On the efficient numerical computation of covariant Lyapunov vectors

TL;DR

This work tackles the practical problem of determining when to terminate the forward and backward transient phases in Ginelli et al.'s covariant Lyapunov vector algorithm for chaotic Hamiltonian dynamics. It introduces two convergence tests—the direct method and the faster indirect method—based on subspace distance and demonstrates that the indirect test yields nearly identical results with lower computational cost. A center-subspace instability during long backward evolutions is identified and resolved via a center-correction that orthonormalizes the two center CLVs at each backward step, dramatically improving accuracy. Through two Hamiltonian models (the Hénon–Heiles system and a three-degree-of-freedom system), the paper provides concrete termination criteria and shows that the center-correction yields robust convergence for the critical center subspace, enabling more reliable CLV computations in practice.

Abstract

Covariant Lyapunov vectors (CLVs) are useful in multiple applications, but the optimal time windows needed to accurately compute these vectors are yet unclear. To remedy this, we investigate two methods for determining when to safely terminate the forward and backward transient phases of the CLV computation algorithm by Ginelli et al.~\cite{GinelliEtAl2007} when applied to chaotic orbits of conservative Hamiltonian systems. We perform this investigation for two prototypical Hamiltonian systems, namely the well-known Hénon-Heiles system of two degrees of freedom and a system of three nonlinearly coupled harmonic oscillators having three degrees of freedom, finding very similar results for the two methods and thus recommending the more efficient one. We find that the accuracy of two-dimesnional center subspace computations is significantly reduced when the backward evolution stages of the algorithm are performed over long time intervals. We explain this observation by examining the tangent dynamics of the center subspace wherein CLVs tend to align/anti-align, and we propose an adaptation of the algorithm that improves the accuracy of such computations over long times by preventing this alignment/anti-alignment of CLVs in the center subspace.

Figures (12)

• Figure 1: Diagram depicting the numerical integrations required to implement the direct and indirect methods discussed in Sects. \ref{['sec:m2']} and \ref{['sec:m3']} (respectively) for measuring the convergence between estimates $G_i$ and $\tilde{G}_i$ of the filtration subspaces $\Gamma_i^-$ during the forward transient phase of the GC algorithm, but both methods can also be applied to the backward transient phase. The time $T$ denotes a time in the future, while $-T_{\infty}$ denotes a distant time in the past.
• Figure 2: The time evolution of the distance $\Delta$\ref{['eq:distance']} between estimates of $\Gamma_i^-$ for $i=1,2,3$, computed using the (a) direct and (b) indirect methods of Sect. \ref{['sec:converge']} during the forward transient phase of the GC algorithm for a chaotic orbit of the Hénon-Heiles system \ref{['eq:hamiltonian']} with initial condition \ref{['eq:ic']} and $H_2=1/6$. Each thin curve represents results obtained for one of 20 sets of random initial deviation vectors, while the thicker curves represent the average $\Delta$ values in log scale over these 20 simulations. Note that the thick blue ($i=1$) and red ($i=2$) curves practically overlap each other. Both panels are in log-linear scale, and the black dashed lines represent $\Delta=10^{-12}$.
• Figure 3: The backward time evolution of the distance $\Delta$ between estimates of $\Omega_i$ for $i=2,3$, computed using the (a) direct and (b) indirect methods of Sect. \ref{['sec:converge']} during the first $1000$ time units of the backward transient phase of the GC algorithm for the same orbit used in Fig. \ref{['fig:forward']}. Each thin curve represents results obtained for one of 20 sets of random initial deviation vectors, while the thicker curves represent the average $\Delta$ values in log scale over these 20 simulations. Both panels are in log-linear scale, and the black dashed lines represent $\Delta=10^{-12}$.
• Figure 4: Similar to Fig. \ref{['fig:back_lin']}, where panels (a) and (b) correspond between the figures. This figure, however, is in log-log scale and the backward time evolution of $\Delta$ over the entire backward transient interval of $10^7$ is shown here. The black dotted line in (b) denotes a function $\propto t_b$.
• Figure 5: The backward time evolution of the distance $\Delta$ between the two linearly independent CLV estimates $\hat{\boldsymbol{\mathrm{c}}}_2$ and $\hat{\boldsymbol{\mathrm{c}}}_3$ in the computed center subspace $C_2$. The black dotted line denotes a function $\propto t_b^{-1}$. The figure is in log-log scale.