I am working on a problem out of "An Introduction to Applied Nonlinear Dynamical Systems and Chaos" by S. Wiggins. Section 3 is all about local bifurcations and I am asked to describe the bifurcation of the origin of the following systems, and to compare them.
$$\dot{x}=x-2y+\epsilon x$$ $$\dot{y}=3x-y-x^2$$
I put them in normal form, $$\begin{bmatrix}\dot{x}\\ \dot{y}\end{bmatrix}=\begin{bmatrix} 1 & -2 \\ 3 & -1 \end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}+\begin{bmatrix}\epsilon x\\-x^2\end{bmatrix}.$$
I think this is the case because the linear part has eigen values with zero real parts. I'm supposed to introduce a parameter $\dot{\epsilon}=0$ to find the equation of the stable manifold using the standard partial differential equation $$\mathcal{N}(h(x,\epsilon))=0=D_xh(x,\epsilon)\left[Ax+f(x,h(x,\epsilon),\epsilon)\right]-Bh(x,\epsilon)-g(x,h(x,\epsilon),\epsilon)$$.
I think $A$ is the linear part of the above equation, $B$ is zero, $g$ is zero and $f$ is the nonlinear part. I am unsure of how to fully use this equation. My professors lecture is incomplete and I cannot find anything online.
The system that I am comparing it with is $$\dot{x}=x-2y+\epsilon x^2$$ $$\dot{y}=3x-y-x^2$$
In this case all turns out to be simpler. Note first that the equation can be written in the form $$ \begin{bmatrix}\dot{x}\\ \dot{y}\end{bmatrix}=\begin{bmatrix} 1+\epsilon & -2 \\ 3 & -1 \end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}+\begin{bmatrix}0\\-x^2\end{bmatrix}. $$ The eigenvalues of the matrix $$\begin{bmatrix} 1+\epsilon & -2 \\ 3 & -1 \end{bmatrix}$$ are $$ \frac{1}{2} \left(\epsilon\pm\sqrt{\epsilon^2+4 \epsilon-20}\right)\approx \frac{\epsilon}{2} \pm i\sqrt{5} $$ for $\epsilon$ very close to zero.
This implies that you pass from a stable focus to an unstable focus as $\epsilon$ travels from some negative value close to zero to some positive value close to zero (since the eigenvalues have nonzero real part the term $-x^2$ does not change the topological behavior of the linear part in a small neighborhood of the origin).
These are roughly the phase portraits in a neighborhood of the origin:
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for small $\epsilon<0$ and
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for small $\epsilon>0$.