Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation
The paper contains a study of the limit cycles of planar quadratic differential systems \(dx/dt=P(x,y)\), \(dy/dt=Q(x,y)\), in which P and Q are quadratic in x and y. In particular, for the quadratic system \[ (1)\quad dx/dt=(\lambda -\delta -1)x+y-x^ 2+(1+\delta)xy-y^ 2,\quad dy/dt=2x+2/3x^ 2-xy \] with two parameters \(\lambda\) and \(\delta\), it is shown that the system has a homoclinic loop surrounding a limit cycle for some values of \(\lambda\),\(\delta\). The following theorem is obtained. The system (1) has a bifurcation of codimension 2 which is the intersection of a Hopf bifurcation and a homoclinic loop bifurcation on the line \(\lambda =0\). It happens for \(\delta\) in (-1,\(\infty)\). (Numerical simulation gives \(\delta\cong -0.4.)\) So there is an angular region starting from this point in which the system has two limit cycles. The region is inside the larger region \(\lambda <0\), \(\lambda -\delta - 1<0\). Also, the author shows that the planar quadratic system \[ dx/dt=(\lambda -\delta -5)x+y-x^ 2+(5+\delta)xy-y^ 2 \] \[ dy/dt=(9- \epsilon +2\delta)x+x^ 2+(-8+\epsilon -2\delta)xy, \] with three parameters \(\lambda\),\(\delta\) and \(\epsilon\), has a Hopf bifurcation of order 3 at \(\lambda =\delta =\epsilon =0\) and a Hopf bifurcation of order 2 at \(\lambda =\epsilon =0\), \(\delta\cong 1.785\) with a homoclinic bifurcation loop.
scientific article; zbMATH DE number 4017369
scientific article