## Dynamical systems in the plane

Content: This page contains an applet for the vsiualisation of planar dynamical systems and instructions for its use. The applet allows one to visualise vector fields and solution curves of autonomous and non-autonomous differential equations.

Applet

Theory

Analysis for Computer Scientists, Chapter 20 and 21

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### Vector fields

To draw the vector field of an (autonomous) differential equation select the tab Vector field and enter the components of the vector field into the fields x'= and y'=. Alternatively, a number of predefined examples can be loaded by using the combo box Load example. With the slider it is possible to determine the number of arrows that are drawn for visualising the vector field. The options Equal length and By speed specify whether the arrows are drawn with the same length or proportional to their natural length. The domain of the vector field must be entered in the fields Area. The minimal x-coordinate is entered in the left field, the maximal x-coordinate is entered in the right field. The minimal y-coordinate is entered in the lower field, the maximal y-coordinate is entered in the upper field. By pressing the button Draw the vector field is displayed on the screen. A mouse click at a point in the vector field generates a plot of the solution curve that passes through that point.

### Solution of autonomous differential equations

In the tab Solve autonomous ode a differential equation must be defined. In particular, the initial values (x(0), y(0)) and the integration interval must be given. The input format is illustrated by means of the predefined examples. After Pressing Compute the solution curve is displayed.

### Solution of non-autonomous differential equations

In the tab Solve non-autonomous ode a differential equation can be entered and the solution of that equation is displayed. The input format is illustrated by means of the predefined examples.

### Scaling of the axes

After the curve has been drawn, a number of options are available to change the scaling of the axes. The option Scale uniformly uses the same scale on the x- and y-axis. This ensures that a circle appears as a circle. The option Scale independently ensures that both axes are adjusted so that the curve fits into the window. Both options also allow for manual adjustment of the axes by using the "+" and "-" buttons. If a vector field is drawn the rescaling of the axes is deactivated.

With the options Variable and Box the appearance of the axes can be changed.

### Options

The numerical method for solving the ode is determined in the tab Options. The desired local error is set in the field Local error. If the option Dense output is activated, the solution is also displayed at intermediate points. Without this option the discrete numerical solution values are conneted by straight lines (which gives the wrong impression of the solution for large step sizes).

### Numerical method used

For computing the solution of the differential equation, a Runge-Kutta method of order 5 with an embedded method of order 4 is used. (The embedded method serves to determine an appropriate step size sequence.) The step size is adjusted so that the error in every step is at most as large as the local error specified. Beginning from the initial values a maximum of 2000 steps (in time) are taken to reach the endpoint of the integration interval. If the maximal number of steps is exceeded or the step size becomes too small (e.g., for stiff differential equations) an appropriate message is displayed.

### 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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