Classification of catastrophes in dynamical systems

Consider a system of ordinary differential equations dependent on the parameters c  

The critical (bifurcation) value of the parameters, c0, is defined as their values such that in the vicinity of c0 there is the point c1( such that the phase portraits determined by f(x c0) and /(x ct) are qualitatively different. On the basis of the analysis carried out in Section 5.4.5 the critical value c0 is known to correspond to eigenvalues of the system of equations obtained by linearizaton of the system (5.58) for which Re(A) = 0. 

Two general problems associated with a catastrophe (bifurcation) of this type can be formulated  

The above problems are closely related to the problem of structural stability, in analogy with the problems of elementary catastrophe theory. 

We shall examine the class of bifurcation problems for which the system (5.58) has the stationary state (x0 c0), that is f(x0 c0) = 0, for which the condition Re ) =. .. = Re(Ap) = 0 is satisfied. Such a stationary state is sensitive. As we have established previously, a catastrophe involving a change in the phase portrait near a stationary point takes place on the system crossing a sensitive state in which the stability matrix has the eigenvalue (eigenvalues) equal to zero or a pair (pairs) of purely imaginary eigenvalues. 

---