Formula used
The largest Lyapunov exponent quantifies how fast nearby trajectories separate.
- Direct divergence: \(\lambda = \frac{1}{T}\ln\!\left(\frac{d(T)}{d_0}\right)\). Use consistent distance units, and convert time to seconds.
- Paired time series (simplified): compute \(d_k = |x_2(k) - x_1(k)|\), then \(\lambda_k = \frac{1}{t_k}\ln\!\left(\frac{d_k}{d_0}\right)\). The calculator reports the average of \(\lambda_k\) over your selected index range.
Positive \(\lambda\) suggests sensitive dependence, while negative \(\lambda\) indicates convergence.
How to use this calculator
- Select a method: Direct divergence or Paired time series.
- Enter values using consistent units for each field.
- For time series, paste equal-length datasets and set \(dt\).
- Choose an averaging range to reduce transient effects.
- Click Calculate to view results above the form.
- Use Download CSV or Download PDF to export.
Example data table
| Scenario | d0 | d(T) | T (s) | λ (1/s) | Meaning |
|---|---|---|---|---|---|
| Growing separation | 0.001 | 0.010 | 5 | 0.460517 | Often indicates chaotic-like divergence. |
| No net change | 0.002 | 0.002 | 4 | 0.000000 | Neutral separation for this interval. |
| Converging separation | 0.010 | 0.003 | 6 | -0.200714 | Convergent-like behavior, often stable. |
Tip: for real data, choose \(T\) and ranges that avoid measurement noise floors.
Lyapunov exponent guide
1) What the exponent measures
The Lyapunov exponent \(λ\) describes the average exponential rate at which two nearby trajectories separate in a dynamical system. If the separation follows \(d(t)\approx d_0 e^{λ t}\), then λ captures sensitivity to initial conditions. In practical measurements, the largest exponent is most useful because it sets the fastest predictability loss rate, while smaller exponents mostly refine geometry.
2) Direct divergence mode in this calculator
Use Direct divergence when you can estimate an initial separation d0, a later separation d(T), and the elapsed time T. The calculator applies λ = (1/T) ln(d(T)/d0) using the natural logarithm, after converting time to seconds. This mode works well for controlled perturbations, repeated trials, and simulations with a well-defined distance metric. Choose T long enough to average out short transients, but not so long that the separation saturates or hits a measurement floor.
3) Paired time series mode for measured signals
Use Paired time series when you have two recordings x1(k) and x2(k) sampled at the same dt. The tool forms d_k = |x2(k) − x1(k)| and estimates pointwise growth rates from log ratios relative to a baseline separation, then reports an average over your selected index range. If early differences are zero or near-zero, the baseline shifts to the first nonzero value to keep the calculation stable. For best results, pre-align signals in time and apply the same filtering to both series.
4) Choosing averaging indices and time windows
Early samples may include transients, actuator settling, or filtering artifacts. Averaging from a later index helps focus on steady behavior. If your data saturates or hits a noise floor, late indices can bias \(\lambda\) downward. A practical approach is to try several windows and check whether the result remains consistent. Also watch for noise floors: if separations stop growing or hit zero, log ratios become unstable and can bias the estimate.
5) Units and scaling considerations
This calculator outputs λ in 1/s after converting your time inputs to seconds. If you rescale the time axis (for example, milliseconds instead of seconds), the numerical value of λ changes accordingly. Rescaling signal amplitude affects d0 and d(T) but cancels in the ratio when both are measured consistently. If you want a per-sample growth rate, multiply λ by dt; if you want a characteristic divergence time, use 1/|λ| as a rough scale.
6) Typical physics applications
Lyapunov exponents appear in fluid and plasma turbulence, nonlinear oscillators, laser dynamics, driven pendulums, coupled resonators, and N-body simulations. They help compare models, validate numerical integrators, and detect regime changes (for example, when a control parameter crosses a stability threshold). In experiments, treat the result as an estimate tied to sampling rate, filtering, sensor resolution, and the specific state or signal difference you used as “distance.” Reporting your window and dt makes results reproducible.
7) Interpreting results in practice
Beyond the sign of λ, translate the value into time scales. A rough doubling time for divergence is \(t_2=\ln 2/λ\) when λ>0. When λ<0, the magnitude indicates how quickly differences shrink. Compare several averaging windows, keep preprocessing consistent, and report λ with your \(dt\) and units.
FAQs
1) What does a positive Lyapunov exponent mean?
A positive largest exponent suggests sensitive dependence on initial conditions. Small differences grow exponentially over time, which is often associated with chaotic-like behavior and reduced long-term predictability.
2) Can I compare \(\lambda\) from different experiments directly?
Only if the time units, sampling, filtering, and the definition of distance or signal difference are consistent. Changes in \(dt\) or preprocessing can change the estimated exponent.
3) Why do I sometimes get negative or near-zero values?
Negative values indicate contraction or convergence, typical of stable regimes. Near zero can occur for periodic motion, short windows, or when measurement noise dominates the separation dynamics.
4) What should I use for \(d_0\) and \(d(T)\)?
Use a meaningful distance between two trajectories: state-space distance, position difference, or a consistent norm of parameter perturbations. Keep units consistent so the ratio \(d(T)/d_0\) is well-defined.
5) How many samples do I need for the paired time series method?
More is usually better, but you need enough samples to observe growth above noise and beyond transients. Start with tens to hundreds of samples, then test stability across multiple averaging windows.
6) Does this replace full phase-space reconstruction methods?
No. This tool provides practical estimates using divergence inputs or paired series. For rigorous largest-exponent estimation from a single signal, use embedding, neighbor tracking, and validated algorithms.
7) Why does the CSV export include a table?
The table records intermediate values used in the estimate, which helps you audit calculations, plot \(\lambda_k\) versus time, and document the exact window and assumptions used.