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MathidiaMathematics Academy
Dynamical systems
Lesson 33 of 75
Standard practice

Lyapunov methods

Objective. Prove stability without explicit solutions.

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  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

Lyapunov methods establish stability of a fixed point without solving the differential equation, by finding a scalar function that decreases along all trajectories near that point. A function V(x,y) is a Lyapunov function at a fixed point if V is positive everywhere nearby except zero at the point itself, and its derivative along trajectories, V̇ = ∇V·(ẋ,ẏ), is negative (or non-positive) there. If such V exists, the fixed point is stable, and strict negativity of V̇ away from the point gives asymptotic stability. Example: for ẋ = −x³, the function V(x) = x² gives V̇ = 2xẋ = −2x⁴ ≤ 0, proving the origin is asymptotically stable.

Worked example

Use a Lyapunov function to show the origin is asymptotically stable for ẋ = −x + xy², ẏ = −y − x²y... actually use ẋ = −x³, ẏ = −y³.

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Illustrated card showing ?
  1. 1Choose the candidate V(x,y) = x² + y², which is positive everywhere except zero at the origin.
  2. 2Compute V̇ = 2xẋ + 2yẏ = 2x(−x³) + 2y(−y³) = −2x⁴ − 2y⁴.
  3. 3Since x⁴ ≥ 0 and y⁴ ≥ 0, V̇ ≤ 0, with equality only when x = y = 0.
  4. 4As V is positive definite and V̇ is negative definite away from the origin, the origin is asymptotically stable.

Answer: Origin is asymptotically stable, using V = x² + y²

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Common mistake. Students often pick V(x,y) = x² + y² automatically for every system; this only works if V̇ turns out negative for that specific system, and for many nonlinear systems a different or weighted Lyapunov function is needed.

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