cusp2 **The Catastrophe Machine**

## 2. An algebraic version of the double well

It is awkward to calculate precisely the angle at which the catastrophe happens in the double-well example. There is a more algebraic version of the phenomenon which is easier to work with.

| Imagine that the ball is constrained to roll on the graph of the function `y=x`^{4}-x^{2}. Instead of tilting the graph we perturb the function by adding a linear term (`ax`) so as to raise one well and lower the the other. |

| `a=0.3` Here is the graph of `y=x`^{4}-x^{2}+.3x. If the ball had started on the right, it would still be on the right. |

| `a=0.6` Here is the graph of `y=x`^{4}-x^{2}+.6x. The ball would have rolled over to the left. The exact point at which this happens can be reckoned (easy calculus exercise) to be `a=(4/3)*(1/6)`^{1/2}=.5443... |

The corresponding negative values `a=-.3, a=-.6` give graphs where the left well grows higher than the right. These values and the initial `a=0` generate a family of figures exactly analogous to the the configurations 1 -- 9 of our original double well. The perturbation parameter `a` plays the role of the angle of tilt.

The catastrophes take place when `a=(4/3)*(1/6)`^{1/2} and when `x=(1/6)`^{1/2}=0.4082... Looking closely at the graphs right near the catastrophe point, for values of `a` just above and just below the critical value `a`_{0}=.5443..:

`a=a`_{0}+0.00001 | `a=a`_{0} | `a=a`_{0}-0.00001 |

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makes it plausible that the graph can be reparametrized near the catastrophe point as `y=x^3`, and that this catastrophe can be explained mathematically as a perturbation of the function `y=x^3` so as to create, or destroy, a local minimum near `x=0`.

`y=x`^{3}+x | `y=x`^{3} | `y=x`^{3}-x |

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Here we can see the reason for the name "Fold Catastrophe." If the graph is projected onto the `y`-axis, the catastrophe corresponds to the appearance or disappearance of a pair of folds.