The Unfolding of a Cusp

This section of the tutorial assumes that you have completed the section on computing level sets of the default model. We also assume that you have completed the previous section on Conic Sections. In this section we animate the level sets of a two-parameter family of functions (called the "unfolding of a cusp") and compute the singular set for this family.

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

Setting up the Model

Select the Cubic 2D model from the Model menu on the Main Panel.

We will study the two-parameter family of quadratic polynomials given by
x^3 - y^2 + ax + b
as the parameters a and b are varied. To begin, set all of the parameters in the Model Panel to zero, except for the parameters coef_x^3 (set equal to 1) and coef_y^2 (set equal to -1).

Computing an Initial Level Set

We wil compute the level set of x^3 - y^2 using the Predictor-Corrector Algorithm, paying particular attention to the presence of singularities in the model.

Change the parameter Singular_Points on the Predictor-Corrector Control Panel to read 1. This sets up a 1 x 1 array of initial points which Pisces will use to attempt to detect singularities in the current model. Pressing the Go button on the Predictor-Corrector Control Panel will cause Pisces to detect the singularity at the origin, to locally represent the structure of the singularity, and then to trace the remainder of the level set.

Two Parameter Animation

The mathematical question behind this section of the tutorial is "How do small perturbations of a function affect level sets for that function?" In general, small perturbations will not change the topology of a nonsingular level set, whereas small perturbations may change the topology of singular levels. In this section we will perturb the function whose level set is a cusp in order to gain insight into how the level set changes under perturbation.

We will once again use two-parameter animation and let parameters vary along a circle in parameter space. Open the Parametric Animation Panel by selecting Parametric Animate from the Utilities menu of the Main Panel. Change the very first entry marked Steps so that we will take 16 animation steps. About halfway down the panel are two menus that allow the user to select the parameters to be animated. Select Model.coef_x and Model.const_coef. Under the menu labeled Algorithm, select Predcorr, then press the button marked Go .

What you will see is a single curve transform into a pair of curves, and then back again.

Computing Parameters Corresponding to Singular Curves

As in the previous section, we will compute a curve in (x,y,a,b)-space; each point on the curve corresponds to a function (indexed by (a,b)) whose level set is singular (at the point (x,y).

As in the previous problem:

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 Planar Singularity, 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 variables coef_x and const_coef to the domain. (Don't forget to press the OK button in order to dismiss the Permutation Panel and 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 solution curve onto (a,b)-space. This curve should look like a cusp.
The curve separates the parameter space into two regions. For level curves indexed by parameters in the region containing (a,b)=(1,0), the topology of the level set is that of a line; for level curves indexed by parameters in the region containing (a,b)=(-1,0) are topologically equivalent to a circle and a line. For parameter values on the separating curve, the corresponding level set is singular. This information is summarized in the image below.

Level sets for Perturbations of the Cusp.

Ending the Session

When you are finished, end this Pisces session by selecting Quit from the File menu on the Main Panel.

Next: The Pitchfork Bifurcation

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