Condition for generating limit cycles by bifurcation of the loop of a saddle-point separatrix

Abstract

In paper [1] (On the stability of a saddle-point separatrix loop and analytical criterion for its bifurcation limit cycles Acta Mathematica Sinica Vol. 28. No. 1, 55–70, Bejing China 1985), we considered the problem of generating limit cycles by the bifurcation of a stable or an unstable loop of a saddle-point separatrix. We gave for the first time a criterion for the stability of the loop as following:L0 is stable (unstable) if\(\int_{ - \infty }^\infty {(P'_{0x} + Q'_{0y} )dt 0)} \) wherex=ϕ(t),y=ϕ(t) then a sufficient condition for the bifurcation which generates limit cycles. This paper generalizes the result of [1] to the case where the loop contains a center or the loop tends to an infinite saddle-point, and removes the restriction that the saddle-point should be an elementary singular point. Applying the results of this paper, the author studies a two-parameter system $$\left\{ \begin{gathered} x = lx^2 + y^2 - y + 5\varepsilon xy \hfill \\ y = (3l + 5)xy + x + \varepsilon x^2 \hfill \\ \end{gathered} \right.$$