Uniqueness of limit cycles for quadratic vector fields
Copyright policy )Uniqueness of limit cycles for quadratic vector fields
This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x'= a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y'= x+a_1 y + a_2x^2+(2 a_3+a_4)xy -a_2y^2$. In particular, we study the semi-varieties defined in terms of the parameters $a_1,a_2,\ldots,a_6$ where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.
24 pages
- University of Valladolid Spain
Bifurcation theory for ordinary differential equations, algebraic variety, Ciclos límite, Solución periódica, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, periodic solution, Abel equation, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Ecuación de Abel, limit cycle, Limit cycles, Solución en forma cerrada, Periodic solution, Mathematics - Classical Analysis and ODEs, Closed solution, Classical Analysis and ODEs (math.CA), FOS: Mathematics, closed solution
Bifurcation theory for ordinary differential equations, algebraic variety, Ciclos límite, Solución periódica, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, periodic solution, Abel equation, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Ecuación de Abel, limit cycle, Limit cycles, Solución en forma cerrada, Periodic solution, Mathematics - Classical Analysis and ODEs, Closed solution, Classical Analysis and ODEs (math.CA), FOS: Mathematics, closed solution
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