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A complete classification of quadratic differential systems according to the dimensions of \(\text{Aff}(2,\mathbb{R})\)-orbits. (English) Zbl 1202.34070
Authors’ abstract: We consider the action of the group \(\mathrm{Aff}(2,{\mathbb R})\) of affine transformations and time rescaling on real planar quadratic differential systems. Via affine invariant conditions, we give a complete stratification of this family of systems according to the dimension \(\mathcal D\) of affine orbits proving that \(3\leq \mathcal D \leq6\). Moreover, we give a complete topological classification of all the systems located on the orbits of dimension \(\mathcal D \leq5\) constructing the affine invariant criteria for the realization of each of the 49 possible topologically distinct phase portraits.
34C14 Symmetries, invariants of ordinary differential equations
34A26 Geometric methods in ordinary differential equations
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms