of reconstruction. I. INTRODUCTION There is now a great interest in the study of the theory of nonlinear dissipative dynamical systems, with applica- tions to several miscellaneous fields, including, for in- stance, mechanics of structures, hydrodynamics, general physics, chemistry, biology, ecology, epidemiology, and economics. Indeed, the number of actual and potential applications of nonlinear dynamics is tremendously big because most systems existing in nature may be described by mathematical models such as nonlinear maps and Rows. To gain a background knowledge on such topics, the reader may consult textbooks such as those by Guck- enheimer and Holmes, ' Thompson and Stewart, Deva- ney, or Berge, Pomeau, and Vidal, and also increasingly prolific literature. Particular attention is paid to systems producing strange and chaotic attractors, the words strange and chaotic here referring to metric properties, i.e. , fractal di- mensions, and to dynamical properties, i.e. , sensitivity to initial conditions, respectively. In such cases, dissipative phenomena previously attacked in terms of high- dimensional phase spaces and/or of noise sources may be actually understood in low-dimensional phase spaces as the consequence of deterministic chaos. When studying experimental systems, a single scalar variable is usually recorded versus time. Therefore a great deal of effort has been devoted to the quantitative characterization of un- derlying attractors when our knowledge of the system is limited to a recorded numerical scalar time series. There exists a host of invariants to quantify attractors, such as natural measure, pointwise and partial dimen- sions, Hausdorff dimension, or Lyapunov exponents (see Refs. 5 — 7, for instance). Now the most popular quanti- ties might be generalized dimensions D, generalized en- tropies K, and their associated singularity spectra. Also, there are now several well-known algorithms available to derive these quantities from numerical scalar time series relying, for instance, on fixed-radius or fixed-mass ap- proaches, or on the determination of unstable periodic orbits that are dense in the attractor (see Refs. 8 — 19, for instance, and references therein). The above evaluations are purely numerical and re- quire the reconstruction of attractors in phase spaces of dimensions n usually much bigger than the minimal di- mension no. There is one theoretical and one practical reason, implying that n must usually be bigger than no. Theoretically, we have a theorem of Mani and Takens ' stating that the points of the attractor may be parametrized by n real coordinates if n nT in which nT is given by the Takens criterion n T = 2D + 1. Here, D can be taken as being the Hausdorff dimension DH or the capacity Dz of the set. One recalls that, for every com- pact set S, DH(S) ~Dz(S). In practice, D is often taken as being the correlation dimension D2 of the attractor, which may be rather easily evaluated. A rough but not too misleading evaluation is to take D as being the dimen- sion no of a minimal embedding phase space. Takens's criterion, however, gives a sufFicient condition because many manifolds and attractors contained in them having a typical dimension D can be embedded in less than nT- dimensional phase spaces. Although this paper is devot- ed to fIows, we stress that the above discussion extends to the case of maps. Practically however, for a phase space of dimension n satisfying Takens s criterion, the quality of the recon- struction is not warranted. Sophisticated procedures like singular value decomposition or redundancy analysis may be used to approach the best reconstruction, but there also exists a simple pragmatic method relying on the observation that the value of the so-called window length is determinant in the quality of the projection pro- cess. ' ' We then find that the dimension n of the phase space in which the attractor is reconstructed must be in practice much bigger than nT to avoid biased evaluations such as severe underestimations. For instance, we may consider an attractor generated by a simple model of thermal lens oscillations. The original phase space is of dimension 3 and generalized dimensions D of the attrac- tor are equal to about 2. Therefore, nT is about 5. The optimal phase-space dimension n relying on the 43 5321 1991 The American Physical Society