## 6. ConclusionsIn the following items we give a summary of the main results obtained by the presented dimension analysis of several types of solar radio events, based on the correlation dimension and the local pointwise dimension method. The results are relevant concerning the physics of the analyzed events as well as the different methods applied. -
The analysis does not enable to claim low-dimensional determinism in the time series. This outcome is in agreement with the results obtained by Isliker (1992b) and Isliker & Benz (1994a, 1994b), who also, among others, investigated type I storms, type IV events, and spikes. We cannot confirm the results of Kurths & Herzel (1986, 1987), Kurths & Karlicky (1989), and Kurths et al. (1991), who obtained finite dimension values for decimetric pulsations. However, the outcome of the present paper does not exclude deterministic chaos in the analyzed time series but makes pure low-dimensional determinism, characterized by few free parameters, rather improbable. -
The analyzed time series are not fully stochastic, i.e. white noise. This fact we infer from the distinctly slower increase of the dimension values with increasing embedding dimension than expected for fully stochastic processes, which always fill the whole phase space, i.e. . Moreover, the surrogate data analysis suggests that the time series do not represent linear stochastic processes. -
For most of the analyzed data sets we have evidence that nonlinearity in the time series is present (given on a level by means of a surrogate data test). -
A comparison of the two different methods used for the determination of fractal dimensions reveals that the local dimension method is more stable and enables more physical insight than the classical correlation dimension method. The local dimension analysis can provide a statistically significant quantity for systems, which cannot be characterized by invariants of the dynamics, probably since they are in fact not purely deterministic. Such quantities can be of special interest for comparative studies, investigating interrelations between different time series (which, e.g., can be useful for classificational purposes) or investigating intrarelations in between one time series (in order to detect dynamical changes). -
The retrieved pointwise dimension values can be interpreted in terms of complexity of the underlying physical system. In this frame our analysis indicates that spikes and fast pulsations are the signature of systems of higher complexity than pulsations, sudden reductions and type I storms.
In relation with other kind of analysis of solar radio bursts the presented results might give further ideas on the physics of the events. In the following we present a short discussion in this respect, applied to pulsation and spike events, which are quite striking features associated with solar flares. Spikes have been intensively studied during recent times. Their short duration and small bandwidth gives rise to the evidence that they are associated with the energy fragmentation process in solar flares (Benz 1985, 1986). Based on this connection, Schwarz et al. (1993) performed a nonlinear analysis by means of symbolic dynamics methods, interpreting the spikes appearance in the frequency-time domain as spatio-temporal patterns. This analysis gives indications that the simultaneous appearance of spikes at different frequencies is not a purely stochastic phenomenon but may be caused by a nonlinear deterministic (not necessarily low-dimensional) system or by a Markov process, compatible with a scenario in which spikes at nearby locations are simultaneously triggered by a common exciter, i.e. the localized sources are causally connected. In the present paper we find evidence for the spike events analyzed, that they do not represent a purely stochastic phenomenon in their temporal order either, even if the degree of freedom of the related physical system is expected to be quite high. Interpreting this result in the frame of the scenario suggested by Schwarz et al. (1993), it might give indications that the triggering of successive spikes by a localized source is not caused by a fully stochastic process, but reveals some (possibly weak) kind of nonlinear causal connection. However, this inference is restricted to the assumption that the spikes time series rather reflect the physical conditions of the triggering mechanism than those of the emission. Pulsations, although a rather marginal phenomenon in the course of solar flares, have reached a wealth of attention, especially due to the very regular features they sometimes reveal (for a review see Aschwanden 1987). In previous investigations of the dimensionality of solar pulsations (Kurths & Herzel 1986, 1987; Kurths & Karlicky 1989; Kurths et al. 1991) the presence of low-dimensional determinism is reported, with dimensions . Moreover, for one single event a dynamical evolution from a limit cycle to a low-dimensional chaotic behavior was found (Kurths & Karlick 1989). Although we cannot confirm these results, we want to stress that our analysis suggests that pulsation events, especially quasi-periodic pulsations, represent the least complex phenomena among the analyzed types of radio events, i.e. their degree of freedom is expected to be lower than that of other burst types, even if not low-dimensional. The inferred high-dimensionality and the nonlinear structures detected do not match with linear MHD oscillation models for pulsations (e.g., Rosenberg 1970; Roberts et al. 1984), in which only a few eigenmodes are excited, but rather favor models of self-organizing systems of plasma instabilities, which comprise periodic as well as low- and high-dimensional chaotic behavior. Such a self-organizing model for the electron-cyclotron maser instability, based on a Lotka-Volterra type equation system, is discussed in Aschwanden & Benz (1988), however restricted to limit cycle solutions. © European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |