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The bifurcation set for the \(1:4\) resonance problem. (English) Zbl 0828.34027

Since the pioneering work of F. Takens and V. I. Arnol’d it is conjectured that the principal part of a \(\mathbb{Z}_4\)-equivariant planar vector field \[ \dot z= \varepsilon z+ Az|z|^2+ B\overline z^3\qquad (\varepsilon, A, B\in \mathbb{C})\tag{1} \] is a versal unfolding. This problem is related to the 1:4 resonance problem for a closed orbit of a vector field in \(\mathbb{R}^3\).
The present paper describes the bifurcation set of a scaled version of (1), that is \[ \dot z= e^{i\alpha} z+ e^{i\varphi} z|z|^2+ b\overline z^3\tag{2} \] with \(\alpha\in (-\pi; \pi]\), \(\varphi\in \left[\pi, {3\pi\over 2}\right]\) and \(b\in [0, 1]\). A model of this bifurcation set is presented, which divides the \((\alpha, \varphi, b)\)- space into fifteen regions of generic phase portraits. All bifurcation phenomena seem to unfold generically for \(\varphi\neq {\pi\over 2}\) and \(\varphi\neq {3\pi\over 2}\).
Reviewer: L.Recke (Berlin)

MSC:

34C23 Bifurcation theory for ordinary differential equations
37C80 Symmetries, equivariant dynamical systems (MSC2010)
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations

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References:

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