Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities (Q1671209)

This paper studies the number of limit cycles of the system \[ \dot x= ax-y+P_n(x,y),\, \dot y=x+ay+Q_n(x,y), \] where \(a\) is a real parameter and \(P_n\) and \(Q_n\) are real homogeneous polynomials of degree \(n,\) under some additional hypotheses. More concretely, if we introduce the following functions: \(f(\theta)=P_n(\cos \theta,\sin\theta)\cos\theta+Q_n(\cos\theta,\sin\theta)\sin\theta,\) \(g(\theta)=Q_n(\cos \theta,\sin\theta)\cos\theta-P_n(\cos\theta,\sin\theta)\sin\theta,\) \(A(\theta)=ag(\theta)-f(\theta)\) and \(B(\theta)=(n-1)(2ag(\theta)-f(\theta))+g'(\theta)\), the main results of the paper are that the planar system has at most one limit cycle if one of the following conditions holds: (i) there exist two constants \(c_1,c_2\) such that \(c_1^2+c_2^2\neq0\) and \(c_1A(\theta)+c_2B(\theta)\geq0\), or (ii) there exist two nonnegative constants \(d_1\) and \(d_2\) such that \((ad_1)^2+d_2^2\neq0\) and \(A(\theta)(d_1ag(\theta)-d_2f(\theta))\leq0.\) The functions \(f\) and \(g\) appear writing the system in polar coordinates \(r\) and \(\theta\), while the functions \(A\) and \(B\) appear because transforming this expression into an Abel differential equation, by using the so-called Cherkas-Liouville transformation, \(\rho=r^{n-1}/(1+g(\theta)r^{n-1}),\) we arrive to \[ {d\rho}/{d\theta}=(n-1)A(\theta)g(\theta)\rho^3-B(\theta)\rho^2+(n-1)a\rho. \] These results are proved by using that the hypotheses provide some useful curves without contact or invariant by the flow together with the construction of suitable Dulac functions, either for the planar system, or for its expression as an Abel differential equation.