The loop quantities and bifurcations of homoclinic loops (Q872007)

The paper considers a planar vector field \[ \dot x=f(x),\quad x\in\mathbb R^2, \tag{1} \] where \(f:\mathbb R^2\to\mathbb R^2\) is a \(C^{\infty}\) function. It is assumed that (1) has a homoclinic loop consisting of a homoclinic orbit and a hyperbolic saddle point. It is known that for the homoclinic loop, one can define a sequence of quantities (homoclinic loop quantities) that determines 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. Formulas for the first three and the fifth loop quantities are available. In the paper, the formula for the fourth loop quantity is found for both the single and double homoclinic loops. Since the stability and bifurcations of a homoclinic loop are closely related to the limit cycles, as applications of the main results, for two special planar systems the limit cycles which can bifurcate from the homoclinic loops are studied.