Phase Plane Analysis

Phase plane analysis studies autonomous systems of two first-order ODEs qualitatively, using geometric methods to understand the global behavior of solutions without solving the equations explicitly.

Autonomous Systems

Definition6.1Autonomous system

An autonomous system in $\mathbb{R}^2$ is

$\frac{dx}{dt} = f(x, y), \qquad \frac{dy}{dt} = g(x, y)$

where $f, g$ do not depend explicitly on $t$ . The phase plane is the $(x,y)$ -plane, and solutions trace out trajectories (or orbits) in this plane. The vector field $(f,g)$ determines the direction of motion at each point.

Definition6.2Equilibrium point

An equilibrium point (or critical point, fixed point) is $(x_0, y_0)$ where $f(x_0, y_0) = g(x_0, y_0) = 0$ . At an equilibrium, the constant function $\mathbf{x}(t) = (x_0, y_0)$ is a solution.

ExampleSimple pendulum

The undamped pendulum $\theta'' + \sin\theta = 0$ gives the system $x' = y$ , $y' = -\sin x$ . Equilibria: $(n\pi, 0)$ for $n \in \mathbb{Z}$ . Even points ( $n$ even) are centers; odd points ( $n$ odd) are saddle points. The phase portrait shows closed orbits around centers connected by separatrices through saddle points.

Classification of Linear Systems

Theorem6.1Classification of linear equilibria

For the linear system $\mathbf{x}' = A\mathbf{x}$ with eigenvalues $\lambda_1, \lambda_2$ of $A$ :

| Eigenvalues | Type | Phase portrait | |------------|------|----------------| | $\lambda_1 > \lambda_2 > 0$ | Unstable node | Trajectories radiate outward | | $\lambda_1 < \lambda_2 < 0$ | Stable node | Trajectories converge to origin | | $\lambda_1 > 0 > \lambda_2$ | Saddle | Hyperbolic trajectories | | $\alpha \pm i\beta$ , $\alpha > 0$ | Unstable spiral | Outward spirals | | $\alpha \pm i\beta$ , $\alpha < 0$ | Stable spiral | Inward spirals | | $\pm i\beta$ (pure imaginary) | Center | Closed elliptical orbits | | $\lambda_1 = \lambda_2 \neq 0$ , 2 eigenvectors | Star node | Straight-line trajectories | | $\lambda_1 = \lambda_2 \neq 0$ , 1 eigenvector | Degenerate node | Tangent to eigenvector |

Nullclines and Direction Fields

Definition6.3Nullclines

The $x$ -nullcline is $\{(x,y) : f(x,y) = 0\}$ (where $dx/dt = 0$ , trajectories are vertical). The $y$ -nullcline is $\{(x,y) : g(x,y) = 0\}$ (where $dy/dt = 0$ , trajectories are horizontal). Equilibria are intersections of nullclines.

ExampleCompeting species model

$x' = x(3-x-2y)$ , $y' = y(2-x-y)$ . Nullclines: $x = 0$ , $x+2y = 3$ ( $x$ -null); $y = 0$ , $x+y = 2$ ( $y$ -null). Equilibria: $(0,0), (3,0), (0,2), (1,1)$ . Analysis of the direction field in each region reveals the competitive exclusion dynamics.

RemarkQualitative analysis strategy

Find all equilibria (intersect nullclines).
Classify each equilibrium by linearization (compute $Df$ and its eigenvalues).
Draw nullclines and determine flow directions in each region.
Sketch trajectories consistent with the local and global information.