• 대한수학회지
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• 제36권3호
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• pp.459-467
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• 1999

A complete analysis of the equation x'=y, y'=$\beta$y-$\alpha$x2+$\alpha$x2+$\delta$xy, where $\alpha$ and $\beta$ small, describing a particular unfolding of the Takens-Bogdanov singularity is presented.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제4권1호
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• pp.61-69
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• 1997

In this paper, we study the dynamics of a two-parameter unfolding system $\chi$' = y, y' = $\beta$y＋$\alpha$f($\chi\alpha\pm\chiy$＋yg($\chi$), where f($\chi$,$\alpha$) is a second order polynomial in $\chi$ and g($\chi$) is strictly nonlinear in $\chi$. We show that the higher order term yg($\chi$) in the system does not change qulitative structure of the Hopf bifurcations near the fixed points for small $\alpha$ and $\beta$ if the nontrivial fixed point approaches to the origin as $\alpha$ approaches zero.