# Why do these limit cycles appear?

Consider the system of ODEs \begin{align*}x'&=y+x(\varepsilon+\ell_1(x^2+y^2)+\ell_2(x^4+y^4)) \\y'&=-x+y(\varepsilon+\ell_1(x^2+y^2)+\ell_2(x^4+y^4)).\end{align*}

Going through the linearization process at the equilibrium point $(0,0)$, we find the eigenvalues $\lambda_{1,2}(\varepsilon)=\varepsilon\pm i$. Using the Hopf Bifurcation theorem (p. 5-6), it is easy enough to show that a Andronov-Hopf Bifurcation appears when $\varepsilon=0$.

The issue is, if $\ell_1=0$, we fail to satisfy the conditions of the Hopf Bifurcation theorem, but a bifurcation still appears.

For example, suppose $\ell_1=0$ and $\ell_2=-1$. The following two images show our system for $\varepsilon=-0.1$ and $\varepsilon=0.1$, respectively.

So my first question is:

How would we show, analytically, that the system \begin{align*}x'&=y+x(\varepsilon+\ell_2(x^4+y^4)) \\y'&=-x+y(\varepsilon+\ell_2(x^4+y^4))\end{align*} has a Hopf Bifurcation at $\varepsilon=0$?

Secondly, if $\ell_1$ and $\ell_2$ have different signs, then we can have two limit cycles, depending on $\varepsilon$. For example, suppose $\ell_1=1$ and $\ell_2=-1$. The next two images show when $\varepsilon=0.1$ and $\varepsilon=-0.1$, respectively.

Now, these two limit cycles will eventually collapse on each other as $\varepsilon$ decreases, but otherwise, there is always at least one limit cycle. So my second question is:

How would we show, analytically, that the system \begin{align*}x'&=y+x(\varepsilon+(x^2+y^2)-(x^4+y^4)) \\y'&=-x+y(\varepsilon+(x^2+y^2)-(x^4+y^4)) \end{align*} has a single limit cycle for all $\varepsilon>0$?

• +1: Might I ask how you created these good-looking phase portraits :D? – MrYouMath Sep 26 '17 at 15:56
• @MrYouMath Thank you! It was made with MatLab. – Bonnaduck Sep 27 '17 at 5:20

• Just to make sure I understand. I found that $\frac{dr}{d\theta}=-\varepsilon r-\ell_1r^3-\ell_2(\cos^4\theta+\sin^4\theta)r^5$. Does this immediately imply that the second and third terms are my first and second Lyapunov values? – Bonnaduck Mar 19 '17 at 20:48
• Okay. So, using the notation you used, I have $R_1(\theta)=-\varepsilon$, $R_3(\theta)=-\ell_1$, and $R_5(\theta)=-\ell_2(\cos^4(\theta)+\sin^4(\theta))$, with $R_i(\theta)=0$ for all other $i$. I am still confused on how you proceed from here. I get equating $u'_1(\theta)=R_1(\theta)u_1$, etc., but after that I am a bit lost. – Bonnaduck Mar 20 '17 at 0:39