Introduction to Phase Portraits

A phase portrait is a visual tool used to understand the long-term behaviour of a system of coupled differential equations , without needing to solve the equations explicitly. Instead of plotting $x$ or $y$ against time, we plot the trajectory of the point $(x (t), y (t))$ directly in the $x y$ -plane (called the phase plane).

In AHL 5.17, we focus on linear systems of the form:

$\begin{cases} \dfrac{dx}{dt} = ax + by \$ 6pt] \dfrac{dy}{dt} = cx + dy \end{cases}$$

where $a, b, c, d$ are real constants. This can be written compactly in matrix form:

$\frac{d}{d t} (x y) = (a c b d) (x y) = A x$

The matrix $A$ is called the coefficient matrix of the system, and its eigenvalues and eigenvectors completely determine the qualitative behaviour of the phase portrait.

Analogy

Think of the phase portrait like a wind map. At every point $(x, y)$ in the plane, the system tells you the "wind direction" , the direction and speed at which the state of the system is moving. The trajectories are the paths a leaf would follow if dropped into that wind field.

Equilibrium Points

Equilibrium Point: A point $(x^{*}, y^{*})$ where $\frac{d x}{d t} = 0$ and $\frac{d y}{d t} = 0$ simultaneously. At an equilibrium point, the system is stationary , there is no change over time.

For a linear system $\dot{x} = A x$ , if $A$ is invertible (i.e. $det (A) \neq = 0$ ), the only equilibrium point is the origin $(0, 0)$ .

The nature of this equilibrium , whether trajectories approach it, move away from it, or orbit around it , depends entirely on the eigenvalues of $A$ .

Example

For the system:
$\begin{cases} \dfrac{dx}{dt} = 2x - y \$ 6pt] \dfrac{dy}{dt} = x + y \end{cases}$$

Set both equations to zero: $2 x - y = 0$ and $x + y = 0$ . Solving gives $x = 0, y = 0$ . The only equilibrium point is $(0, 0)$ .

Note

In AHL 5.17, we only consider systems with distinct, non-zero eigenvalues. This ensures the origin is always the unique equilibrium point and covers the most instructive range of behaviours.

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Introduction to Phase Portraits

In AHL 5.17, we focus on linear systems of the form:

$\begin{cases} \dfrac{dx}{dt} = ax + by \$ 6pt] \dfrac{dy}{dt} = cx + dy \end{cases}$$

where $a, b, c, d$ are real constants. This can be written compactly in matrix form:

$\frac{d}{d t} (x y) = (a c b d) (x y) = A x$

The matrix $A$ is called the coefficient matrix of the system, and its eigenvalues and eigenvectors completely determine the qualitative behaviour of the phase portrait.

Analogy

Equilibrium Points

Equilibrium Point: A point $(x^{*}, y^{*})$ where $\frac{d x}{d t} = 0$ and $\frac{d y}{d t} = 0$ simultaneously. At an equilibrium point, the system is stationary , there is no change over time.

For a linear system $\dot{x} = A x$ , if $A$ is invertible (i.e. $det (A) \neq = 0$ ), the only equilibrium point is the origin $(0, 0)$ .

The nature of this equilibrium , whether trajectories approach it, move away from it, or orbit around it , depends entirely on the eigenvalues of $A$ .

Example

For the system:
$\begin{cases} \dfrac{dx}{dt} = 2x - y \$ 6pt] \dfrac{dy}{dt} = x + y \end{cases}$$

Set both equations to zero: $2 x - y = 0$ and $x + y = 0$ . Solving gives $x = 0, y = 0$ . The only equilibrium point is $(0, 0)$ .

Note

In AHL 5.17, we only consider systems with distinct, non-zero eigenvalues. This ensures the origin is always the unique equilibrium point and covers the most instructive range of behaviours.

Free preview

AHL 5.17—Phase portrait

Introduction to Phase Portraits

Equilibrium Points

11 more sections in this topic

AHL 5.17—Phase portrait

Introduction to Phase Portraits

Equilibrium Points

11 more sections in this topic