New Method for Computing Finite-Time Lyapunov Exponents

We present a novel method for computing finite-time Lyapunov exponents and vectors, via generalizing a correction given by Goldhirsch, Sulem, and Orszag [Physica (Amsterdam) 27D, 311 (1987)] into higher-order corrections. This method is a generalized $LR$ method, which is, in contrast to the existing methods, applicable to multidimensional systems with degenerate spectra. The efficiency and accuracy is demonstrated by applying it to multidimensional dynamical systems. Without these corrections, we could not accurately detect, as an example, the coexistence of qualitatively different Lyapunov instabilities along a trajectory for a multidimensional oscillator system.

Figure 1

The finite-time errors for $K=1.5$, $\mathbf{(}x(0),y(0)\mathbf{)}=(1.1\pi ,0)$: (a) The errors in the $QR$ method, $|{\lambda }_2^{QR}(t,0)-{\lambda }_2(t,0)|$, are plotted against $t$. (The solid line is $10/t$ for eye guidance.) The $k$th corrected errors in the smallest exponents are plotted: (b) $|{\lambda }_2^{(k)}(t,0)-{\lambda }_2(t,0)|$ vs $t$ for $k=0,1,2,25$; (c) $|{\lambda }_2^{(k)}(t,0)-{\lambda }_2(t,0)|$ vs $k$ for $t=1,11,15,20$.

Figure 2

${\mu }_1(t,t_i)$ against $t$, $t_i<t<t_i+1000$, for an initial condition [$H\mathbf{(}p(0),q(0)\mathbf{)}=300$]: (a) $t_i=73000$ and (b) $t_i=20000$. The thick and the thin lines are our corrected results and the approximate results of the standard $QR$ method, respectively. The thick line in (c) is ${\mu }_1$ for $t_i=24000$. The thin lines are the fitted lines in the early linear and in the latter exponential stage, respectively. $T_L\sim 24300$ is the time at the intersection of these fitted lines.