Hope this isn't a duplicate.
I was trying to solve the following problem,
Consider the following non-linear system : $$ \dot x = y-a_0-a_1 x - a_2 x^2$$ $$\dot y = -x$$ , where $a_0,a_1,a_2 \in \Bbb R \text{ and } a_2 \ne 0$ . Then prove that it has no isolated periodic orbits.
My attempt :
I considered the associated Linear system at the origin i.e.
$$Df(0) = \begin{bmatrix} -a_1 & 1 \cr -1 & 0 \end{bmatrix}$$
And then considering the eigenvalues I ended up having$$\lambda = \frac{-a_1 \pm \sqrt{{a_1}^2 -4}}{2}$$ and 3 cases : (i) ${a_1}^2 > 4$ , (ii) ${a_1}^2=4$ i.e. $a_1 = \pm 2$ and (iii) ${a_1}^2 < 4$ .
$(ii) \implies$ it (the Linear system) is either a stable or unstable node.
$(iii) \implies$ it is a center .
$(i) \implies$ it is either a stable node or an unstable node or a saddle.
All the observations $(i),(ii),(iii)$ are the expected behaviour of the Linear system at the origin $Df(0)$ .
Am I on the right track ? What do I have to do next ?
The fact that the origin is the only critical point of the non-linear system, would it help me to argue?
Thanks in advance for help.