We consider the Vander Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$

It is well known that this equation has a unique limit cycle $\gamma$ which is an attractor closed orbit. Moreover the eigenvalue $\lambda$ of the derivative of the Poincare return map, corresponding to a one dimensional local section on $\gamma$, satisfies $0<\lambda <1$

If we replace the real variable $(x,y)$ with complex variables $(z,w), \;\;z=x_1+ix_2,\;w=y_1+iy_2$, the equation (1) becomes the following complex vector field $$(2) \;\;\;\;\;\begin{cases} z'=w-(z^3-z)\\ w'=-z\end{cases}$$ with the following real representation in $4$ dimensional space $(x_1,x_2,y_1,y_2)$

$$(3) \;\;\;\;\;\begin{cases} x_1'=y_1-x_{1}^3+3x_{1}x_{2}^2+x_{1}\\ x_{2}'=y_{2}-3x_{1}^2x_{2}+x_{2}^3+x_2\\ y_1'=-x_1\\ y_2'=-x_2\end{cases}$$

Now $\gamma$ is counted as a periodic orbit of system (3) (put $x_{2}=y_2=0)$. Note that $\gamma$ is a non isolated closed orbit of (3). The reason is that $\gamma$ lies on a unique leaf $L$ of complex singular foliation (2). On the other hand the complex vector field (2) defines a real vector field on $L$ which can not possess an isolated closed orbit. So we have a band of closed orbits in $L$ containing $\gamma$.

The conclusion is that $\{1, 1, \lambda\} $ are the three eigenvalues of $DP$, the derivative of the Poincare return map $P$ defined on a $3$ dimensional local section $S$ on $\gamma $. Because, assuming $T$ is the period of $\gamma$, we have $$Det( D\phi_{T})=e^{\int_0^T Div(X_{3})(\gamma(t))dt}=e^{\int_0^T Div(X_{1})(\gamma(t))dt}=\lambda$$ where $X_{1}, X_{3}$ are vector fields associated with (1) and (3), respectively and $\phi$ is the flow of (3).

So we have a degenerate eigenvalues for the derivative of Poincare return map.In fact an eigenvalue estimate can not help us to understand the behavior of solutions of (3) near $\gamma$.

Question: With this degeneracy of eigenvalues what can be said about the stability of $\gamma$? Is $\gamma$ a stable periodic orbit of (3), in the sense that there is a bounded open set containing $\gamma$ which is invariant under the positive flow of (3)?

What about if we consider an $\epsilon$ multiplier in the original vector field, that is we consider $$(1)_{\epsilon} \;\;\;\;\;\; \begin{cases} x'=y-\epsilon(x^3-x)\\ y'=-x\end{cases}$$ and corresponding $4$ dimensional system

$$(3)_{\epsilon} \;\;\;\;\;\begin{cases} x_1'=y_1-\epsilon(x_{1}^3+3x_{1}x_{2}^2+x_{1})\\ x_{2}'=y_{2}-\epsilon (3x_{1}^2x_{2}+x_{2}^3+x_2)\\ y_1'=-x_1\\ y_2'=-x_2\end{cases}$$

What can be said about the stability of $\gamma_{\epsilon}$, the unique limit cycles of $(1)_{\epsilon}$, as a periodic orbit of $(3)_{\epsilon}$?


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