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Astron. Astrophys. 357, 337-350 (2000)

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1. Introduction

Nonlinear time series analysis based on the theory of deterministic chaos has turned out to be a powerful tool in understanding complex dynamics from measurements and observational time series. In particular it can provide descriptions and interpretations for irregular times series, which nevertheless might not be governed by a stochastic physical process and which are only poorly understood by linear methods. A number of recent reviews and conference proceedings shows the great interest in the field of nonlinear time series analysis (see, for instance, Grassberger et al. 1991; Casdagli & Eubank 1992; Weigend & Gershenfeld 1993; Kugiumtzis 1994a, 1994b; Abarbanel 1996; Kantz & Schreiber 1997; Schreiber 1999).

Since the development of chaos theory, it is well known that even simple dynamical systems, described by few nonlinear differential equations, can reveal a complex and quasi-irregular behavior. A central concept to characterize such systems is the so-called attractor . Under the dynamics of a deterministic system the trajectories do not cover the whole phase space, but, after all transient phenomena have faded out, converge to a subset of the phase space, the attractor. The attractor itself is invariant to the dynamical evolution. Simple examples of attractors are fixed points and limit cycles. However, when the related dynamical system is chaotic , the attractor can have a complex geometry with a fractal , i.e., non-integer, dimension. Different invariant parameters exist to describe the geometry and the dynamics of an attractor, such as dimensions, Lyapunov exponents and entropies. Besides the fractal geometry, chaotic systems have the striking property that initially neighboring trajectories diverge exponentially under the dynamics, and the growth rate is given by the Lyapunov exponent . This phenomenon results from the folding and stretching of the trajectories under the dynamics, the folding leading to the convergence of the trajectories to the attractor and the stretching to the divergence in certain directions. While the average stretching rate is given by the Lyapunov exponent, the loss of information due to the folding is quantified by the entropy . Lyapunov exponents and entropies characterize the dynamics on the attractor, and the dimension characterizes its geometry. The physical meaning of the dimension of an attractor is that it corresponds to the degree of freedom of the related dynamical system, i.e., in a deterministic case, the minimum number of ordinary differential equations needed to fully describe the system. Deterministic systems are characterized by a finite dimension. Deterministic chaotic systems have the additional characteristic that their dimension is fractal. Contrary to that, a stochastic system is characterized by an infinite dimension, indicating its infinite degree of freedom. Therefore, the determination of the dimension of an attractor enables to discriminate whether a dynamical system is deterministic or stochastic.

Previous papers exist concerning the investigation of fractal dimensions of solar radio bursts. It has to be noted, that the time series used by the different authors are not directly comparable as they represent different types of radio events. Kurths & Herzel (1986, 1987), Kurths & Karlicky (1989), and Kurths et al. (1991) analyzed decimetric pulsations and ascertained finite dimension values. Contrary to that, Isliker (1992b) and Isliker & Benz (1994a, 1994b) investigated different types of solar radio bursts in the metric (m) and decimetric (dm) wavelength range (type I storms, type II bursts, type III bursts, type IV events, and narrowband spikes), which did not reveal any hints for low-dimensional determinism. 1

However, these investigations rely all on the correlation dimension method. The present paper additionally introduces a complementary dimension analysis, motivated by the fact that solar radio bursts represent real-world systems, which implies some major restrictions. First, the time series cannot be expected to be stationary, and second, pure determinism is rather unlikely to be realized. Therefore we do not only concentrate on the usual way of looking at the problem: "Does the analyzed time series represent a deterministic or a stochastic system?" but in particular focus the question: "What statistical description can be extracted from a dimension analysis of the time series?" With such refined formulation of the problem, we try to make use of the concepts and tools of nonlinear time series analysis even in cases in which the determination of invariants of the dynamics, as, e.g., attractor dimensions, possibly fails.

The paper is structured as follows. Sect. 2 explains the used methods and discusses critical points in the determination of fractal dimensions from time series. In Sect. 3 the investigated data sets are characterized and the analysis procedure is described. Sect. 4 presents the results of the dimension analysis, which are discussed in Sect. 5. Finally, the conclusions are drawn in Sect. 6.

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000