Normal forms near a symmetric planar saddle connection. (English) Zbl 1342.34057
This paper studies vector fields of the form
\[
\dot{x}=\left(q/2 + O(1-x^2)\right)(1-x^2)+O(y),\quad \dot{y} = \left(p x + O(1-x^2)\right) y + O(y^2),
\]
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.
The 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 \[ \left((1-x^2) y\right)^n\quad \text{ and } \quad x\left((1-x^2)y\right)^2 \] for the case \(p=q=1\), and \[ \left((1-x^2)^p y^q\right)^n\quad \text{ and } \quad x\left((1-x^2)^py^q\right)^2 \] when \((p,q)\not=(1,1)\).
The 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 \[ \left((1-x^2) y\right)^n\quad \text{ and } \quad x\left((1-x^2)y\right)^2 \] for the case \(p=q=1\), and \[ \left((1-x^2)^p y^q\right)^n\quad \text{ and } \quad x\left((1-x^2)^py^q\right)^2 \] when \((p,q)\not=(1,1)\).
Reviewer: Jinzhi Lei (Beijing)
MSC:
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 37G05 | Normal forms for dynamical systems |
| 37C10 | Dynamics induced by flows and semiflows |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
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