Showing a trajectory is attracted to an invariant line

190 Views Asked by At

Given the system $x' = ax^c - \phi x$, $y' = by^c - \phi y$, $\phi = ax^c + b^c$

Part a was just showing that the derivative is $0$, which I was able to just fine.

The second part of the question says: Show that all trajectories starting in the positive quadrant are attracted to the invariant line $x + y =1$ found in part (a). Thus, the analysis of the system’s longterm dynamics boils down to figuring out how trajectories flow along this invariant line

I have no idea how to even start so any help would be appreciated.

Thanks