Phase Plane Analysis
Trajectories of autonomous systems.
Key idea
Trajectories of autonomous systems.
In depth
For autonomous 2D systems, plot trajectories in the (x, y) plane without time. Equilibria and the eigenvalues of the Jacobian classify behaviour: node, saddle, spiral, centre.
Worked example
For x′ = −x, y′ = −2y, all trajectories approach the origin: a stable node.
Real-world uses
- Visualising whether a pendulum settles, swings or spins in mechanics and biology
Common mistake
Reading the plot as if the horizontal axis were time.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Phase Plane Analysis?
Trajectories of autonomous systems. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
Where is Phase Plane Analysis used in real life?
Real-world examples: Visualising whether a pendulum settles, swings or spins in mechanics and biology.
What should I learn before Phase Plane Analysis?
Make sure you are comfortable with Systems of Linear ODEs, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Phase Plane Analysis?
Reading the plot as if the horizontal axis were time.
How can I practise Phase Plane Analysis?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Phase Plane Analysis?
Move on to Equilibria and Stability next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.