Andronov-Pontryagin theorem

Rough systems on a plane. Andronov-Pontryagin theorem... 

This condition may be weakened so that a finite number of points of a quadratic contact with the vector field can be allowed on dG. In such a case, the fourth condition that neither periodic orbits nor separatrices pass through these contact points should be added to the Andronov-Pontryagin Theorem. 

It follows immediately from the Andronov-Pontryagin theorem that a rough system may possess only a finite number of equilibrium states and periodic orbits in G. 

The proof of sufficiency of the conditions of the Andronov-Pontryagin theorem relies heavily on the Poincare-Bendixson theory which gives a classification of every possible type of trajectories in two-dimensional systems on the plane (see Sec. 1.3). We refer the reader to the books [11, 12] for further details. 

Rough systems are also dense in the space of systems on two-dimensional orientable compact surfaces for which the necessary and sufficient conditions of roughness are analogous to those in the Andronov-Pontryagin theorem. The theory of such systems was developed by Peixoto [107]. The key element in this theory proves the absence of unclosed Poisson-stable trajectories in rough systems (they may be eliminated by a rotation of the vector field). 

For rough systems on a plane, the Andronov-Pontryagin theorem gives a = 1. The case where a = 2 takes place in systems which has a loop of separatrix F to a saddle O, the loop is the limit trajectory for nearby orbits (see Fig. 7.2.1) and is non-wandering. Here, Aii = F U O. On the second step of the above procedure, one obtains Ai2 = O, i.e. the center of the region G is minimized to the equilibrium state. 

Any modification of the phase portrait of a system may occur when the system becomes structurally unstable. By the Andronov-Pontryagin theorem, such a system must necessarily possess either ... 

Structurally stable systems can be identified in the Banach space Be of dynamical systems on a plane using conditions involving only inequalities (see Andronov-Pontryagin theorem). However, systems of first-degree of... 

Theorem 7.1. (Andronov-Pontryagin) A system X is rough in the region G, if and only if,... 

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