PHYSICAL REVIE% A VOLUME 14, NUMBER 6 Kolmogorov entroyy anti numerical exyeriments DECEMBER 1976 Giancarlo Benettin Istituto di Fisica dell'Universita, and Gruppo Wazionale di Struttura della Materia del Consiglio Nazionale delle Ricerche, Padova, Italy LMgi GRlgani Istituto di Matematica and Istituto di Rsica dell'Universita, Milano, Italy Jean-Marie Strelcyn Departement de Mathematiques Centre Scienti fique et Poly technique, Universite Paris-Nord, Paris, France (Received 8 June l976) Numerical investigations of dynamical systems allow one to give estimates of the rate of divergence of nearby trajectories, by means of a quantity which is usually assumed to be related to the Kolmogorov (or metric) entropy. In this paper it is shown first, on the basis of mathematical results of Oseledec and Piesin, how such a relation can be made precise. Then, as an example, a numerical study of the Kolmogorov entropy for the Henon-Heiles model is reported. I. INTRODUCTION In recent years many attempts have been made in order to investigate the so-called stochasticity properties of dynamical systems, in particular Hamiltonian systems, by numerical computations. However, stochasticity is generally defined and tested in a rather qualitative may, and the connec- tion between the empirical parameters introduced to describe it and rigorous theoretical concepts is fa, r from being clear. Qne of the most powerful empirical tools has always been the study of the divergence of nearby trajectories in phase space, Such a method allows one to define a quantitative parameter (the "en- tropylike quantity"), which is supposed to be strictly related to the Kolmogorov (or metric) entropy for associated f lorn. ' 4 The aim of the present paper is to analyze this entropylike quantity, deriving its precise connec- tion with the metric entropy, and to explain cer- tain properties observed in the numerical compu- tations. This connection turns out to be particu- larly simple for the case of Hamiltonian systems with two degrees of freedom. As an example, for one of them, the well-known Henon-Heiles mod- el, '' we compute the entropylike quantity and test its properties; moreover, we are able to draw a, tentative curve for the entropy itself as a function of energy. In Sec. II me collect first the necessary mathe- matical tools, i.e. , the results of Oseledec' and the fundamental results of Piesin. '" [We are very grateful to Dr. A. B. Katok (Moscow) for the communication of the latter results. ] We then re- call the definition of the entropylike quantity and explain its empirically observed properties. The numerical example for the Henon-Heiles model is treated in Sec. III. This paper has been written, as far as possible, in a self -contained way; however, a certain famil- iarity mith ergodic theory and in particular with entropy is necessary (see, for example, Hefs. 11 and 12). The elementary notions on differentiable manifolds used here can be found, for example, in Ref. 13. II. THEORETKAL ANALYSIS OF THE NUMERKAL COMPUTATIONS A. Mathematical preliminaries: Lyapunov characteristic numbers and entropy Let us give first the main definitions and fix the notation. Let M be a differentiable, n-dimensional, com- pact, connected Riemannian manifold of class C'. If x&M, the tangent space to M at x and the norm induced in it by the Riemannian metric on M will be denoted by E„and ~[ . ~[, respectively. Let X be a vector field of class C' defined on M and tT'j the flow induced by X, i.e. , for any t let T'x=x(t), where (x(t)) is an integral curve of the vector field X such that x(0) =x. The tangent mapping of E, onto E~~„ induced by the diffeomorphism T' mill be denoted by dT„'. It will also be assumed that the flow (T'j preserves a normalized measure p which is equivalent to the Lebesgue measure on M and whose density in local coordinates is of class C', i.e. , that the flow tT']admits an inte. gral invar- iant of order n and class C'. The following theorems A and 8, mhich partially summarize theorems 2 and 4 of Ref. 8, are the basis for all further considerations of the present paper.