Centers and limit cycles for a family of Abel equations (Q2396880)

This paper deals with the Abel equation \[ x'=A(t)x^m+B(t)x^n,\tag{1} \] where \(m,n>1\) are given integers and \(A,B\) are trigonometric polynomials of the following form: \[ A(t)=a_1 A_1(t)+a_2 A_2(t),\quad B(t)=a_3 A_3(t)+a_4 A_4(t), \] with \(A_1,\ldots,A_4\) being trigonometric monomials such that the signs of \(A_1,A_3\) coincide with that of \(\sin t\) whereas the signs of \(A_2,A_4\) agree with that of \(\cos t\). Let \(V\) be the set in \(\mathbb{R}^4\) of \(4\)-ple \((a_1,\ldots,a_4)\) such that every bounded solution of (1) is periodic. The authors prove that \(V\) has co-dimension equal either to \(1\) or \(2\) and provide necessary and sufficient conditions to discriminate between these two situations.