Elementary catastrophes for functions of two variables

We shall now proceed to the examination of catastrophes modelled by Thom potential functions of two state variables. This will be done using the notions of the catastrophe manifold M, singularity set I and bifurcation set 

employed in Section 2.2 to investigate potential functions of one variable. In the case of functions of two variables the catastrophe manifold M is given by the equation (cf. equation (2.1a)) 

The set Z is such a subset of the set M that the function V has degenerate critical points at points belonging to Z. The sufficient condition for a critical point to be degenerate is vanishing of the Hessian matrix determinant, i.e. vanishing of at least one eigenvalue of the Hessian matrix (the catastrophe functions in two variables have both the eigenvalues equal to zero). 

Points in the control parameters space, for which the function V(x, y, cv. ck) has a degenerate critical point, belong to the bifurcation set B. Hence, the set B has the form  

The description of catastrophes corresponding to functions of two variables will be begin with a hyperbolic umbilic catastrophe Z)4+, which is represented by the potential function 

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