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Theory · 02

Lyapunov exponents

Definition

For a flow \(\dot{\mathbf{y}} = f(\mathbf{y})\), an infinitesimal perturbation \(\mathbf{w}\) evolves under the variational equations

\[ \dot{\mathbf{w}} = J(\mathbf{y}(t))\, \mathbf{w}, \qquad J_{ij} = \frac{\partial f_i}{\partial y_j}, \]

linearized along the trajectory. The Lyapunov exponents are the long-time average exponential growth rates of volumes spanned by such perturbations:

\[ \lambda_i = \lim_{t \to \infty} \frac{1}{t} \ln \frac{\|\mathbf{w}_i(t)\|}{\|\mathbf{w}_i(0)\|}, \]

ordered \(\lambda_1 \ge \lambda_2 \ge \dots\) For a map \(\mathbf{x}_{n+1} = f(\mathbf{x}_n)\) the same holds with the Jacobian product \(J(\mathbf{x}_{n-1}) \cdots J(\mathbf{x}_0)\) in place of the flow's fundamental matrix. A positive \(\lambda_1\) is the practical definition of chaos; flows additionally carry one exactly-zero exponent along the trajectory direction.

The QR / re-orthonormalization algorithm

Naively evolving \(k\) deviation vectors fails: all of them collapse onto the fastest-growing direction, and their norms overflow. The standard fix (Benettin, Galgani, Giorgilli & Strelcyn 1980) is to re-orthonormalize periodically with a QR decomposition: after each interval, factor the evolved frame \(W = QR\), continue with \(Q\), and accumulate \(\ln |R_{ii}|\) — the log of how much the \(i\)-th orthogonal direction stretched. Then

\[ \lambda_i \approx \frac{1}{T} \sum_{\text{intervals}} \ln |R_{ii}|. \]

Comparisons across method variants (Geist, Parlitz & Lauterborn 1990) established this as the robust default, and it is what TSDynamics uses for maps (the compiled QR tangent-map kernel behind TangentSystem) with the exact _jacobian at every iterate.

What each family actually solves

  • ODEs — the right-hand side is differentiated symbolically and the extended variational system (state ⊕ tangent vectors) is integrated on the Rust engine, QR re-orthonormalizing per chunk; the integrator returns local exponents per sampling interval. Because the adaptive stepper makes interval lengths uneven, TSDynamics averages them weighted by interval duration — an unweighted mean would bias the estimate toward whatever the step controller did. A burn_in discards the transient during which the deviation frame aligns with the attractor's Oseledets subspaces.

  • Maps — pure QR as above, in a single forward pass alongside the trajectory.

  • DDEs — the tangent space of a delay system is the infinite-dimensional history space \(C([-\tau_{\max}, 0])\); there is a full spectrum of infinitely many exponents. The DDE estimator approximates the leading few by building an extended delay system — the base state plus deviation states whose dynamics are the symbolic variational equations — and integrating it on the engine in delay-window chunks, with a function-space QR over the deviation history segment. This is why k must be chosen consciously, why the estimates converge more slowly than ODE ones, and why TangentSystem refuses delay systems outright.

The two-trajectory estimator

lyapunov_spectrum falls back to the older and simpler estimator when the system carries no Jacobian to differentiate (Benettin, Galgani & Strelcyn 1976): evolve the system and a copy displaced by \(d_0\), measure the separation \(d\) after a short interval, accumulate \(\ln(d/d_0)\), renormalize the displacement back to \(d_0\), repeat. It needs no Jacobian at all — only the ability to step and to overwrite a state — making it the right tool for non-smooth systems, at the price of estimating only \(\lambda_1\).

The Kaplan–Yorke conjecture

Kaplan & Yorke (1979) conjectured that the information dimension of an attractor equals the Lyapunov dimension

\[ D_{KY} = j + \frac{\sum_{i=1}^{j} \lambda_i}{|\lambda_{j+1}|}, \qquad j = \max\Big\{ k : \sum_{i=1}^{k} \lambda_i \ge 0 \Big\} \]

— the dimension at which an interpolated volume neither grows nor shrinks. For Lorenz, \(D_{KY} \approx 2 + 0.906/14.57 \approx 2.06\). kaplan_yorke_dimension computes exactly this, returning 0.0 when all exponents are negative and len(spectrum) when the sum never turns negative (the spectrum is incomplete — compute more exponents).

References

  • G. Benettin, L. Galgani, J.-M. Strelcyn, Kolmogorov entropy and numerical experiments, Phys. Rev. A 14, 2338 (1976).
  • G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Meccanica 15, 9–30 (1980).
  • K. Geist, U. Parlitz, W. Lauterborn, Comparison of different methods for computing Lyapunov exponents, Prog. Theor. Phys. 83, 875 (1990).
  • J. L. Kaplan, J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Mathematics 730, Springer (1979).

See also