JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. A9, PAGES 15,241-15,254, SEPTEMBER 1, 1993 Fractal Behavior of Cosmic Ray Time Series' Chaos or Stochasticity? M. AGLIETTA, 1,2 B. ALESSANDRO, 2 F. ARNEODO, 2 L. BERGAMASCO, 2,3A. CAMPOS FAUTH,TM C. CASTAGNOLI, 1,2,3 A. CASTELLINA, 1,2 C. CATTADORI, 5 A. CHIAVASSA, 2 G. CINI, 2,3 a. D'ETTORRE PIAZZOLI, 6 W. FULGIONE, •,2 P. GALEOTTI, 2,3P. L. GHIA, •,2 G. MANNOCCHI, 1,2 C. MORELLO, •,2 G. NAVARRA, 2,3A. R. OSBORNE, •,3 L. RICCATI, 20. SAAVEDRA, 2,3M. SERIO, 2,3G. C. TRINCHERO, 1,2 P. VALLANIA, •,2 AND $. VERNETTO •,2 This paper presents resultson the fractal and statistical behaviorof cosmic ray time series detected in an air showerexperimentlocatedat 2000-maltitude above the underground Gran Sasso Laboratory, Italy. We consider singleparticles (muons), corresponding to primary energiesof ->10 GeV, and air showers, corresponding to primary energiesof ->80 TeV. For all time series the analysisindicates a clear stochastic monofractal, non-Gaussian character;comparing theseresultswith thoseobtainedfor undergroundmuons and for neutron monitors, we conclude that these properties likely belong in general to cosmic ray time series,irrespective of the nature of the particles and the energiesof their progenitors. In particular, the air shower time series from high-energy primaries have a fractal dimension larger than the single-muon time series originating from low-energyprimaries. 1. INTRODUCTION The understanding and description of the many processes which influence the flux of galactic cosmic rays throughout their journey from their source to an Earth-baseddetector is still far from being achieved. Among the open issues is whether the cosmic ray propagation mechanismsmay be better described by a relatively small set of differential equationsor a statisticalapproach. Over the past decadethe field of dynamical systems has emphasized that the former may be associated with the presenceof deterministicchaos (with the possible existenceof a strange attractor), while the latter is related to the study of stochastic systems. Dynam- ical systems theory has been the sourceof several new time series analysis techniques aimed at distinguishingbetween dynamic and stochastic effects. Among the various methods availablefor extractingphase spaceinformation from experimental data, the calculation of the fractal dimension has received the widest attention. The assessment of chaotic effects in experimental data (inevita- bly affected by noise) by means of the fractal dimension is, however, not easy. For most cases, where only a small sampleof data points is available, the ultimate precision may be limited, and the sole observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics [Osborne and Provenzale, 1989]. l Istitutod{ Cosmogeofisica del Consiglio Nazionale delle Ricerche, Turin, Italy. 2Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, Italy. 3Istituto di Fisica Generale dell'Universith diTorino, Turin, Italy. 4Istituto di Fisica, Universidade Estadual de Campinas, Campi- nas, Brazil. 5Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milan, Italy. 6•)ipartimento di Scienze Fisiche dell'Universith di Napoli e Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Naples, Italy. Copyright 1993 by the American GeophysicalUnion. Paper number 93JA00956. 0148-0227/93/93 J A-00956505.00 The assessment of stochastic effects is relatively simpler. We are concerned here wi•h time series which have the properties of "colored random noise," signals for which there is a one-to-onecorrespondence between the power law dependence of the spectrum and fractal behavior, e.g., when the spectral analysis yieldsa power law power spectrumf -• (with a between 1 and 3) and uniformly distributed Fourier phases. For self-affine signals of this type, evaluations of the fractal dimension by differeat methods generally agree within all experimental and computational uncertainties. Additional information may be sd•pliedby the surrogate data test [Theiler, 1991] and b•, the evaluation of the K 2 entropy [Grassberger and Procaccia, 1983a]. It is impor- tant to remember that colored random noise does not nec- essarily arise from erroi's or insufficient statistics in the experimental measurements but can have a physical,dynam- ical origin, arising from solutions to nonlinearly coupled ordinary differential equations which undergo a non- Gaussian fractal random walk in phase spaceover a substan- tial range of large scales[Osborne and Pastorello, 1993]. The search for deterministic chaos in interplanetary dy- namics has only recently been attempted but has quickly developed into an active area. The analysis of sunspot numbers by Mundt et al. [1991]concludes that the sunspot cycle is chaotic and low dimensional. The solar wind seems instead to have a stochastic behavior, and the analysis of AE andAI geomagnetic indices, which yields a fractal dimension between2.2 and 4.2, nevertheless doesnot provide a definite conclusion about the presenceof truly differentiabledynam- ics [Roberts, 1991]. Multifractal structures at different scales have been recognized in time series of the interplanetary magnetic field strength, temperature, and density and in solar wind velocity [Burlaga, 1991a, b, c, 1992]. In the field of cosmic ray particles, yearly time series of under- ground muons have been shown to be stochastic, with a correlation dimension oscillating between 4.$ ñ 0.4 at solar maximum and 6.2 ñ 0.6 at solar minimum, and these results have been interpreted as an indication that the field-particle interactions inside the solar cavity may play a role in reducing the value of the correlation dimension [Bergatn- asco et al., 1992]. Neutron monitor time series have also 15,241