Curves of separatrix cycles of a quadratic system. The case of three antisaddles and one saddle
Consider the differential system \[ \frac{dx}{dt}=-(1+x)y+\alpha Q(x,y),\quad \frac{dy}{dt}=x+\lambda y+ax^2+b(x+1)y+cy^2\equiv Q(x,y),\tag{*} \] where \(a,b, c,\alpha,\lambda\) are real parameters. The parameters \(\alpha,b,\lambda\) rotate the vector field (*). The authors prove that under some conditions there are exactly two values of the parameter \(\lambda\) such that (*) has a heteroclinic orbit with two saddles at infinity. The proof is based on the bifurcation theory and the theory of rotated vector fields.
scientific article; zbMATH DE number 1916042