Nonlinear Time Series AnalysisThe paradigm of deterministic chaos has influenced thinking in many fields of science. Chaotic systems show rich and surprising mathematical structures. In the applied sciences, deterministic chaos provides a striking explanation for irregular behaviour and anomalies in systems which do not seem to be inherently stochastic. The most direct link between chaos theory and the real world is the analysis of time series from real systems in terms of nonlinear dynamics. Experimental technique and data analysis have seen such dramatic progress that, by now, most fundamental properties of nonlinear dynamical systems have been observed in the laboratory. Great efforts are being made to exploit ideas from chaos theory wherever the data displays more structure than can be captured by traditional methods. Problems of this kind are typical in biology and physiology but also in geophysics, economics, and many other sciences. |
Contents
Determinism andpredictability 4 1 Sources ofpredictability 4 2 Simple nonlinear predictionalgorithm | |
Lyapunov exponents | |
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algorithm amplitude Appendix approximation autocorrelation function average behaviour bifurcation bythe canbe chaos chaotic attractor chaotic systems Chapter component compute correlation dimension correlation integral correlation sum curves data set delay embedding delay vectors dependence determined deterministic discussed distribution dynamical systems electric resonance embedding dimension embedding space entropy equations Example experimental exponential Figure filter finite fixed point flow data fluctuations fractal frequency Gaussian Grassberger Hence Hénon map initial conditions interval inthe invariant iteration Kantz length scales limit cycle linear Lyapunov exponents manifold Markov Markov chain method neighbourhood neighbours NMR laser data noise level nonlinear noise reduction nonstationary observed ofthe onthe oscillations parameters periodic orbits phase space Phys plot Poincaré Poincaré map power spectrum prediction error quantities reconstruction recurrence plot sampling scalar Section selfsimilarity series analysis signal stationary statistical stochastic processes surrogates thatthe thedata timeseries TISEAN tothe trajectory typical unstable yields