Periodical solutions of this system of differential equations We have the system of differential equations:
$$x'=(1+m)y+x(1-(x^2+y^2))(4-(x^2+y^2)),$$
$$y'=-x+y(1-(x^2+y^2))(4-(x^2+y^2)),$$
with $m>0$.

  
*
  
*How do I show that $(0,0)$ is the only (instable) critical point?  
  
*How do I show that there is an $m_0$ such that for $0<m<m_0$ this system has TWO periodic (not-constant) solutions?
  

What I have done so far: I wrote the sytem above in polar coordinates:
$$r'=mr\sin\theta\cos\theta+r(1-r^2)(4-r^2),$$
$$\theta'=-1-m\sin^2\theta.$$
But I don't know how this can help me.
For question 2. I must use Poincare-Bendixson, but I don't see how exactly.
 A: First of all the proof that the origin is the only critical point can be proved following the comments of @Hans Engler. I will skip the details.
For the second question there are possibly other ways to prove this by I will present one that looks appealing to me. Consider the function 
$$V:=r^2=x^2+y^2$$
The time derivative of $V$ is given by 
$$\dot{V}=-2m\: xy+2V(1-V)(4-V)$$
that can be written equivalently as a quadratic form
$$\dot{V}=\left[\matrix{x & y}\right]\left[\matrix{2(1-V)(4-V) & -m\\-m & 2(1-V)(4-V)}\right]\left[\matrix{x \\ y}\right]$$
From the above equation $\dot{V}< 0$ iff
$$(i)\quad 1<V<4\\(ii)\quad m<2(V-1)(4-V)$$
and $\dot{V}>0$ iff
$$(i)\quad 0<V<1\quad or \quad V>4\\(ii)\quad m<2(1-V)(4-V)$$
Note that the maximum value of $2(V-1)(4-V)$ when $V\in(1,4)$ is $4.5$ and is achieved if $V=2.5$. Thus, for $0<m<m_0=4.5$ we have 
$$\dot{V}<0\text{   whenever  } V=2.5$$
Also, if $0<m<m_0=4.5$ then for any $\epsilon\in\left(0,\frac{5-3\sqrt{2}}{2}\right]$ we have that 
$$\dot{V}>0 \text{   whenever  } V=\epsilon$$
This means that if initially $V(0)\in[\epsilon,2.5]$ then $V(t)\in[\epsilon,2.5]$ for all $t\geq 0$. Define now the bounded closed subset of the plane  $R:=\{(x,y)|\epsilon\leq V \leq 2.5\}$. Obviously $R$ does not contain fixed points (the origin) and all trajectories starting in $R$ are confined in $R$. Thus, the Poincare-Bendixson theorem can be directly applied to prove  the existence of a periodic trajectory inside $R$. 
