Equilibria and Stability
Classify fixed points as stable, unstable or saddle.
Key idea
Classify fixed points as stable, unstable or saddle.
In depth
An equilibrium has x′ = 0. It is stable if nearby solutions stay near, asymptotically stable if they converge. For x′ = f(x), stable if f′(x*) < 0; for systems, all Jacobian eigenvalues must have negative real parts.
Worked example
x′ = x(1 − x): at x = 0, f′ = 1 > 0 (unstable); at x = 1, f′ = −1 < 0 (stable).
Real-world uses
- Checking whether an economy or an ecosystem returns to balance after a shock
Common mistake
Linearising when the eigenvalue real part is zero; the test is inconclusive.
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 Equilibria and Stability?
Classify fixed points as stable, unstable or saddle. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
Where is Equilibria and Stability used in real life?
Real-world examples: Checking whether an economy or an ecosystem returns to balance after a shock.
What should I learn before Equilibria and Stability?
Make sure you are comfortable with Phase Plane Analysis, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Equilibria and Stability?
Linearising when the eigenvalue real part is zero; the test is inconclusive.
How can I practise Equilibria and Stability?
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 Equilibria and Stability?
Move on to Predator–Prey (Lotka–Volterra) 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.