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Algebraic and geometric aspects of the theory of polynomial vector fields. (English) Zbl 0790.34031

Schlomiuk, Dana (ed.), Bifurcations and periodic orbits of vector fields. Proceedings of the NATO Advanced Study Institute and Séminaire de Mathématiques Supérieures, Montréal, Canada, July 13-24, 1992. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 408, 429-467 (1993).
This is a survey article on systems \(dx/dt=P(x,y)\), \(dy/dt=Q(x,y)\) where \(P,Q\) are polynomials. Early work of Poincaré, Lyapunov, Darboux is discussed in detail, in particular the problems of finding algebraic solutions of \(Pdy-Qdx=0\) and the problem of whether a given point is a “center” of the system, the trajectories of nearby points being closed. Methods of Poincaré for dealing with the center were refined by Shi Songling and others so that they are well suited to computer algebra. The problem of a center is solved for quadratic systems and for some kinds of cubic systems but is open in the general cubic case. Existence of elementary integrals is also discussed.
For the entire collection see [Zbl 0780.00040].

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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