Does there exist a system which has closed orbit but no constant one? Can you please give me an example of an ODE system which has no constant orbit or fixed point but it has closed orbit?
Thank you very much
 A: Consider following ODE in $\mathbb{R}^3$ where $\beta\ne 0$:
$$\begin{cases}
\dot{x} &= -\beta y + 2zx\\
\dot{y} &= \;\;\;\beta x + 2zy\\
\dot{z} &=   4 - x^2 - y^2 + z^2
\end{cases}
$$
We have 
$$\dot{x}^2 + \dot{y}^2+\dot{z}^2 = \beta^2 (x^2+y^2) + 16z^2 + (x^2+y^2+z^2-4)^2 \ne 0$$
over all $\mathbb{R}^3$. This flow doesn't have any fixed point at all.
On the other hand, if $\beta \in \mathbb{Q}$, then every orbit of it except the one passing through the origin is a closed orbit. To see this, embed $\mathbb{R}^3$ into $S^3 \subset \mathbb{R}^4 \sim \mathbb{C}^2$ through the mapping:
$$\mathbb{R}^3 \ni (x,y,z) \quad\longrightarrow\quad (U,V) = \bigg(\frac{x+iy}{1+\frac{r^2}{4}},
\frac{z + i(1 - \frac{r^2}{4})}{1+ \frac{r^2}{4}}\bigg) \in \mathbb{C}^2$$
The ODE can be rewritten as
$$\begin{cases}\dot{U} &= \;\;\;i\beta U\\\dot{V} &= -i4V\end{cases}$$
The corresponding flow is a rotation in $U$ direction with speed $\beta$ and $V$ direction with speed $-4$. If $\beta$ is rational, then this defines a closed orbit in $S^3$ and hence one in $\mathbb{R^3}$ (except the one with $|U| = 0$ which escapes to infinity).
