Study of the cyclicity of some degenerate graphics inside quadratic systems (Q1016654)

The authors prove that at most finitely many limit cycles can bifurcate from certain graphics possessing a line of critical points under small perturbation of parameters in the quadratic family \[ \dot x = \lambda x - \mu y + a_1 x^2 + a_2 xy + a_3 y^2, \] \[ \dot y = \mu x + \lambda y + b_1 x^2 + b_2 xy + b_3 y^2, \] \[ (\lambda, \mu) \in \mathbb{S}^2,~ (a_1,a_2,a_3,b_1,b_2,b_3) \in \mathbb{S}^5. \] In particular, they thoroughly treat graphics \((DF_{1a})\) and \((DF_{2a})\) of the Dumortier-Roussarie-Rousseau program, except for a small region in parameter space. A key technique employed is blow-ups of parametrized families, but with some special refinements for these difficult cases.