I am a Master 1 student studying dynamical systems. I am new to it. There's a problem, I have with invariant sets. Excuse me, I didn't know how to start the following question.
I have the following system of ODEs
$$\begin{aligned} x_{1}' &= -\gamma_{1}x_{1} + (1-\alpha)\phi(x_{3})\\ x_{2}' &= -\gamma_{2}x_{2} + \alpha\phi(x_{3})\\ x_{3}' &= \gamma_{2}x_{2} - \phi(x_{3}) + u\end{aligned}$$
where $0 < \alpha < 1$ is a given parameter, $u > 0$ is a control input, $\gamma_{1},\gamma_{2}$ are positive constants and
$$\phi(x_{3}) = k_{1} x_{3} e^{-k_{2} x_{3}}$$
with $k_{1}, k_{2} > 0$ are given constants. Show that the set of $x=(x_{1},x_{2},x_{3}) \in \Bbb R_{\geq 0}^3$ such that,
$$\begin{aligned} (1-\alpha)\phi(x_{3}) &\leq \gamma_{1}x_{1} < u\\ \alpha\phi(x_{3}) &\leq \gamma_{2} x_{2}\\ \phi'(x_{3}) &< 0\end{aligned}$$
is an invariant set.
Attempted Solution:
Let $f_{1}=-\gamma_{1}x_{1} + (1-\alpha)\phi(x_{3})$, $f_{2}=-\gamma_{2}x_{2} + \alpha\phi(x_{3})$ and $f_{3}=\gamma_{2}x_{2} - \phi(x_{3})+u$ .\
We have $$ div (f)= \dfrac{\partial f_{1}}{\partial x_{1}}x'_{1}+\dfrac{\partial f_{2}}{\partial x_{2}}x'_{2}+\dfrac{\partial f_{3}}{\partial x_{3}}x'_{3}$$
I tried to evaluate $div(f)$ along the boundary of the set of interest:
$$\begin{aligned} (1-\alpha)\phi(x_{3}) &= \gamma_{1}x_{1} = u\\ \alpha\phi(x_{3}) &=\gamma_{2} x_{2}\\ \phi'(x_{3})&=0\end{aligned}$$
And i got: $div(f)=0$.\ It means that there is a balanced set of outings and entries with respect to the vector fields.
Please is the method correct? If yes, what can i deduce from the answer? Does if mean that the set is invariant? Thanks