Help me understand this equality? (assessing stabilty of critical points of a dynamic system)

Question

I'm reading through the solution to a question I became stuck on, and I'm struggling to understand why the following is true:

$$ -kx_{\pm}-k_{1}x^{3}_{\pm}-k\xi-3k_{1}x^{2}_{\pm}\xi = 2k\xi $$

Any help would be greatly appreciated, thank you!

Answer 1

The dynamical system is: $\dot{x}=F(x)$

The equation you are asking about concerns the study of critical points (i.e. points where $F(x)=0$) when $k<0$.

The three solution of $F(x)=0$ in this case are $x_0,x_{\pm}$.

You have to exploit the fact that:

• $F(x_{\pm})=0$
• $\vert k\vert = -k \quad$ (we know $k$ is negative)
• $x_{\pm} =\pm \sqrt{\dfrac{\vert k \vert}{k_1}}$, therefore $x_{\pm}^2=\cdots$