Physics Letters A 169 (1992) 25—30 PHYSICS ET North-Holland L TE R S A Detecting chaos with local associative memories J. Jim~nez, J. Moreno, G.J. Ruggeri Grupo de Dindmica de Sistemas Complejos e Inteligencia ArtifIcial, Departamento de Fisica, Facultad de Ciencias, Universidad Central de Venezuela, AP 52120, Caracas 1050-A, Venezuela and A. Marcano Departamento de Fisica, Facultad de Ciencias, Universidad Central de Venezuela, AP 52120, Caracas 1050-A, Venuzuela and Fundación Instituto de Ingenieria, Caracas, Venezuela Received 30 March 1992; revised manuscript received 8 June 1992; acceptedfor publication 11 July 1992 Communicated by AR. Bishop A technique to discriminate complex signals associatedwith deterministic chaos from those of random origin is presented. The method applies a new, computationally efficient, prediction procedure based on the associative memory concept. This procedure was used as a method to model non-linear series. Its performance was analyzed for several time series, including simulated nu- merical data from the logistic map and from the Mackey—Glass delay equation. Also experimental data from a dripping faucet in a chaotic regime and from low-temperature thermal fluctuations of the voltage across a resistor are considered. The problem of deciding if the information avail- in order to see if convergence can be achieved. able about the evolution of a system of interest cor- Among the main drawbacks of these methods we responds to an underlying stochastic process or if, on should mention that they require great volumes of the contrary, it reflects the existence of a determin- data [81, which in many interesting cases are not istic chaotic mechanism is an important one. This is available. Another set of methods avoid these prob- a problem which has received recently a great deal lems through the construction of deterministic of attention not only in physics but also in other areas models which satisfactorily reproduces the available such as epidemiology, economy and meteorology [1— information [2,4—61. If the fitting achieved through 6]. In all these cases it is crucial to discriminate, for the model can be judged substantially better than the instance, between random measurement errors and ones obtained using probabilistic models, it is rea- a possible chaotic dynamics controlling the process sonably assumed that a deterministic dynamics gov- under study. Only after a correct decision has been ems the process under study. made concerning these two broad possibilities one is In this paper we present a method which allows in a position to choose the appropriate mathematical one to discriminate if the phenomenon under study tools to analyze the problem at hand, is controlled by some deterministic mechanism or if, A number of techniques have been proposed to on the contrary, it is better to assume the existence distinguish between randomness and determinism, of some underlying stochastic process. The method In a set of them one tries to estimate some dynamical works well even for time series as short as 500 points. invariants which could exist, such as generalized As it has been pointed out by some authors [5,91 fractal dimensions of a hypothesized strange attrac- the smallest embedding dimension for which good tor [71. These programs require one to compute predictions can be made is an upper bound on the those invariants for different embedding dimensions number of degrees of freedom. Additionally Wales 0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved. 25