Investigations of System's Dynamics

Examples

The dynamics of systems is usually represented by its phase space portrait. However, a phase space could not have a dimension which allows it to be pictured (two or three). Higher dimensional phase spaces can only be visualized by projection into the two or three dimensional sub-spaces. Here helps the method of recurrence plots. This method can also be applied to non-deterministic data.

Typical examples of various systems: (A) uniformly distributed noise, (B) super-positioned harmonic oscillations, (C) drift (logistic map corrupted with a linearly increasing term) and (D) non-stationarity (Brownian motion).

Recurrence plot of the January insolation at Italy for the last 100000 years. From such plots we can learn about symmetry and time relations in the data.

References:

Thiel, M., Romano, M. C., Read, P. L., Kurths, J.: Estimation of dynamical invariants without embedding by recurrence plots, Chaos, 14(2), 2004, 234-243, doi:10.1063/1.1667633, Article (PDF).

Marwan, N., Kurths, J.: Cross Recurrence Plots and Their Applications, in: Mathematical Physics Research at the Cutting Edge, Hauppauge, Nova Science Publishers, 2004, 101-139.