Andronov-Pontryagin system

The principal feature of Morse-Smale systems which distinguishes them from Andronov-Pontryagin systems is that the former may have infinitely many special heteroclinic trajectories. As an example, let us consider a two-dimensional diffeomorphism with three fixed points of the saddle type denoted by Oi, O and O2- Suppose that O Wq 0 and n Wq 0, the... 

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

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

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... 

In the same paper, Andronov and Pontryagin had presented the necessary and sufficient conditions of roughness for systems on the plane. Consequently,... 

The notion of roughness/structural stability can be extended to the highdimensional case without any problem. However, some other problems do arise here when we need to find out explicitly the necessary and sufficient conditions for roughness. We have remarked that Andronov and Pontryagin, as well as Peixoto, had used the classification of proper two-dimensional systems in an essential way. So, we must stop here to get acquainted with some basic notions and facts from the general theory of dynamical systems. 

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