# 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 T_{m} and corresponding mapping frequency ω_{m} = 2π/T_{m}. If two trajectories are observed

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

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

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

ω_{0} = p/q ω_{m}

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

## Chaotic Systems - References

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

## Machine Learning - OptFinderML

Package for machine learning - OptFinderML.