Lyapunov analyzer

Choose one of five maps or four flows, set the initial coordinates and parameters, then Compute. Alignment steps advance the orbit and tangent basis without contributing finite logarithms to the measured average. Measurement steps determine the finite observation window. Maps report rates per iteration; flows report rates per model time unit. For flows Δt=.006 for Lorenz and .02 for Rössler, Duffing and Van der Pol. A cancellable worker reports actual completed steps; resize and curve selection redraw cached data.

Analytical Jacobians avoid finite differences across the standard map’s 2π coordinate jumps. Two-dimensional map QR uses logarithmic determinants to obtain the second growth rate without subtracting nearly parallel vectors. ODEs integrate the state and variational equation together by RK4, then reorthogonalize the tangent basis. Detected numerical failures produce errors, not zero exponents. An exactly annihilated map tangent direction remains collapsed, including loss during alignment, and is listed as −∞. Tent uses its selected one-sided branch slope at its cusp; finite precision can send its orbit to endpoints.

Duffing is augmented with a periodic forcing phase: ẋ=v, v̇=γcosφ−δv−αx−βx³, φ̇=ω. The full three-dimensional spectrum includes its phase direction. Lorenz and Rössler have editable z₀; Duffing has φ₀ in radians. The exponent table is ordered from largest to smallest. Curves show ordered running estimates of the largest exponent, the full spectrum or their sum; nonfinite values are omitted from a Cartesian curve. Σλ is not repeated for one-dimensional maps.

D_KY is the Kaplan–Yorke formula on the finite ordered spectrum, not a certified fractal dimension. Positive finite-time growth alone does not prove chaos: an unstable fixed point can have positive exponents. Zero does not certify neutrality. Increase alignment and measurement windows and inspect convergence; fixed time steps, initial direction, rank loss, rounding, periodicity and short trajectories affect results. No automatic stability label, confidence interval or guaranteed asymptotic/ODE accuracy.

PNG exports the selected convergence curve with its units and parameters. Valid settings may be stored locally; reload waits for Compute. Start coordinates outside the supported range and invalid counts/parameters are rejected rather than clamped.

Scholarpedia · Lyapunov · Chirikov–Shepelyansky · Standard map