On a codimension 3 bifurcation of plane vector fields with \({\mathbb{Z}}_ 2\) symmetry. (English) Zbl 0724.34014

The paper deals with a 3-parameter family of plane vector fields \((1)\quad \dot x=f(x,\lambda):=A(\lambda)x+h(x,\lambda),\) \(x\in {\mathbb{R}}^ 2\), \(\lambda \in {\mathbb{R}}^ 3\), \(f=(f_ 1,f_ 2)\) in \(C^{\infty}\), with \(h(0,\lambda)=0\) for all \(\lambda\), \(-h(- x,\lambda)=h(x,\lambda)\) for all \(\lambda\),x. Under suitable assumptions on A(\(\lambda\)) and f, (0,0) is a singularity of codimension 3. Using normal forms (1) is reduced to a simple system. Then, it is shown the existence of a homoclinic trajectory as well as periodic trajectories surrounding three equilibrium points and periodic trajectories surrounding single equilibrium points.


34C23 Bifurcation theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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