Local Limit Cycles of Degenerate Foci in Cubic Systems

Abstract

The problem of determining the stability of a weak focus in a quadratic or cubic system has been the focus of much research. Here we outline a simple but imperfect approach to the study of degenerate foci and use the method to give an example of a cubic system with four local limit cycles about a degenerate focus.

Appendix: Mathematica 8

Mathematica was used interactively to produce the results in Sect. 2.3. Firstly the base functions are put in place:

$$ \mathrm{P}2=cx\hat{\mkern6mu} 2+Dxy+F\hat{\mkern6mu} 2\vspace*{-8pt} $$

$$ \mathrm{Q}2=Axy+By\hat{\mkern6mu} 2\vspace*{-8pt} $$

$$ \mathrm{P}3=Nx\hat{\mkern6mu} 2y+Qxy\hat{\mkern6mu} 2+Ry\hat{\mkern6mu} 3\vspace*{-8pt} $$

$$ \mathrm{Q}3=-x\hat{\mkern6mu} 3+Kx\hat{\mkern6mu} 2y+Lxy\hat{\mkern6mu} 2+ My\hat{\mkern6mu} 3\vspace*{-8pt} $$

$$ \mathrm{V}2=1/2y\hat{\mkern6mu} 2\vspace*{-8pt} $$

$$ \mathrm{V}3=-\mathrm{Integrate}\left[\mathrm{Q}2,x\right] $$

After this each iteration of the algorithm has a sequence of similar steps. The first set is as follows:

$$ T4=-D\left[\mathrm{V}3,x\right]\mathrm{P}2-D\left[\mathrm{V}3,y\right]\mathrm{Q}2-D\left[\mathrm{V}2,x\right]\mathrm{P}3-D\left[\mathrm{V}2,y\right]\mathrm{Q}3\vspace*{-8pt} $$

$$ \mathrm{Collect}\left[\%,\left\{x,y\right\}\right]\vspace*{-8pt} $$

$$ \mathrm{X}4=\mathrm{Coefficient}\left[\%,x\hat{\mkern6mu} 4\right]\vspace*{-8pt} $$

$$ \mathrm{V}5=\mathrm{Simplify}\left[\left(\mathrm{T}4-\mathrm{X}4\ x\hat{\mkern6mu} 4\right)/y\right] $$

Each X terms give us a focal value η, and the V terms give us the homogeneous pieces of the Liapunov function that we are constructing. We continue through as many of these steps as is necessary.