Exercise :
Consider the following system of differential equations that is dependent by two parameters $κ$ and $λ$ : $$x'=-x^3-κxy^3 + xy$$ $$y'=-y+yx^2-x^2+λ$$ Find the curve of the branch around the origin of the $(κ,λ)$ space for stationary points around $O(0,0)$. Moreover draw phase portraits for every area of the parameter space $(κ,λ)$
Discussion/Question :
This is an exercise under the title Zero Eigenvalue : Stability and Branches. I know how to work around systems with a zero eigenvalue and non-hyperbolic stationary point as described in another question of mind, here.
In this case though, things get a little complicated since the stationary point isn't the origin $O(0,0)$. As one can see easily, a stationary point is $(0,λ)$.
My first question is exactly the first part of the exercise : How would one find the branch curve around the origin of $(κ,λ)$ ? I know how to find a branch point in a $1\times 1$ system but not in higher dimensions. Also, what does the phrase : origin of $(κ,λ)$ space mean ? Does it mean the space generated by a $2 \times 2$ vector as $\text{span}[(κ,λ)]$ ?
My second question is : The phrase "around $(0,0)$ seems too broad. Is $(0,λ)$ considered "around" $(0,0)$ for $ λ << 1$ ? Finally, how would one draw phase portraits with respect to a space that yields infinite pairs ?
I would really appreciate if someone could elaborate thoroughly on how to proceed on such a problem, as I'm trying to cover every possible exercise for my upcoming term exams in 9 days.