Autonomous non-gradient systems

In the analysis of autonomous non-gradient systems the methods of gradient system theory have proved useful. The notions such as a stationary (critical) point, degenerate stationary point, structural stability (instability), morsification, phase portrait can be directly transferred to autonomous systems. A qualitative description of dynamical autonomous systems is constructed analogously with the description of gradient systems. 

A general program of examination of autonomous dynamical systems may be formulated, in the spirit of elementary catastrophe theory, as follows  

Further analysis of non-gradient systems will be carried out for the case of a system of two equations (n = 2)  

Stationary states of a non-gradient system fulfil (equivalent) equations 

States of a system may be classified into three groups (a) the state beyond a stationary point, F(x, y) 0 (b) nondegenerate stationary state, F(x, y) = 0, det(dFi/dxj) 0 (c) degenerate stationary state, F(x, y) = 0, det(dFJdxj) = 0 analogously with the classification of gradient systems. 

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