Han, MaoanZhu, Huaiping2007-01-312007-01-312007-01-31M. Han and H. Zhu, “The loop quantities and bifurcations of homoclinic loops,” Journal of Differential Equations, vol. 234, no. 2, pp. 339–359, Mar. 2007.http://hdl.handle.net/10315/902https://doi.org/10.1016/j.jde.2006.11.009The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.209750 bytesapplication/pdfen© 2003 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0.homoclinic loopssaddle quantitieslimit cyclesstabilitybifurcationArticle