Normal form of a nonlinear system near a critical point

A rough system containing a parameter may become a non-rough system at a critical value of the parameter. Reduction of such a system to the ordinary (linear) normal form is impossible without special features (such as poles) in the change of variables. It is shown that a (nonlinear) generalization of the concept of the normal form allows us to use only smooth (by the independent variables and parameter) transformations. Integration of the proposed normal form is reduced to a series of quadratures. Such a formulation of the problem is caused by the need to study oscillatory biochemical systems, especially the variety of types of oscillatory excitation.