Chaos in ODEs - Diagnostics


Plots the time evolution of the variables X,Y,Z. The full dynamics needs to show the three variables as they vary in time. On the computer screen it is easiest to show various projections such as the Z-X projection. The two variables plotted are chosen by the x-ax and y-ax parameters with 1 standing for X, 2 for Y and 3 for Z.

To study the sensitivity to small changes in initial conditions, set dX0, dY0, or dZ0 to a small value. Two traces will be seen, corresponding to the two orbits starting from the initial points (X0,Y0,Z0) and (X0+dX0,Y0+dY0,Z0+dZ0).


Plots the power spectrum of the variable defined by y-ax. The power spectrum is generated by successively accumulating Points number of points spaced by Interval time steps dt, performing a fast Fourier transform on these points, and calculating the magnitude squared to give the power spectrum of this segment of data. This is then averaged in with the power spectrum of previous segments. The maximum frequency in the power spectrum is the Nyquist frequency /(Interval*dt) and the resolution on the frequency axis is 2/(Points*Interval*dt).

Segmenting the data introduces apparent discontinuities in the data to be transformed, which leads to broadening of the transform even for periodic signals and to high frequency tails. These effects are reduced by multiplying each segment by a window function that goes continuously to zero at the beginning and end of each data segment. The windowing function used is set by the parameter WinNum:

WinNum Window Description Singularity in:
0 None Top Hat Function
1 Bartlett Tent First derivative
2 Welch Parabolic First derivative
3 Hanning Sinusoidal Second derivative

For more details see "Numerical Recipes" by W.H. Press et al.

Poincare section

Plots the intersection points of the orbit with a two dimensional plane. The plane is defined by X=const, Y=const, or Z=const depending on the choice of Variable, with const set by the parameter Section

Return Map

Plots the return map of the variable on one axis of the Poincare section e.g. Xn+1 against Xn, with Xn the value of X on the nth intersection with the Poincare section. The variable plotted is determined by x-ax.


Plots the return map of successive maxima of one variable (chosen by Variable). This type of plot was introduced by Lorenz in his original study.


Reconstructs trajectory using time delay coordinates. A reconstructed phase space is constructed from V(t), V(t+T), V(t+2T), with V either X,Y, or Z depending on the value of Variable, and T the delay time given by Delay x dt. The projection of this 3-dimensional space plotted remains determined by plot_x and plot_y.

[Equations] [Instructions]
Last modified Tuesday, December 30, 1997
Michael Cross