Elementary analysis of dynamical catastrophes

A description of the method will be begun with the analysis of a catastrophe occurring in the van der Pol system 

the stationary point (0,0) for small (as regards the absolute value) e is a stable focus and for small positive e is an unstable focus. When the parameter e changes sign, a catastrophe — a change in the nature of trajectories, takes place in the system. In addition, At 2(0) = +i hence, the state of the system corresponding to e = 0 is a sensitive state typical for the Hopf bifurcation. 

Since e is a small parameter, it may be presumed that, to a first approximation, the function H also corresponds to the energy of the perturbed system (5.96). Let us thus compute how the function H varies with time on the trajectories of the perturbed system (5.96). In view of equations (A. 16), (5.96) we obtain 

We may now calculate the change in energy of the system, A H, in the course of its evolution along the phase trajectory  

Not knowing the exact trajectories of the system (5.96) we may determine an approximate value of AH by computing the integral (5.103) on the trajectories of the system (5.99), i.e. on the circles of radius R 

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