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Local normal forms for constrained systems on 2-manifolds. (English) Zbl 0799.58071

A system of the form \(A(x)\dot x = v(x)\), where \(x \in \mathbb{R}^ n\), \(A(x)\) is a matrix-valued function and \(v(x)\) is a vector field is said to be a constrained system or constrained vector field. A point 0 is regular if \(\text{det }A(0) \neq 0\), otherwise 0 is called an impasse point.
The main result of the paper is the following Theorem. Let \(E\) be a generic constrained system on a 2-manifold, 0 be an impasse point of \(E\). The germ at 0 of the phase portrait of \(E\) is equivalent to the phase portrait of one and only one of 5 systems of normal forms. These systems and the phase portraits of them are listed.

MSC:

37G05 Normal forms for dynamical systems
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

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