Calculator
Notes: For flows, keep dt small and total time large. For maps, use large iterations and a sizable transient. The convergence table helps spot under-sampling.
Formula used
Lyapunov exponents measure average exponential separation rates of nearby trajectories. For a flow
ẋ = f(x), tangent dynamics follow δ̇ = J(x) δ, where J(x) is the Jacobian.
Evolving an orthonormal basis of tangent vectors and repeatedly re-orthonormalizing gives the spectrum.
With a renormalization interval, we update a tangent matrix W, perform Gram–Schmidt to obtain diagonal
stretch factors Rii, then accumulate:
λi ≈ (1/T) Σ log(Rii).
For maps, replace time with iteration count and use the map Jacobian each step.
This implementation uses RK4 for state and variational equations in flows, and QR-like Gram–Schmidt on tangent columns.
How to use this calculator
- Select a system (flow or map) from the dropdown.
- Set total time or iterations, and a transient to discard.
- Choose a renormalization step count for re-orthonormalization.
- For flows, pick a small dt and longer total time.
- Click Calculate Spectrum to compute results above the form.
- Use CSV/PDF buttons to export the computed spectrum.
Example data table
| System | Parameters | Settings | Expected spectrum (approx.) |
|---|---|---|---|
| Logistic map | r=4, x0=0.2 | Total=2000, Transient=200, Renorm=10 | λ1 ≈ 0.693 |
| Hénon map | a=1.4, b=0.3, (x0,y0)=(0.1,0.1) | Total=20000, Transient=2000, Renorm=10 | λ1 ≈ 0.42, λ2 ≈ −1.62 |
| Lorenz-63 | σ=10, ρ=28, β=8/3 | dt=0.01, Total=50, Transient=10, Renorm=10 | λ ≈ (0.9, 0, −14.6) |
| Rössler | a=0.2, b=0.2, c=5.7 | dt=0.01, Total=200, Transient=20, Renorm=10 | λ1 > 0 for chaotic regimes |
Exact values vary with dt, total time, and transient choices.
Professional article
1) Why the Lyapunov spectrum matters
A single Lyapunov exponent can flag sensitivity, but the full spectrum describes how volumes stretch and contract in phase space. For a three‑dimensional flow, the three exponents explain whether trajectories expand along one direction, remain neutral along the flow, and contract transversely. This calculator reports the full ordered set and highlights the largest exponent, which is the most common chaos indicator.
2) Interpreting signs and magnitudes
Positive values indicate exponential divergence of nearby states. A value near zero often appears for continuous systems because perturbations along the trajectory neither grow nor shrink on average. Strongly negative values represent rapid contraction. For example, classic Lorenz‑63 settings frequently yield one positive exponent near 0.9, one close to 0, and one strongly negative around −14, depending on step size and averaging time.
3) Discrete maps versus continuous flows
For maps, exponents are “per iteration” and arise from the Jacobian of the map at each step. In the logistic map at r=4, the largest exponent is expected near ln(2) ≈ 0.693. For flows, exponents are “per time unit” and are estimated by integrating both the state and its variational dynamics.
4) Renormalization and numerical stability
Tangent vectors quickly align with the most expanding direction. To recover the entire spectrum, the algorithm re‑orthonormalizes the tangent basis every N steps. Each re‑orthonormalization produces stretch factors (diagonal terms) whose logarithms are accumulated. Smaller N improves stability, while larger N can reduce overhead for long runs.
5) Transients and effective averaging time
Early motion can be dominated by initial conditions rather than the attractor. The transient setting discards the first part of the trajectory so averages reflect long‑term dynamics. If results drift across the convergence table, increase the total time or iterations. Stable estimates typically require thousands of map iterations or tens to hundreds of time units for flows.
6) Derived outputs: KS entropy and Kaplan–Yorke dimension
The sum of positive exponents is often used as an estimate of the Kolmogorov–Sinai entropy rate in smooth systems. The Kaplan–Yorke dimension uses partial sums of ordered exponents to approximate attractor dimension. For the Hénon map near a=1.4, b=0.3, typical outputs include one positive exponent around 0.4 and a negative exponent near −1.6, giving a fractal dimension between 1 and 2.
7) Practical parameter strategies
Start with published “benchmark” parameters (like Lorenz σ=10, ρ=28, β=8/3) and verify that the largest exponent is positive. Then vary one parameter at a time and monitor how the spectrum changes. A transition to chaos is usually visible when the largest exponent crosses from negative to positive and the Kaplan–Yorke dimension increases.
8) Quality checks and reporting
Use the convergence samples to confirm stationarity: estimates should settle rather than trend. For flows, test two smaller step sizes to ensure results are not dominated by integration error. Export CSV for notebooks and reports, and export PDF for quick sharing. Document your dt, transient, and total time alongside the spectrum.
FAQs
Q1. What does a positive largest exponent mean?
It indicates nearby trajectories separate exponentially on average, implying sensitive dependence on initial conditions. In many nonlinear systems, this is treated as strong evidence of chaos.
Q2. Why do I see an exponent close to zero for flows?
Continuous-time systems usually have one exponent near zero because perturbations along the trajectory direction neither grow nor shrink on average over long times.
Q3. How many iterations or time units are enough?
It depends on the system and parameters. As a rule, increase total runtime until the convergence samples stabilize. Maps may need thousands to tens of thousands of iterations.
Q4. How should I choose the renormalization steps value?
Smaller values improve numerical stability and help recover the full spectrum. Typical choices are 5–50 steps. If estimates fluctuate, reduce the interval or increase total runtime.
Q5. Why does step size affect my flow results?
Flow spectra rely on numerical integration of both the state and variational equations. A large time step can introduce error, shifting exponents. Decrease dt and compare results for consistency.
Q6. What is the Kaplan–Yorke dimension used for?
It provides a practical estimate of attractor dimension from the ordered exponents. It is widely used to summarize how many effective degrees of freedom contribute to chaotic dynamics.
Q7. Can the spectrum detect stability changes across parameters?
Yes. As parameters vary, exponents can cross zero, signaling bifurcations or loss of stability. Tracking the largest exponent and the sum of positive exponents is especially informative.