DynaLab Screen Shots

Below is an image of the DynaLab screen with the graph vs time views of a Duffing mechanical oscillator in a chaotic mode.

DMO graph vs time


Here are two projected orbit views. Projected orbits are projected along the time axis onto the non-time phase space dimensions. In the case of two-dimensional systems the projection is a plane figure. For three-dimensional systems the projection is a solid figure which may be pitched and yawed.

2D orbit projection
3d orbit projection


Next we have a full orbit including the time dimension. Since this system is driven by a periodic forcing function we close the time axis on itself in one period so it appears circular in this view.

full orbit


The full orbit view may be sliced across the time axis to reveal the internal structure of the orbit. An orbit section view is shown below. You might recognize that the cross sections are fractals.

orbit sections


The vector field view shows a projection of the trajectory of the system through phase space similar to the projected orbit view. Here though the track is displayed against a background showing the direction field, the nullclines where the rate of change of the variables are zero, the fixed points where the nullclines intersect and the stable and unstable manifolds for any saddle points. Here is a vector field view for the system whose two-dimensional projected orbit we saw above.

vector field view


Finally we offer the basin of attraction view. This view color codes each point in the plotting area depending on the x coordinate of the system after a cycle of so of the forcing function, starting from that point. The system here is the simple pendulum.

basins of attraction view