Normal forms near a symmetric planar saddle connection (Q264482)

This paper studies vector fields of the form NEWLINE\[NEWLINE\dot{x}=\left(q/2 + O(1-x^2)\right)(1-x^2)+O(y),\quad \dot{y} = \left(p x + O(1-x^2)\right) y + O(y^2),NEWLINE\]NEWLINE which contain a separatrix connection between hyperbolic saddles with opposite eigenvalues where the connection is fixed. The author provide smooth semi-local normal forms in the vicinity of the connection, both in the resonant and non-resonant case. First, a conjugacy is constructed near the separatrix. Then, a smooth change of coordinates is realized by generalizing known local results near the hyperbolic points.NEWLINENEWLINEThe author applies the method of two-step procedure in the study of local normal forms near singularities: first establish a ``formal normal form'', and latter eliminate the flat terms after applying Borel's theorem. In particularly, the resonance monomials are taken as NEWLINE\[NEWLINE\left((1-x^2) y\right)^n\quad \text{ and } \quad x\left((1-x^2)y\right)^2NEWLINE\]NEWLINE for the case \(p=q=1\), and NEWLINE\[NEWLINE\left((1-x^2)^p y^q\right)^n\quad \text{ and } \quad x\left((1-x^2)^py^q\right)^2NEWLINE\]NEWLINE when \((p,q)\not=(1,1)\).