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The loop quantities and bifurcations of homoclinic loops
Authors:Maoan Han  Huaiping Zhu
Institution:a Department of Mathematics, Shanghai Normal University, Shanghai 200030, China
b LAMPS/LIAM, Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
Abstract:The 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.
Keywords:Homoclinic loops  Saddle quantities  Limit cycles  Stability  Bifurcation
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