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1994 The bifurcation set for the {$1:4$} resonance problem
Bernd Krauskopf
Experiment. Math. 3(2): 107-128 (1994).


We study the bifurcation set in $(b,\phi,\alpha)$-space of the equation $\dot{z} = e^{i\alpha} z +e^{i\phi} z \,|z|^2 + b \bar{z}^3$. This $\mZ_4$-equivariant planar vector field is equivalent to the model equation that has been considered in the study of the 1:4 resonance problem.

We present a three-dimensional model of the bifurcation set that describes the known properties of the system in a condensed way, and, under certain assumptions for which there is strong numerical evidence, is topologically correct and complete. In this model, the bifurcation set consists of surfaces of codimension-one bifurcations that divide $(b,\phi,\alpha)$-space into fifteen regions of generic phase portraits. The model also offers further insight into the question of versality of the system. All bifurcation phenomena seem to unfold generically for $\phi \neq \pi/2, \, 3\pi/2$.


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Bernd Krauskopf. "The bifurcation set for the {$1:4$} resonance problem." Experiment. Math. 3 (2) 107 - 128, 1994.


Published: 1994
First available in Project Euclid: 3 September 2003

zbMATH: 0828.34027
MathSciNet: MR1313876

Primary: 34C23
Secondary: 58F14

Rights: Copyright © 1994 A K Peters, Ltd.


Vol.3 • No. 2 • 1994
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