Phase portraits of the quadratic vector fields with a polynomial first integral
Copyright policy )Phase portraits of the quadratic vector fields with a polynomial first integral
The main goal of the paper is the topological classification of the phase portraits of quadratic vector fields having a polynomial first integral. By using the compactification of Poincaré it is proved that all such fields are topologically equivalent to the one of \(25\) phase portraits on the Poincaré disc. The paper also contains the proof of the fact that all these cases can be realized by Hamiltonian systems of degree \(2\). Since a similar situation occurs in the case of linear vector fields the authors state the following problem: Are all the phase portraits of polynomial differential systems of degree \(n\) having a polynomial first integral realizable by Hamiltonian systems of degree \(n\)?
Explicit solutions, first integrals of ordinary differential equations, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Equivalence and asymptotic equivalence of ordinary differential equations, Polynomial first integral, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Phase portraits, Quadratic vector fields
Explicit solutions, first integrals of ordinary differential equations, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Equivalence and asymptotic equivalence of ordinary differential equations, Polynomial first integral, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Phase portraits, Quadratic vector fields
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