The phase plane is a two-dimensional space where we plot one variable on the horizontal axis and the other on the vertical axis. A solution to the system becomes a trajectory โ a path through the plane traced as time flows.
To see the โpushโ of the system without fully solving it, we can draw a direction field (or slope field): at each point \((x, y)\text{,}\) we sketch a small arrow showing the vector \((dx/dt, dy/dt)\text{.}\)
In the phase plane, every trajectory moves straight toward the origin โ \(x\) and \(y\) are simply decaying independently, so the arrows point inward along both axes.
For more interesting systems โ like ones with partial or full coupling โ the direction field can show curved paths, spirals, or saddle-shaped flows. This visual approach helps us predict system behavior even before we dive into equations.
So far, weโve seen examples of systems where variables evolve independently or influence each other. Now letโs step back and look at how to visualize a system as a whole.
The phase plane is a two-dimensional space where we plot one variable on the horizontal axis and the other on the vertical axis. A solution to a system like
To understand how the system behaves without solving it, we can draw a slope field or direction field: a grid of arrows showing the direction of motion at each point.
This interactive slope field shows the direction of motion for the system \(x' = x + y\text{,}\)\(y' = -x + y\text{.}\) Each arrow represents the vector \((dx/dt, dy/dt)\) at that location.
Some systems push every solution toward a single point (stable equilibrium). Others send trajectories outward (unstable). Some cause spirals, as if the solution is both rotating and growing or shrinking at the same time.
Later, weโll connect these patterns to the algebra of the systemโs coefficients. But even now, these โbig pictureโ behaviors help us think about what the math is saying.
