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Bifurcations of the Riccati quadratic polynomial differential systems. (English) Zbl 1471.37026

The authors characterize the global phase portrait of the Riccati quadratic polynomial differential system \(\dot{x}=\alpha _2(x)\), \(\dot{y}=ky^2+\beta _1(x)y+\gamma _2(x)\), where \((x,y)\in \mathbb{R}^2\), \(\gamma _2(x)\) is nonzero (otherwise this is a Bernoulli differential system), \(k\neq 0\) (otherwise this is a Liénard system), \(\beta _1(x)\) is a polynomial of degree at most \(1\), \(\alpha _2(x)\) and \(\gamma _2(x)\) are polynomials of degree at most \(2\), and the maximum of the degrees of \(\alpha _2(x)\) and \(ky^2+\beta _1(x)y+\gamma _2(x)\) is \(2\). They give the complete description (with the help of \(74\) figures) of the phase portraits in the Poincaré disk (i.e., in the compactification of \(\mathbb{R}^2\) adding the circle \(\mathbb{S}^1\) of the infinity) modulo topological equivalence.

MSC:

37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37G05 Normal forms for dynamical systems
37G10 Bifurcations of singular points in dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
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References:

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