3D Attractors
3D Attractors
3DAttractors is a Mac application to explore and visualize the Lorenz and Roessler attractors in 3 dimensions. I have tested it in Mac OS X 10.5 and 10.6.
Controls
• Rotate the attractor with the left mouse button.
• Translate the attractor with the right mouse button.
• Zoom in pressing the ‘z’ key and moving the mouse forward.
• Increase parameter X by an amount dX by setting this amount in the Parameter rollouts (see figure) and then pressing the space bar. This feature is useful to pinpoint bifurcations.
• “Static” rendering shows the trajectory from tmin to tmax all at once.
• “Dynamic” rendering shows the evolution of the trajectory from tmin to tmax and then restarts.
• “Segment” shows a segment of a trajectory, adding the new position and deleting the last position of the segment each time. The length of the segment can be adjusted by changing tmax or by switching from “dynamic” to “segment” rendering at the desired time.
• “Twin trajectory” creates another trajectory with a slightly different initial condition (10^(-9) in each coordinate). Pressing ‘s’ with this option turned on reinitializes this second trajectory as a perturbation of the current position of the primary trajectory. This option is best used in combination with “dynamic” or “segment” rendering. When this option is selected, a plot of log|delta(t)| vs t is rendered, where delta(t) is the euclidean distance between the two trajectories. I recommend setting tmin to zero when using this option.
• “Random Initial Condition” restarts the system with an initial condition randomly chosen in [-20,20]x[-20,20]x[10,20]. The twin trajectory, if applicable, is also restarted as a perturbation of the main trajectory.
• The differential equations are solved using a 4th order Runge-Kutta method. the timestep dt can be adjusted by the user and this is useful sometimes to slow down fast rendering.
• “Maximum map” shows a variant of the Lorenz map.
• There are various options for coloring and displaying the attractor. The one I prefer is “red ghost” in which the latest part of the trajectory is colored yellow and the rest is red.
Bugs or features to correct in future versions:
• If the program window is too small, or when using the program on a projector, sometimes the control buttons become unresponsive. When using a projector, you should maximize the program window.
• There is a rogue point in the “Maximum map” option.
• The apparent speed of trajectories does not correspond to the real speed, but is determined by computation time. In particular, rendering becomes slower as a longer trajectory is drawn.
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