## 1. IntroductionNonlinear 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
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. 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. © European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |