Lyapunov exponents, covariant vectors and ensembles

What is computed

The Lyapunov spectrum is that of the discrete map \(\Phi_{dt}\) of the time stepper (an implicit– explicit Runge–Kutta scheme at fixed step), not of a separately discretised variational equation. Its tangent is exact: for one stage,

\[ (M + dt\,H_{ii} L)\,\delta X_i = M\,\delta X_0 + dt \sum_{k} A_{ik}\, F'(X_k)\,\delta X_k - dt \sum_k H_{ik}\, L\,\delta X_k , \]

with the stage states \(X_k\) of the trajectory and \(F'\) the symbolic Fréchet derivative of the nonlinear terms, evaluated on the same dealiased grid. The exponents of \(\Phi_{dt}\) converge to those of the flow as \(dt \to 0\). Multistep schemes are refused: their tangent lives in an augmented state and carries the parasitic roots of the scheme.

Algorithm

  • Benettin / QR: \(k\) orthonormal tangent vectors are advanced with the trajectory and re-orthonormalised by QR every few steps (positive diagonal of \(R\)); \(\lambda_i = \frac1T \sum \log R_{ii}\). Transients of the trajectory and of the tangent vectors are discarded before averaging, both with declared lengths.

  • Statistical error: the average is split into blocks; the standard error of the block means is reported together with the lag-one correlation of consecutive blocks, which must be small.

  • Covariant Lyapunov vectors (Ginelli et al. 2007): the \(R\) factors are stored, a backward iteration with upper-triangular matrices gives the coefficients of each covariant vector in the Gram–Schmidt basis; principal angles between the unstable and stable subspaces measure hyperbolicity.

  • Kaplan–Yorke dimension \(D_{KY} = j + \sum_{i\le j}\lambda_i/|\lambda_{j+1}|\), with an error propagated from the exponents.

  • An invariant subspace can be imposed (for example zero mean, or the Galilean direction removed) by a declared projection before each QR.

Ensembles

An ensemble is \(B\) states of one discretised problem advanced together: per stage, one evaluation of the nonlinear terms for all members, one sparse product by \(M\) and \(L\), one stacked banded solve. Parameters that enter the nonlinear terms become per-member coefficients; a parameter inside \(L\) gives per-member factorisations. The same machinery advances \(B\) trajectories times \(k\) tangent vectors at once, on the CPU or the GPU.

The fate of each member (relaminarised, transitioned, in which basin) is decided by declared criteria, and transition probabilities come with a Wilson confidence interval.

References

  • G. Benettin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems, Meccanica 15 (1980) 9–30.

  • I. Shimada and T. Nagashima, A numerical approach to ergodic problem of dissipative dynamical systems, Prog. Theor. Phys. 61 (1979) 1605–1616.

  • F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett. 99 (2007) 130601.

  • J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, Lecture Notes in Math. 730, Springer (1979).

  • E. B. Wilson, Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927) 209–212.