Phase Plane Analysis

You will learn how to sketch and interpret phase portraits, classify equilibrium points, and understand the long-term behavior of solutions to systems of differential equations.

For a system of two first-order equations

$$x' = f(x, y)$$ $$y' = g(x, y)$$

a phase portrait is a plot in the $xy$-plane (called the phase plane) showing many solution trajectories. Each point $(x, y)$ has a direction vector $(f(x,y), g(x,y))$ drawn as a short arrow.

Key features we look for:

• Equilibrium points: Where $f(x,y) = 0$ and $g(x,y) = 0$
• Trajectories: Curves that follow the direction field
• Stability: Whether nearby solutions move toward or away from equilibria

The visual shows a saddle point phase portrait. Trajectories approach the origin along one eigenvector direction (stable manifold) and flee along the other (unstable manifold).

• Stable Node: All nearby solutions approach the point (both eigenvalues negative)
• Unstable Node: Solutions move away (both eigenvalues positive)
• Saddle Point: Solutions approach along one direction and leave along another (eigenvalues of opposite sign)
• Center: Closed orbits around the point (purely imaginary eigenvalues)
• Stable Spiral: Solutions spiral inward (complex eigenvalues with negative real part)
• Unstable Spiral: Solutions spiral outward (complex eigenvalues with positive real part)

Example 1: Classifying Equilibria

Analyze the system $x' = x - 2y$, $y' = 3x - 4y$.

Solution: The coefficient matrix is $A = \begin{bmatrix} 1 & -2 \\ 3 & -4 \end{bmatrix}$.

Characteristic equation: $\lambda^2 + 3\lambda + 2 = 0 \implies \lambda = -1, -2$

Both eigenvalues negative → Stable Node.

Example 2: Saddle Point

Analyze $x' = 2x - y$, $y' = 3x - 2y$.

Solution: $A = \begin{bmatrix} 2 & -1 \\ 3 & -2 \end{bmatrix}$. Eigenvalues: $\lambda^2 - 1 = 0 \implies \lambda = 1, -1$

Opposite signs → Saddle Point. Solutions approach along the eigenvector for $\lambda = -1$ and diverge along the eigenvector for $\lambda = 1$.

Example 3: Spiral

For the system with $A = \begin{bmatrix} -1 & 2 \\ -2 & -1 \end{bmatrix}$, eigenvalues are $-1 \pm 2i$.

Complex with negative real part → Stable Spiral. Solutions spiral inward toward the origin.

Phase plane analysis is used extensively in biology (predator-prey cycles), chemistry (reaction dynamics), economics (market stability), and engineering (stability of control systems). Ecologists use it to predict whether two competing species will coexist or if one will drive the other to extinction. Engineers use it to ensure that mechanical or electrical systems remain stable.