Determine perturbation around saddles for 2D system

Question

Consider the system $$ \dot{x} = \mu + x^2 - xy\\ \dot{y} = y^2 - x^2 -1 $$ with $\mu \neq 0$ and small. I need to determine the Taylor/perturbation expansion of the two saddles $a^+$ and $a^-$ up to quadratic terms in $\mu$.

I don't understand how I can determine this perturbation in this sytem. In the one dimensional case, I would write $\phi(t) = x^* + \epsilon h(t)$, where $x^*$ is a fixed point, and I would then work with the differential equation for $h$. But in this case, the system is two dimensional, so I don't really know how I should define this $h$. I can look at the linearised sytem, but this isn't a perturbation.

I already determined the coordinates of the saddles for what it's worth, namely $\left(\tfrac{\mu}{\sqrt{1-2\mu}}, \tfrac{1-\mu}{\sqrt{1-2\mu}}\right)$ and $\left(\tfrac{-\mu}{\sqrt{1-2\mu}}, -\tfrac{1-\mu}{\sqrt{1-2\mu}}\right)$, so they are approximately $(0, 1)$ and $(0, -1)$.

Answer 1

You have already determined the saddles, giving $$ \left(\tfrac{\mu}{\sqrt{1-2\mu}}, \tfrac{1-\mu}{\sqrt{1-2\mu}}\right)\quad\text{and}\quad\left(\tfrac{-\mu}{\sqrt{1-2\mu}}, -\tfrac{1-\mu}{\sqrt{1-2\mu}}\right) $$ (perhaps you want to take $\mu<1/2$?). So you only need to compute their Taylor series, and you don't need the differential equation for that.

Answer 2

You want to write $x$ and $y$ as asymptotic series in powers of $\mu$: $x=x_0+\mu x_1+\mu^2 x_2+\ldots$, and similar for $y$. Substituting into the equations gives (to $O(\mu^2)$) $$\mu+x_0^2+2\mu x_0x_1+\mu^2\left(x_1^2+2x_0x_2\right)-x_0y_0-\mu\left(x_1y_0+x_0y_1\right)-\mu^2\left(x_0y_2+x_2y_0+x_1y_1\right)=0$$ and $$y_0^2+2\mu y_0y_1+\mu^2\left(y_1^2+2y_0y_2\right)-x_0^2-2\mu x_0x_1-\mu^2\left(x_1^2+2x_0x_2\right)-1=0.$$

At $O(1)$, the equations are $$x_0^2-x_0y_0=0$$ $$y_0^2-x_0^2-1=0$$ with solution $x_0=0$ and $y_0=\pm1$.

At $O(\mu)$, the equations are, $$1+2x_0x_1-x_1y_0-x_0y_1=0\Rightarrow1\mp x_1=0$$ $$2y_0y_1-2x_0x_1=0\Rightarrow\pm2y_1=0$$ so $y_1=0$ and $x_1=\pm1$.

At $O(\mu^2)$, the equations are, $$x_1^2+2x_0x_2-x_0y_2-x_2y_0-x_1y_1=0\Rightarrow1\mp x_2=0$$ $$y_1^2+2y_0y_2-x_1^2-2x_0x_2=0\Rightarrow\pm2y_2-1=0$$ so $x_2=\pm1$ and $y_2=\pm1/2$.

So, the $O(\mu^2)$ approximation for the two saddles are $$a^\pm=\pm\left(\mu+\mu^2,1+\frac{\mu^2}{2}\right).$$