• Journal of the Korean Mathematical Society
• /
• v.36 no.3
• /
• pp.459-467
• /
• 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.

• The Pure and Applied Mathematics
• /
• v.4 no.1
• /
• pp.61-69
• /
• 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.