bifurcation with more than parameter

Question

Problem: Consider the scalar differential equation depending on the parameters $\alpha_1, \alpha_2$ ∈ $\Re$

$x˙ = \alpha_1 + \alpha_2 x − x^2$.

Find a change of coordinates $y = \phi(x)$ such that $y˙ = μ − y^2$ for some constant $μ$ = $μ$($\alpha_1$, $\alpha_2$) ∈ $\Re$.

Could you guys help me to do this problem?

thanks,

Answer 1

Note that

$x^2 - \alpha_2 x - \alpha_1 = (x - \dfrac{\alpha_2}{2})^2- (\alpha_1 + (\dfrac{\alpha_2}{2})^2); \tag{1}$

set

$y = x - \dfrac{\alpha_2}{2} \tag{2}$

and

$\mu = \alpha_1 + (\dfrac{\alpha_2}{2})^2; \tag{3}$

then

$\dot y = \dot x = -x^2 + \alpha_2 x + \alpha_1 = \mu - y^2, \tag{4}$

as some simple algebra will verify.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!