By Cees Diks
Tools of nonlinear time sequence research are mentioned from a dynamical structures viewpoint at the one hand, and from a statistical viewpoint at the different. After giving a casual evaluate of the speculation of dynamical platforms correct to the research of deterministic time sequence, time sequence generated through nonlinear stochastic structures and spatio-temporal dynamical structures are thought of. a number of statistical equipment for the research of nonlinear time sequence are offered and illustrated with purposes to actual and physiological time sequence.
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Extra info for Nonlinear Time Series Analysis: Methods And Applications
One usually deals with a nonlinear system far from equilibrium and even if the microscopic equations are simple, the behavior of the system can be very complex. In the study of turbulence often a transition from regular to complex behavior can be observed if a control parameter is changed. The behavior just after loss of stability of the equilibrium solution can be described well by low-dimensional models. In the turbulent regimes, however, the behavior is far more complex. New scaling relations may be discovered in these regimes, such as scaling laws relating characteristics of the dynamics to the system size.
For low noise levels, the reconstruction measures associated with the cleaned attractor should be close 32 3. STOCHASTIC TIME SERIES to the reconstruction measures of the observed time series. Generally speaking, the noise level should not be too high in order to get reliable results. After successfully cleaning a time series by a noise reduction scheme one may proceed by applying the usual methods for the analysis of deterministic time series. However, if the primary interest lies in estimating dynamical invariants such as the correlation dimension Di and the correlation entropy K-2, one may skip the noise reduction step altogether and adopt a more direct approach.
Aj > AT > . . > A/t. For continuous-time dynamical systems, the spectrum of Lyapunov exponents is defined similarly. In the spectrum of continous-time dynamical systems there is always one Lya punov exponent identically zero, since perturbations along the direction of the flow have a mean growth rate of zero. If the evolution equations of a dynamical system are known, good numerical estimates of the spectrum of Lyapunov exponents are usually feasible. How- 28 2. NONLINEAR DYNAMICAL SYSTEMS ever, it is notoriously difficult to obtain reliable estimates of the spectrum of Lyapunov exponents for a dynamical system that is reconstructed from an ex perimental time series.
Nonlinear Time Series Analysis: Methods And Applications by Cees Diks