## 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.

### MSC:

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

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