Stability of a feedback system Take the following feedback system:
$\dot{x} = (\theta - k_1) x - k_2  x^3$
Now my book says:
For $\theta > k_1$, the equilibrium $x = 0$ is unstable. I wonder why...
Furthermore my book indicates that it is easy to see that $x(t)$ will converge to one of the two new equilibria $\pm \sqrt{\frac{\theta-k_1}{k_2}}$. Again, how did they obtain this result?

Question 1: From stability theory it is known, that 

a fixed point $x^*$ of $\dot{x}=f(x)$ is stable $\Leftrightarrow$ all eigenvalues of the jacobian $f'(x^*)$ have negative real parts. 

For $f(x)=(\theta - k_1) x - k_2  x^3$ and $x^*=0$ this restricts to $\theta<k_1$
Question 2: You can use the same argument.
For $\theta>k_1$ the new equilibria become stable, as (assuming $k_2>0$) we have
$$f'\left(\pm \sqrt{\frac{\theta-k_1}{k_2}}\right)=\theta-k_1-3k_2\frac{\theta-k_1}{k_2}=-2(\theta-k_1)<0$$
This fundamental change of the dynamics at $\theta=k_1$ is known as a pitchfork bifurcation.
