Cubic systems and Abel equations. (English) Zbl 0911.34020
Consider autonomous differential systems of the type
\[
dx/dt = \lambda x + y + P_2 (x,y)+ P_3 (x,y), \qquad dy/dt =-x+ \lambda y + Q_2 (x,y)+ Q_3 (x,y),\tag \(*\)
\]
where \(P_i (x,y)\) and \(Q_i (x,y)\), \(i=2,3\), are homogeneous polynomials of degree \(i\). In polar coordinates \(r,\Theta\), \((*)\) is equivalent to \((**)\) \(dr/d\Theta = R(r,\Theta)\).
The authors consider systems \((*)\) such that there is a transformation \(\rho (\Theta) = u(r)(-1+B(\Theta) u(r))^{-1}\) with \(u(0)=0\) and \(B(\Theta)\) of period \(2 \pi\), so that \((**)\) is equivalent to an equation of Abel’s form \[ d \rho / d\Theta = \alpha_1 (\Theta) \rho + \alpha_2 (\Theta) \rho^2 + \dots + \alpha_N (0) \rho^N,\tag \(***\) \] where \(\alpha_\kappa (\Theta)\) are of period \(2\pi\). For these systems they derive necessary and sufficient conditions such that for \(\lambda =0\), \(x=y=0\) is a center, and especially an isochronous center. Moreover, the authors prove that up to five limit cycles can bifurcate from the fine focus at the origin.
The authors consider systems \((*)\) such that there is a transformation \(\rho (\Theta) = u(r)(-1+B(\Theta) u(r))^{-1}\) with \(u(0)=0\) and \(B(\Theta)\) of period \(2 \pi\), so that \((**)\) is equivalent to an equation of Abel’s form \[ d \rho / d\Theta = \alpha_1 (\Theta) \rho + \alpha_2 (\Theta) \rho^2 + \dots + \alpha_N (0) \rho^N,\tag \(***\) \] where \(\alpha_\kappa (\Theta)\) are of period \(2\pi\). For these systems they derive necessary and sufficient conditions such that for \(\lambda =0\), \(x=y=0\) is a center, and especially an isochronous center. Moreover, the authors prove that up to five limit cycles can bifurcate from the fine focus at the origin.
Reviewer: K.R.Schneider (Berlin)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
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