Nonlinear and multifractal approaches of the geomagnetic field

Recent nonlinear dynamics techniques have been developed to analyse chaotic time series data. We first summarize the procedure which gives an appropri ate reconstruction of the unknown dynamics from scalar measurements in a ps eudophase space. It permits, firstly, the representation of the trajectorie s of the dynamical system-they define an attractor when the system is dissi pative-by preserving its topological properties. We then present the invari ant measures and ergodic quantities such as the multifractal spectrum and L yapunov exponents which can be estimated on the reconstructed attractor. Th e multifractal analysis provides us with a characterization of the scaling energy of the process whereas the Lyapunov exponent gives another statistic al measure of the stability of the dynamics. The estimation of these quanti ties was tested on synthetic data. The nonlinear and multifractal analyses were finally applied to the hourly mean values of the magnetic field record ed at the Eskdalemuir (ESK) observatory over 79 years (692,520 data measure ments for each component). The estimations of a 5-dimensional pseudo-phase space and a positive Lyapunov exponent confirm the possibility of low-dimen sional deterministic chaos in the magnetic field observations at ESK observ atory. The correlation between the solar activity (the Wolf number), the un stable nature of the magnetic field, and the singularity spectrum points ou t the forcing of the solar cycles on the dynamics of the magnetic field at ESK observatory. (C) 1999 Elsevier Science B.V. All rights reserved.