More Limit Cycles for Complex Differential Equations with Three Monomials
Copyright policy )More Limit Cycles for Complex Differential Equations with Three Monomials
In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, z˙=Azz¯+Bzz¯+Czz¯, being k,l,m,n,p,q non-negative integers and A,B,C∈C. More concretely, if N=max(k+l,m+n,p+q) and H(N)∈N∪{∞} denotes the maximum number of limit cycles of the above equations, we show that for N≥4, H(N)≥N-3 and that for some values of N this new lower bound is N+1. We also present examples with many limit cycles and different configurations. Finally, we show that H(2)≥2 and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles.
polynomial differential equation, Lyapunov quantities, Bifurcation theory for ordinary differential equations, Number of limit cycles, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Polynomial differential equation, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Centre-focus problem, number of limit cycles, centre-focus problem
polynomial differential equation, Lyapunov quantities, Bifurcation theory for ordinary differential equations, Number of limit cycles, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Polynomial differential equation, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Centre-focus problem, number of limit cycles, centre-focus problem
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