Kaplan-Yorke Dimension Calculator

Formula used

Given ordered Lyapunov exponents λ₁ ≥ λ₂ ≥ ... ≥ λ_n, define partial sums:

S_k = Σ_i=1..k λ_i

Let j be the largest index such that S_j ≥ 0 and S_j+1 < 0 (if it exists). The Kaplan-Yorke (Lyapunov) dimension is:

D_KY = j + \frac{S_j}{|\lambda_j+1|}

If S₁ < 0, the calculator reports 0. If all S_k ≥ 0, it reports n.

How to use this calculator

1. Enter your Lyapunov exponents as a list.
2. Keep sorting enabled unless your list is already ordered.
3. Choose a precision for clean display and exports.
4. Click Calculate to view D_KY above the form.
5. Use CSV or PDF buttons to save results.

Example data table

This example yields a fractional attractor dimension between 2 and 3.

Article

1. What the Kaplan-Yorke dimension measures

The Kaplan-Yorke (KY) dimension estimates an attractor's effective fractal dimension from a Lyapunov spectrum. It turns combined stretching and contraction rates into a single value that often lies between two integers, giving a compact complexity metric. It is valuable when you only have spectral estimates.

2. Why the Lyapunov spectrum matters

Lyapunov exponents describe how fast nearby trajectories separate or converge. Positive exponents indicate sensitivity to initial conditions, while negative exponents indicate dissipation. The ordered spectrum acts like a stability signature that can be compared across models and parameter settings.

3. The critical index and partial sums

The calculator forms partial sums S_k = sum(lambda_i) and finds the largest j with S_j >= 0. This j marks the last direction where expansion still dominates. When S_{j+1} becomes negative, the KY dimension falls between j and j+1.

4. Interpreting fractional results

A result such as 2.85 suggests behavior more complex than a smooth surface but not volume-filling in 3D. Fractional values are typical in chaotic maps and flows. Tracking KY across a parameter sweep can highlight bifurcations and shifts in dissipation. Values often drop as damping increases, and rise near chaos.

5. Sorting and input quality

The KY formula assumes exponents are sorted from largest to smallest. Sorting prevents common ordering mistakes that change j and the final estimate. Exponent values can be noisy due to finite data, short integration windows, or measurement error, so consistency matters.

6. Edge cases that signal issues

If the first partial sum is negative, the tool reports 0, indicating immediate net contraction in the supplied spectrum. If all partial sums remain nonnegative, it reports n, which can mean missing negative exponents or an embedding dimension that is too small.

7. Where KY is used

KY dimension is reported in turbulence studies, nonlinear electronics, laser dynamics, driven oscillators, and climate-style models. It complements phase portraits and spectra by providing one comparable number. In control or synchronization work, reduced KY often reflects successful stabilization.

8. Reporting and reproducibility

Use the CSV and PDF exports to capture inputs, cumulative sums, and the final dimension for lab notes or reports. For reproducible comparison, also record the exponent estimation method, sampling rate, time window, and any preprocessing, then keep those settings fixed. Attach the exported step table so collaborators can verify j and arithmetic quickly.

FAQs

1) What inputs does the calculator need?

Enter a list of Lyapunov exponents for your system. Values can be separated by commas, spaces, or new lines. For best results, provide the full spectrum for the embedding dimension you are analyzing.

2) Should I always enable sorting?

Yes, unless you are certain your exponents are already ordered from largest to smallest. Incorrect ordering can change the critical index j and produce a misleading Kaplan-Yorke dimension.

3) What does a fractional KY dimension mean?

A fractional value indicates the attractor occupies a geometry between two integer dimensions. It suggests complex, self-similar structure common in chaotic dynamics, rather than filling the entire phase-space volume.

4) Why does the calculator sometimes return 0?

If the first partial sum is negative, contraction dominates immediately. In that situation, the Kaplan-Yorke estimate is reported as 0 by convention, indicating no sustained expanding directions in the provided spectrum.

5) Why can the output equal the number of exponents?

If all partial sums stay nonnegative, the method never finds a contracting boundary within the supplied list. The calculator then reports n, which can indicate missing negative exponents, insufficient model dimension, or atypical data.

6) How many decimals should I use?

Use 4 to 8 decimals for most research reporting. If exponents are noisy, extra decimals do not add meaning. Pick a precision that matches the uncertainty of your exponent estimation process.

7) Is the Kaplan-Yorke dimension the same as correlation dimension?

No. They are related but computed differently. KY uses Lyapunov exponents, while correlation dimension uses scaling of point-pair distances in reconstructed phase space. They often agree qualitatively, but can differ with finite data and noise.