Lyapunov Diagnostics

Calculate Lyapunov exponents and visualize how small perturbations expand or contract in simulated systems.

Steps

1. Open the Lyapunov workflow in the toolbox.
2. Select the target flow and keep the same parameters used for the trajectory plot.
3. Start with RK4 and a conservative step size; for Lorenz, dt = 0.01 is a practical first value.
4. Use a burn-in interval before accumulating exponent estimates.
5. Run a long enough integration window before interpreting the maximal exponent.
6. Inspect the convergence plot, not only the final number.
7. Report the spectrum as numerical evidence; do not treat a single short run as a proof of chaos.

Theory

Lyapunov exponents measure average exponential rates of separation or contraction of nearby states in phase space. The number of exponents matches the system dimension. A positive maximal exponent is numerical evidence of sensitive dependence on initial conditions.

For a three-dimensional continuous flow, a typical chaotic spectrum has one positive exponent, one exponent close to zero, and one negative exponent. The near-zero direction corresponds to motion along the trajectory itself; the negative direction records contraction.

The convergence curves matter because finite-time estimates can drift. A student should ask whether the estimates stabilize when burn-in, final time, and integration step are refined.

How Parameters Change This Figure

• Burn-in: too little burn-in lets startup behavior bias the exponent estimate.
• Total time: short integrations can give the wrong sign or unstable values.
• Step size: a large step can create artificial expansion or contraction.
• Perturbation size: perturbations must stay small enough to approximate local tangent behavior.
• Renormalization interval: if vectors are renormalized too rarely, they can overflow or collapse into one direction.
• Initial condition: hidden transients or multistability can change the observed finite-time spectrum.