I have the plane autonomous system
$\dfrac{dx}{dt}=x(1-2x-y)$
$\dfrac{dy}{dt}=y(1-x-2y)$
I need to show that the axes of the phase plane and the line $x=y$ are solution trajectories, but I don't know how to do this.
The other part of the question is as follows.
Use the Bendixson-Dulac theorem with $phi=\dfrac{1}{xy}$ to show that there are no closed trajectories in $R=\{(x,y):x>0, y>0\}$ . (I've done this bit) Comment on whether you can prove that there are no periodic functions in the entire phase plane including the origin.
For the last part, I have plotted the phase plane and there are no closed trajectories but how can i prove the last bit properly?
Thanks for any help.
Best Answer
The first part of this question is trivial. The axes of the phase plane correspond to $x=0$ and $y=0$. On substituting $x=0$ in the first equation, we get $\frac{dx}{dt} = 0$. i.e the particle stays on the $y$-axis. Similarly for the $x$-axis.
For $x=y$ line, we find that $\frac{dx}{dt} - \frac{dy}{dt} = 0$. i.e. $v_x = v_y$ and hence, the particle continues on the line.
I am trying the last part of your problem.