In this post, I'll focus on the Lorenz DE's
These have, for various parameters values, chaotic solutions with interesting trajectory paths often shown:
|A trajectory for the standard Lorenz parameters σ=10, β=8/3, ρ=28. Often displayed without mentioning that specific parameters are required.||An attractor (due to Anders Sandberg, Oxford) parameters not specified but seems close to standard.|
The general idea is that you can compare the effect of changing start points, with comparison red/blue trajectories, and also see the very great range of different attractors that result when the parameters are changed, However, the changes are continuous. What I'd like to get to eventually (future post) is a possible relation between an average of trajectories and the attractor. That would help understand how GCM runs can be averaged to get a climate evolution.
So here is the interactive WebGL gadget, starting with the standard parameter values:
The plot works as a trackball; you can drag a virtual enclosing sphere with mouse left button down; there is a little set of unit length colored axes (x-red, y-green, z-blue) in the centre which show what is happening. By dragging vertically with the right button, you can change the scale.
Data and controls are on the right. You can enter parameter values or initial (x,y,z) values. Click Go to show a new plot with values you have entered. By default up to two trajectories will show, one red, one blue. If you enter more, the last two will show. You can remove either using the red/blue X buttons.
The solver works by fourth order Runge Kutta from supplied initial point; N is the number of steps that you can enter. Each trajectory starts out with faint color, then heavier toward the end.
You can click the Orient button at any time to restore the y-axis to vertical. The bottom button with a ↻ rotates between a x-y, y-z and z-x view.
You can start an animation by pressing the "Track" button. The trajectories on show will be traced out. The time interval per frame is in the text box dt in units of millisecs nf is the total number of frames for the trajectory.
So do experiment. I'd suggest start with varying the initial point to compare trajectories, then try varying parameters gradually to change the attractor shape. next in the series will, if I can work out a way, show in magnification how trajectories cluster around an attractor. I'd like to explore whether the attractor does emerge from averaging trajectories. That would help with the question of averaging GCM ensemble results to derive climate.
UpdateThere is some fairly simple maths that picks out a few points on the shapes. The centres of the spirals are where the derivative is zero, and this comes out to a cubic in x:
The solutions are x=y=z=0 and z=ρ-1;x=±sqrt(βz),y=x.
For standard params, z=27, x=y=±sqrt(72) marks the centres. The other root, (0,0,0) is not an attractor point.