Stability of singular limit cycles for Abel equations (Q484412)

Consider Abel equations \[ \dot x = A(t) x^3 + B(t) x^2, \] where \(A\) and \(B\) are continuous functions on an interval \([0, T]\). Let \(u(t, x)\) denote a solution for which \(u(0, x) = x\). It is a closed closed solution if \(u(T, x) = x\). If \(u_x(t,x) \neq 1\) it is simple, otherwise it is singular. The authors obtain a criterion for determining the stability of singular limit cycles (isolated closed solutions), which controls their possible saddle-node bifurcations. Using this result, combined with examination of Hopf-like bifurcation of the solution \(u(t) \equiv 0\) and bifurcation at infinity of a compactification of the equations, they derive upper bounds on the total number of limit cycles. They use these results to prove that the family with \(A(t) = a t (t - t_A)\) and \(B(t) = b (t - t_B)\), \(a, b > 0\) has at most two limit cycles in the region \(x > 0\), and this bound is sharp.