Dynamical systems in space

Content: This page contains an applet for visualising dynamical systems in space and instructions for its use.

Applet

Theory

Analysis for Computer Scientists, Chapter 20 and 21

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Drawing orbits

The easiest way to see how you can use the applet in order to draw orbits of a three-dimensional system of ordinary differential equation is to load one of the predefined examples, accessible via the combo box Examples. Otherwise you have to enter the components of the ode in the fields x', y' and z' (read here about the syntax to use). After entering an initial value and an integration interval in the fields (x(0), y(0), z(0)) and Interval, respectively, you can draw the orbit by pressing the button Solve. The text field at the bottom of the applet will indicate if the integration was successful. For more detailed information (number of steps, minimum step size, etc.) switch to the tab Output. Once the orbit has been drawn you can rotate the curve by moving the mouse while holding the left button pressed.

Output

As already mentioned, you can find information on details of the integration under the tab Output. By clicking the right mouse button a pop-up window is shown, where you can clear the output. Further you can choose whether the output of the last computation should be displayed only.

Options

Under the tab Options you can set the options for the ode solver. (We use a 5th order embedded explicit Runge-Kutta integrator with adaptive step size control.) Under the options General Options you can enter the maximum number of steps the ode solver is allowed to execute. Moreover, you can enable or disable the dense output of the solver, which ensures that the plotted curve looks smooth, even if the solver uses only a few steps. Under the options Step size options the (initial) step size can be altered. If the option Step size control is disabled this step size will be used for the whole integration. If Step size control is enabled the intial step size is adaptively changed. If this option is enabled the desired accuracy for the integration must be entered in the field Local error.

Scaling, style of axes

Under the tab Scaling you can change the scaling of the axes. Moreover, you can choose there if a bounding box and labels should be displayed.

Projection

Under the tab Projection you can switch between perspective and parallel projection. In case of the perspective projection you can adjust the distance between the center of projection and the plane of projection. In the case of parallel projection you can choose between different planes of projection.

Questions

If you have further questions or comments, or if you found a bug, please send us an e-mail.

Financially supported by

University of Innsbruck: New Media and Learning Technologies
Austrian Federal Ministry of Education, Science and Research

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