Sensitive states of autonomous systems

The analysis of properties of gradient systems carried out in terms of elementary catastrophe theory (examination of critical points of the potential V) and of nongradient systems by means of singularity theory (examination of singularities of the vector function F) provides an incentive to investigate the relation between possible catastrophes and the eigenvalues of the stability matrix. 

Note that vanishing of the determinant det(82V/8xidxj) or det dFJdx ), corresponding to the occurrence of a catastrophe, implies vanishing of the stability matrix determinant  

let us characterize mo re exactly, in terms of eigenvalues of the stability matrix, the sensitive state appearing in such a case. 

Consider the system of two equations (5.2), having the stationary point (0, 0) for all values of the control parameters e. The point e = 0 is said to be the point of occurrence of a catastrophe (also called a bifurcation) for the system (5.25), if the qualitative nature of a phase portrait changes in the vicinity of the stationary point (0, 0) on crossing by control parameters the value e = 0. 

It follows from equation (5.9) that the requirement det(ay) = 0 is equivalent to vanishing of at least one eigenvalue, for example 

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