# Coupled differential equations and phase portraits

> IB Mathematics Applications and Interpretation HL · IB Math AI HL
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u4-coupled-differential-equations-and-phase/

This module covers linear 2D coupled differential equation systems, equilibrium classification via eigenvalues, phase portrait sketching rules, and real-world interpretation for population models.

**Prerequisites:** [First order separable differential equations](https://www.owlsprep.com/study/ib-math-ai-hl-u4-first-order-separable-de/); [Eigenvalues and eigenvectors for 2x2 matrices](https://www.owlsprep.com/study/ib-math-ai-hl-u3-eigenvalues-2x2/)

## Learning objectives

- Set up first-order linear coupled differential equations for real-world dynamic systems
- Calculate eigenvalues to classify equilibrium points of 2D linear systems
- Sketch fully labelled phase portraits for all standard system categories
- Interpret trajectory behaviour to predict long-term outcomes for population and epidemic models

## Structure of Linear Coupled 2D Systems

All linear homogeneous 2D coupled systems follow the standard matrix form, where the rates of change of variables x and y are linear combinations of x and y themselves. These systems are widely used to model interacting populations, chemical reaction rates, and compartmental epidemic flows.

**Standard coupled system form** — A is a constant 2x2 coefficient matrix that fully defines the dynamic behaviour of the system

*Notation:* \begin{bmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{bmatrix} = A \begin{bmatrix} x \\ y \end{bmatrix}

*Example:* For two competing species, A may contain positive growth terms and negative inter-species interaction terms

**Worked example:** Write the following coupled system in standard matrix form: dx/dt = 3x - 2y, dy/dt = x + 4y

1. Identify the coefficients of x and y in the dx/dt equation
2. These are 3 and -2, forming the first row of matrix A
3. Identify the coefficients of x and y in the dy/dt equation
4. These are 1 and 4, forming the second row of matrix A
5. $$A = \begin{bmatrix} 3 & -2 \\ 1 & 4 \end{bmatrix}$$

**Check your understanding**

Test your understanding of system structure

1. Which of the following is a valid linear coupled 2D system?

   - dx/dt = 2xy, dy/dt = x + y
   - dx/dt = 3x - y, dy/dt = -x + 2y
   - dx/dt = t + x, dy/dt = y

   *Why:* All terms are linear in x and y, with no t dependence or product terms

## Classifying Equilibrium Points via Eigenvalues

**Derivation:** Find the condition for non-trivial equilibrium points

*Starting from:* A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}

1. The trivial equilibrium at (0,0) always exists for homogeneous systems
2. Non-trivial equilibria exist only if det(A) = 0, otherwise (0,0) is the only equilibrium
3. Calculate eigenvalues of A by solving det(A - \lambda I) = 0

*Conclusion:* The sign and type of eigenvalues (real, complex, repeated) fully define the equilibrium classification

| Eigenvalue type | Equilibrium classification | Stability |
| --- | --- | --- |
| Two distinct positive real | Unstable node | All trajectories move away |
| Two distinct negative real | Stable node | All trajectories converge |
| One positive, one negative real | Saddle point | Unstable, only 2 trajectories converge |
| Complex with positive real part | Unstable spiral | Trajectories spiral outwards |
| Complex with negative real part | Stable spiral | Trajectories spiral inwards |
| Pure imaginary | Centre | Closed periodic orbits, neutral stability |

**Worked example:** Classify the equilibrium at (0,0) for system with A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}

1. Solve characteristic equation det(A - \lambda I) = 0
2. $$(1-\lambda)^2 - 4 = 0 \implies \lambda^2 - 2\lambda -3 =0$$
3. Factor to get eigenvalues \lambda = 3 and \lambda = -1
4. One positive, one negative real eigenvalue means this is a saddle point, unstable

## Sketching Fully Labelled Phase Portraits

**Exam command terms**

IB exam questions use specific command terms for phase portrait tasks

- **Sketch the phase portrait** — You must include eigenvector lines, direction arrows, at least 4 sample trajectories, and label the equilibrium classification

- **Draw the nullclines** — Mark lines where dx/dt=0 and dy/dt=0, where trajectories are horizontal or vertical respectively

> **Exam Sketching Tip**
>
> Always add direction arrows to every trajectory: for stable systems arrows point towards the equilibrium, for unstable systems arrows point away

**Worked example:** Sketch the phase portrait for a stable node with eigenvalues -1 and -2, eigenvectors (1,0) and (0,1)

1. Draw x and y axes, mark the equilibrium point at (0,0)
2. Draw the eigenvector lines along the x and y axes, add arrows pointing towards (0,0)
3. Add curved trajectories that approach the slower eigenvector (eigenvalue -1, x axis) as t \to \infty
4. Label the equilibrium as stable node, add direction arrows to all trajectories

## Interpreting Phase Portraits for Applied Models

For real-world models such as SIR epidemic compartments or competing species populations, phase portraits let you predict long-term outcomes without solving the full analytical solution. You can identify threshold conditions that lead to population extinction, coexistence, or epidemic fade-out.

**Worked example:** For a competing species model with a stable node at (200, 150), interpret the long-term behaviour

1. The stable node means all initial population values will converge to this equilibrium point
2. This tells you the two species will coexist at steady state populations of 200 and 150 respectively
3. No matter the initial non-zero population counts, neither species will go extinct

## Second Order Differential Equations via Coupling

A second order differential equation of the form d²x/dt² = f(x, dx/dt, t) can always be rewritten as a system of two coupled first order equations. This lets you apply the same numerical and phase portrait techniques to second order models, and it is a recurring IB AI HL Paper 3 task, especially for mechanical systems like a damped oscillator d²x/dt² + a dx/dt + bx = 0.

**Reduction to a coupled first order system** — Introduce a new variable y equal to the first derivative dx/dt. The single second order equation then becomes two first order equations in x and y, which you can solve numerically or analyse with eigenvalues.

*Notation:* \text{Let } y = \frac{dx}{dt}, \text{ then } \frac{dx}{dt} = y, \quad \frac{dy}{dt} = f(x, y, t)

*Example:* For \frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 2x = 0, set y = dx/dt to get \frac{dx}{dt} = y and \frac{dy}{dt} = -3y - 2x

**Worked example:** Write \frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 3x = 0 as a system of coupled first order equations, and state its coefficient matrix.

1. Let y = dx/dt, so the first equation is simply dx/dt = y
2. Rearrange the original equation for the highest derivative: d²x/dt² = -4(dx/dt) - 3x
3. Replace dx/dt with y and d²x/dt² with dy/dt to get the second equation
4. $$\frac{dx}{dt} = y, \quad \frac{dy}{dt} = -3x - 4y$$
5. In matrix form the coefficient matrix is:
6. $$A = \begin{bmatrix} 0 & 1 \\ -3 & -4 \end{bmatrix}$$

> **Euler's method and exact solutions**
>
> Once written as a coupled system, solve d²x/dt² + a dx/dt + bx = 0 numerically with Euler's method on your GDC, or classify its long-term behaviour from the eigenvalues of the matrix [[0, 1], [-b, -a]].

**Worked example:** Use Euler's method with step size h = 0.1 to estimate x(0.2) for \frac{d^2x}{dt^2} = -x, given x(0) = 1 and \frac{dx}{dt}(0) = 0. Compare with the exact solution x = \cos t.

1. Reduce to a system: let y = dx/dt, then dx/dt = y and dy/dt = -x
2. Euler update equations: x_{n+1} = x_n + h\,y_n and y_{n+1} = y_n + h(-x_n)
3. Step 1 (t = 0 to 0.1): x_1 = 1 + 0.1(0) = 1, y_1 = 0 + 0.1(-1) = -0.1
4. Step 2 (t = 0.1 to 0.2): x_2 = 1 + 0.1(-0.1) = 0.99, y_2 = -0.1 + 0.1(-1) = -0.2
5. $$x(0.2) \approx 0.99$$
6. The exact solution gives \cos(0.2) \approx 0.980, so Euler's method slightly overestimates here; a smaller step size h would reduce the error.

## Common pitfalls

- **Wrong:** Classifying a saddle point as unstable node
  - Why it fails: Saddle points have one positive and one negative eigenvalue, not two positive eigenvalues
  - Correct: Always check the sign of both eigenvalues before classification
- **Wrong:** Drawing trajectories that cross each other in the phase portrait
  - Why it fails: Uniqueness theorem for ODEs guarantees no two distinct trajectories can intersect
  - Correct: Ensure all trajectories never meet except at equilibrium points
- **Wrong:** Adding direction arrows pointing away from a stable spiral equilibrium
  - Why it fails: Negative real part of complex eigenvalues means trajectories must spiral inwards over time
  - Correct: Verify eigenvalue signs before adding direction arrows
- **Wrong:** Forgetting to label eigenvector lines in the phase portrait
  - Why it fails: IB exam mark schemes explicitly award marks for correctly drawn and labelled eigenvector lines
  - Correct: Mark and label all eigenvector lines before drawing curved trajectories
- **Wrong:** Treating a centre equilibrium as a stable spiral
  - Why it fails: Pure imaginary eigenvalues produce closed periodic orbits, not inward spiralling trajectories
  - Correct: Check that the real part of complex eigenvalues is exactly zero before drawing closed orbits

## Cheatsheet

| Eigenvalue condition | Equilibrium type | Stability | Trajectory shape |
| --- | --- | --- | --- |
| Two distinct real, both negative | Stable node | Asymptotically stable | Curves converge to equilibrium, tangent to slow eigenvector |
| Two distinct real, both positive | Unstable node | Unstable | Curves diverge away from equilibrium |
| Real, opposite signs | Saddle point | Unstable | Two trajectories converge, all others diverge |
| Complex, negative real part | Stable spiral | Asymptotically stable | Trajectories spiral inwards |
| Complex, positive real part | Unstable spiral | Unstable | Trajectories spiral outwards |
| Pure imaginary | Centre | Neutrally stable | Closed elliptical periodic orbits |

## What's next

Mastering linear coupled systems and phase portraits gives you the foundation to tackle far more realistic non-linear dynamic systems that appear frequently in IB AI HL Paper 3 problem sets. You will learn to analyse predator-prey Lotka-Volterra models, which produce closed periodic orbits representing natural population cycles, and extend your understanding to non-linear equilibrium classification via Jacobian matrices. You will also connect these concepts to numerical differential equation solving methods like Euler's method, which you can use to approximate trajectories for systems with no analytical closed-form solution. These skills are heavily weighted for the 20% Paper 3 investigation component of your final exam.

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