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.  
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. 

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..: