The problem is to find the fixed points for the equation:
$ \ddot x + x + \alpha x^²= 0 $ (and then sketch the global flow of the equation) (for $\alpha>0$)
I know that for the autonomous equation $\dot x = f(x) $ the fixed points are given by solving that f(x)=0 (these are the critical points), but, how this would be in the case of an equation involving only a derivative of order two.
And, for example, what soul´d be the fixed points for the system:
$\dot a = a-b $
$\dot b = a+b $ ?
Best Answer
This equation is equivalent to a $2\times 2$ system $$ x_1'=x_2, \\ x_2'=-x_1-ax_1^2. $$ Critical point, is where the flux vanishes, i.e., $$ (x_1,x_2)=(0,0) \quad\text{and}\quad (x_1,x_2)=(0,-1/a). $$