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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

felix gabin Djumene
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  • Along the boundary of the given set, check from which direction the wind is blowing. – Rodrigo de Azevedo Jul 31 '20 at 10:26
  • The set is made up of 4 inequalities, show i test the one by one? Concerning the direction in which the wind is blowing, excuse me but i don't really get the point. – felix gabin Djumene Jul 31 '20 at 11:14
  • Is $u$ a constant or a control input? What else do you know about $\phi$? You need to check whether the wind is always blowing into the candidate invariant set, i.e., if the flow is always into it. It cannot flow out of it. Check the inner product of $\dot x$ and the vector normal to the boundary of the candidate invariant set at each point of the boundary. – Rodrigo de Azevedo Jul 31 '20 at 11:20
  • Ohhh excuse me... $u$ is a control input and $\phi(x_{3})=k_{1}x_{3}e^{-k_{2}x_{3}}$ with $k_{1}, k_{2}$ are positive constants. – felix gabin Djumene Jul 31 '20 at 11:22
  • Is the third equation correct? Most of the system looks like a reaction equation, so should what is removed in the second equation in $-γ_2x_2$ better be added in the equation for the "reaction product" $x_3$? – Lutz Lehmann Jul 31 '20 at 13:33
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    Lutz Lehmann, the equation is correct. I saw the question in a textbook. Could post a screenshot of the question but don't know how to do such.. – felix gabin Djumene Aug 01 '20 at 06:19
  • Ok. Its no problem. I corrected it – felix gabin Djumene Aug 01 '20 at 10:12
  • Please, is my attempted solution correct? – felix gabin Djumene Aug 01 '20 at 10:16

0 Answers0