Numerical chaos

Sensitive nonlinear systems can make nearby states diverge. A numerical chaos experiment must distinguish physical sensitivity from differences introduced by time steps, random seeds, update order, output rounding, or a non-converged integrator. Divergence alone is not a chaos diagnosis; it is a measured quantity with a domain, time horizon, norm, and numerical-control evidence.

Think like a programmer

Build paired runs from one immutable initial-state template. Change exactly one named component by a declared epsilon, while preserving parameters, seed/random state, integrator, step schedule, boundary policy, update ordering, precision, and output times. Define a state-space distance with units or normalization, retain both raw trajectories, and compare fixed-time separation curves under step refinement—not only their final frames.

Model checklist

Inputs: Reference initial state, one-component perturbation and epsilon, identical parameters/seed/random state, integrator, step schedule, precision policy, boundary conditions, state-space norm/normalization, comparison horizon, and output sample times.
State: Paired timestamped trajectories, separation d(t), log-separation where valid, finite-time growth estimate, saturation threshold, and refinement/control results.
Rule: Advance paired states with identical numerical policies except for the declared perturbation; calculate distance at shared physical times; repeat with reduced step and optional precision/epsilon controls.
Output: Separation curve, fitted/local finite-time growth diagnostic, saturation flag, control comparison, and reproducible provenance packet.
Check: At t = 0 separation equals declared epsilon under the chosen norm; zero perturbation gives identical deterministic traces; changing only particle/storage order does not introduce divergence; qualitative early-time growth persists under time-step refinement; a known stable/linear control does not falsely report exponential growth; post-saturation values are not fitted as early-time sensitivity.

\[d(t)=\lVert\mathbf x_\epsilon(t)-\mathbf x_0(t)\rVert,\qquad \lambda_T\approx\frac{1}{T}\ln\frac{d(T)}{d(0)}\]

The norm is part of the claim. Position-only distance, velocity-only distance, and a combined normalized phase-space norm can show different scales. If position is measured in metres and velocity in metres per second, adding components without a scale factor is dimensionally meaningless. State the normalization or report separate distances. A log plot is useful only while separation is positive, above roundoff, and below system-size saturation.

Use controls. A linear harmonic oscillator with a stable integrator is a useful comparison: nearby states can separate periodically but should not show the same sustained nonlinear sensitivity claim. A deliberately unstable large time step can also create rapid growth; reducing the step exposes that as a numerical artifact. Hold comparison horizon and physical output times fixed across Δt, Δt/2, and Δt/4.

Oscillator state sampler

Change stiffness, damping, and displacement. The force is restoring; damping affects velocity once the system is moving.

Stiffness: 20Damping: 1Displacement: 0.40 m

Stiffness 20 N/m; damping 1; displacement 0.40 m; acceleration -8.00 m/s²; natural angular frequency 4.47 rad/s; energy 1.60 J.

Finite-time divergence is not a universal constant for every trajectory. It can depend on initial region, observable/norm, transient duration, and parameter regime. Avoid fitting a line to a logarithmic curve after separation has reached the attractor or domain scale; the initial exponential-like region, if present, is the relevant one. Report uncertainty or variation across several reference states when making a broad claim.

Try this experiment

Prediction: A deterministic update can still show rapidly growing separation, while an unstable time step can mimic it.

Choose an initial-state component and epsilon. Calculate d(0), then record d(t) at shared times for three step sizes. Run a zero-epsilon control and a simple stable control system. Plot log separation only before saturation, estimate one finite-time growth rate, and list the integrator, norm, horizon, and precision policy beside it.

Where this model breaks

Roundoff and unstable integration can mimic sensitivity. Always report the horizon, step size, epsilon, norm, precision, boundary policy, and convergence evidence with a chaos visualization. Finite data cannot prove an asymptotic Lyapunov exponent; noise, stochastic forcing, parameter mismatch, and model inadequacy can cause divergence without deterministic chaos.

Summary

Numerical chaos needs paired, reproducible runs, a dimensional separation metric, controls, and fixed-time step-refinement evidence. Treat divergence as a bounded measurement with saturation and precision limits, not an animation effect or universal label.

Glossary

Epsilon: deliberately tiny initial offset.
Divergence: separation of two trajectories.
Convergence: stability of results as numerical resolution improves.
State-space norm: declared distance rule between complete model states.
Saturation: regime where separation has reached the model’s accessible scale.

Self-check

Which inputs must both runs share beyond model parameters?
Why must a separation metric state its units or normalization?
Why refine the time step at fixed physical output times?
Why should a post-saturation log-separation curve not determine the early growth rate?

Sources

S. H. Strogatz, Nonlinear Dynamics and Chaos.
E. Ott, Chaos in Dynamical Systems.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Numerical Chaos, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Oscillations, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Numerical Chaos as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Numerical Chaos into a test

Run controlled paired simulations to measure sensitivity while checking convergence against solver artifacts.

Name the inputs and units that the oscillations model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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