Phase Planes Show the Geometry of Two-State Systems

For $x'=f(x,y)$ and $y'=g(x,y)$, the vector $(f,g)$ gives the direction of motion at each point of the $xy$-plane. A solution traces a phase-plane trajectory.

Nullclines and equilibria

The $x$-nullcline satisfies $f(x,y)=0$ and the $y$-nullcline satisfies $g(x,y)=0$. Their intersections are equilibria. Nullclines divide the plane into regions where each state component increases or decreases.

Reading trajectories

Closed curves suggest sustained cycles. Inward spirals indicate damped oscillation. Paths approaching along one direction and departing along another indicate saddle behavior.

Build a sign chart from nullclines

For $x'=x(1-y)$ and $y'=y(x-1)$, the nullclines are $x=0$, $y=1$, $y=0$, and $x=1$. In the positive quadrant, test the four regions divided by $x=1$ and $y=1$ to determine left/right and up/down motion. The equilibrium $(1,1)$ is encircled by trajectories in this idealized model.

Check your understanding

Can two trajectories of a smooth autonomous system cross at a non-equilibrium point?

Show the reasoning

No, under uniqueness conditions. A crossing would assign two solution directions or futures to the same state.