Start a new session of Pisces, and open the Model Panel, the Predictor-Corrector Control Panel, and the View Window.

We will study the three-parameter family of vector fields on the line
given by

*x^3 - mu x + alpha + beta mu*

as the parameters *mu, alpha* and *beta*
are varied.

**Create a Derived Model**- Select
**Derived Model**from the**Settings**menu on the Main Panel. The Derived Models Panel will appear. On this panel, select the menu item labeled**Fixed Points ODE**, then press the**Update**button that appears. **Inflate the Model Domain**- At the bottom of the
Model Panel, press the
**Permute**button to bring up the Permutation Panel. Use the Permutation Panel to add the variable**mu**to the domain. Press the**OK**button in order to apply the permutation. **Compute the Zero Set of the Derived Model**- Change the following parameters on the
Predictor-Corrector Control Panel:
**Uniform_Points**= 0**Boundary_Points**= 3**Turning_Points**= 3**Singular_Points**= 1

**Go**button. Change the**Hor**and**Ver**coordinates of the View Window in order to project the solution curve onto*(mu,x)*-space. This curve should look like the pitchfork below (though colors may be different).For

*mu<0*, the vector field has only one (unstable) equilibria; for*mu>0*, there are three equilibria (two unstable; one stable).

To view the symmetry-breaking bifurcation, first make sure that the
Predictor-Corrector parameters are set as indicated above. Open the
Animation Panel
by selecting **Animate** from the
**Utilities** menu of the Main
Panel. Animate the parameter **Model.beta** in eleven (11)
steps within the interval *[-0.3, 0.3]*, using the
**Predcorr** algorithm.

What you will see is a pair of nonsingular curves transform into a singular (symmetric) curve and then back into nonsingular curves.

As in the previous problem:

**Create a Derived Model**- On the Derived Models Panel,
select the menu item labeled
**Saddle Node ODE**, then press the**Update**button that appears. **Inflate the Model Domain**- At the bottom of the
Model Panel, press the
**Permute**button and use the Permutation Panel to add the variable**beta**to the domain. Press the**OK**button to apply the permutation. **Compute the Zero Set of the Derived Model**- Press the
**Go**button in the Predictor Corrector Control Panel. Change the**Hor**and**Ver**coordinates of the View Window in order to project the saddle-node curve onto*(mu,beta)*-space. The saddle-node curve separates the parameter space into four regions. A saddle-node bifurcation occurs for parameter values on the curve.If Geomview is installed on your system, you can view the saddle-node curve in

*(x,mu,beta)*-space. Launch Geomview by selecting the**Geomview**item from the**Output**menu. When Geomview appears, it should show a three-dimensional image of the saddle-node curve. If it does not, press the**Close**button on the Main Panel and then reopen the connection via**Output->Geomview**in order to force Pisces to resend its current graphics buffer.

Comments to: pisces@geom.umn.edu

Last modified: Wed Dec 6 15:06:24 1995

Copyright © 1995 by The Geometry Center, all rights reserved.