Given a cloud of N points in an E-dimensional space, we often need to estimate the intrinsic dimensionality D of this cloud. For example, a set of points in 3-dimensional space all following along a straight line has intrinsic (or fractal) dimensionality D=1. Non-integer fractal dimensionality appears pervasively in nature. In this paper we give a very fast method to estimate the fractal dimens...

Solar bursts with superfine time structure observed at 2.5 and 2.85GHz at the Crimean Astrophysical Observatory have been analyzed by means of statistical methods for nonlinear dynamical systems. Finite values of the correlation dimension have not been detected for all the studied events. The presence of a 1/f component in many of the processed events has been found by means of wavelet analysis...

The most common state space reconstruction method in the analysis of chaotic time series is the Method of Delays (MOD). Many techniques have been suggested to estimate the parameters of MOD, i.e. the time delay τ and the embedding dimension m. We discuss the applicability of these techniques with a critical view as to their validity, and point out the necessity of determining the overall time w...

This paper reports on experimental data on the controlled tuning of chaotic surface spin waves in a magnetic-film active feedback ring. The chaotic behavior of these waves arises through three-wave nonlinear interactions. With a change in the ring gain, two chaotic regimes were observed. One corresponds to the situation where only one ring eigenmode was excited in the ring, while the other corr...

(1.2) C(t, u) ⊂ H is nonempty, closed, and convex for t ∈ [0, T ], u ∈ H. In (1.1), NC(t,u)(x) denotes the normal cone to C(t, u) at x ∈ C(t, u), cf. Section 2 below. We will treat the case of (t, u) 7→ C(t, u) being Lipschitz continuous w.r. to the Hausdorff distance dH with constants L1, L2 ≥ 0, i.e., we require (1.3) dH(C(t, u), C(s, v)) ≤ L1|t− s|+ L2|u− v|, t, s ∈ [0, T ], u, v ∈ H. Note t...

The analysis of redshift surveys with fractal tools requires one to apply some form of statistical correction for galaxies lying near the geometric boundary of the sample. In this paper we compare three different methods of performing such a correction upon estimates of the correlation integral in order to assess the extent to which estimates may be biased by boundary terms. We apply the correc...

The paper deals with the presence of exponential or power-law decay in the power spectra of electroencephalogram (EEG). About 2300 EEG time series recorded during relaxed wakefulness were analysed. The whole spectrum of EEG was studied and power-law decay of about 2.28 prevailing over the exponential falling off was established. Correspondence between spectrum power-law decay and correlation di...

We investigate local correlation dimension-based noise-cleaning of time series, where points having anomalously large dimensions are iteratively removed from the reconstructed attractor. We find an optimal range for the number of iterations in which the algorithm yields good results. Choosing non-local ranges for the linear regression yields a new method for finding nonhyperbolic tangency point...

The analysis of chaotic time series requires proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Using the correlation dimension, we discuss the applicability of the two most common methods of reconstruction, the method of delays (MOD) and the Singular Spectrum Approach (SSA). Contrary to previous dis...

An investigation on collective dynamics of brain during meditation is conducted using the methods of nonlinear dynamics and phase synchronization. The relevance of degree of co ordination of different regions of brain as well as the nonlinear dynamical parameters likes correlation dimension and Kolmogorov entropy is evaluated. It has been found that before meditation each region of the brain in...