Mathematical Software - Chaotic Systems - Poincare Map

Poincare Map

Definition

The Poincare's method is based on collecting values taken from a time series with a mapping period Tm and corresponding mapping frequency ωm = 2π/Tm. If two trajectories are observed

chaos, chaotic systems, dynamical systems, poincare map, ilona kosinska, kosinska, optfinderML, taketechease chaos, chaotic systems, dynamical systems, poincare map, ilona kosinska, kosinska, optfinderML, taketechease

then obtained data form 2D pointal coordinates giving the phase portrait

chaos, chaotic systems, dynamical systems, correlation dimension, ilona kosinska, kosinska, optfinderML, taketechease

whereas Poincare points (obtained with the mapping period Tm ) are depicted on a graph below together with numbers representing their ordering:

chaos, chaotic systems, dynamical systems, poincare map, ilona kosinska, kosinska, optfinderML, taketechease

Both a number of repeated pointal positions seen as different (q) and order of their occurrence matter. Namely, computation of the characteristic frequency is based on the expression

ω0 = p/q ωm

where p is a sum of a number of skipped over positions counted in the counter-clockwise direction between two successive points and 1.

chaos, chaotic systems, dynamical systems, poincare map, ilona kosinska, kosinska, optfinderML, taketechease

Chaotic Systems - References

  1. ^ G. L. Baker, J. P. Gollub, Chaotic dynamics: an introduction, Cambridge University Press, 1996
  2. ^ Vadim S. Anishchenko et al., Nonlinear Dynamics of Chaotic and Stochastic Systems, Springer-Verlag, 2007
  3. ^ Boris P. Bezruchko and Dmitry A. Smirnov, Extracting Knowledge From Time Series, Springer-Verlag, 2010

Machine Learning - OptFinderML

Package for machine learning - OptFinderML.

Genetic algorithms on Facebook Ilona Kosinska products on pinterest