On the behavior of the Generalized Alignment Index (GALI) method for dissipative systems

TL;DR

This work extends chaos-detection techniques to dissipative, non-Hamiltonian dynamics by systematically evaluating the Generalized Alignment Index ($GALI$) across continuous-time and discrete-time models. By comparing $GALI$ with the full Lyapunov spectrum in three representative systems—the 3D Lorenz, a 4D Lorenz hyperchaotic system, and a generalized hyperchaotic Hénon map—the authors reveal that $GALI$ can reliably flag chaotic motion and stable fixed points, but has limited discriminative power for distinguishing stable limit cycles from chaotic and hyperchaotic attractors when using higher-order indices. The key finding is that $GALI_2$ often decays exponentially when the two largest Lyapunov exponents differ and may only remain informative when these exponents are nearly equal, while $GALI_k$ with $k>2$ generally decays for all non-fixed attractors, reducing its utility in dissipative contexts. The results provide practical guidance for applying $GALI$ to dissipative systems and highlight the necessity of supplementing $GALI$ with the Lyapunov spectrum for robust attractor identification in non-Hamiltonian dynamics.

Abstract

The behavior of the Generalized Alignment Index (GALI) method has been extensively studied and successfully applied for the detection of chaotic motion in conservative Hamiltonian systems, yet its application to non-Hamiltonian dissipative systems remains relatively unexplored. In this work, we fill this gap by investigating the GALI's ability to identify stable fixed points, stable limit cycles, chaotic (strange) and hyperchaotic attractors in dissipative systems generated by both continuous and discrete time dynamics, and compare its performance to the analysis achieved by the computation of the spectrum of Lyapunov exponents. Through a comprehensive study of three classical dissipative models, namely the 3D Lorenz system, a modified Lorenz 4D hyperchaotic system, and the 3D generalized hyperchaotic Hénon map, we examine GALI's behavior, and possible limitations, in detecting chaotic motion, as well as the presence of different types of attractors occurring in dissipative dynamical systems. We find that the GALI successfully detects chaotic motion, as well as stable fixed points, but it faces difficulties in distinctly discriminating between stable limit cycles, chaotic attractors, and hyperchaotic motion.

Figures (12)

• Figure 1: (Left column) 3D phase space portraits of trajectories with IC $(x, y, z) = (1, 3, 6)$ [indicated by an orange circle point in (a) and (d), while in (g) is hidden behind the orbit] for the 3D Lorenz system \ref{['eq:3DODE']} with parameters $a=10$, $b=8/3$ and (a) $r=2.1$, (d) $r=1$, and (g) $r=33.3$. The trajectory asymptotically tends to (a) a stable fixed point, (d) a stable limit cycle, and (g) a chaotic strange attractor. In gray we depict the initial part of the trajectory's evolution and in black its asymptotic behavior, while in red, blue and green we show its 2D $xy$, $xz$, and $yz$ projections respectively. (Middle column) The time evolution of the ftLEs of the trajectories depicted in the respective panel of the left column: $\lambda_1$ (red curves), $\lambda_2$ (blue curves), and $\lambda_3$ (green curves). The black line in each panel indicates $\lambda_j=0$ for comparison. Note that in all panels the $\lambda_3$ values have been scaled for visualization purposes. (Right column) The time evolution of the GALI$_2$ (solid blue curves) and the GALI$_3$ (solid red curves in the inset plots) for the orbits depicted in the first panel of each row. Apart from the GALI$_2$ in (c), which oscillates around a constant positive value, all GALIs decay exponentially fast to zero, following the functional forms (dashed curves) given in \ref{['eq:GALI_chaos']} based on the LEs estimations obtained from the result presented in the plots of the middle column of panels. In particular, these values are: (c) $\chi_1=-1.20$, $\chi_2=-1.20$, $\chi_3=-11.26$, (f) $\chi_1 = 0$, $\chi_2=-2.67$, $\chi_3=-11$, and (i) $\chi_1 =1.02$, $\chi_2 = 0$, and $\chi_3=-14.69$.
• Figure 2: The values, at $t=10^4$, of (a) the spectrum of the ftLEs $\lambda_1$, $\lambda_2$, $\lambda_3$ (respectively depicted by red, blue, and green curves), and (b) the GALI$_2$, as a function of $r$ ($r \in [-5,500]$) for the trajectory with IC $(x, y, z) = (1, 3, 6)$ of the 3D Lorenz system \ref{['eq:3DODE']} with $a=10$ and $b=8/3$. Gray vertical dashed lines indicate the values $r=1.3$, $21.3$, $146.9$, $166$, $215.4$ in (a), and the values $r=1.3$ and $21.3$ in (b).
• Figure 3: The parameter space $(r,b)$ of the 3D Lorenz system \ref{['eq:3DODE']} with $a=35$, colored according to the value of (a) the ftmLE $\lambda_1$ (scaled in the interval $[-1,1]$), (b) the index $\Lambda$, and (c) the GALI$_2$ of the trajectory with IC $(x, y, z) = (2, 1, 5)$, at $t=10^4$. In (b) the index $\Lambda$ is $\Lambda=1$ when $\lambda_1>0$, $\lambda_2\leq 0$, $\lambda_3<0$ (blue region), indicating the presence of chaotic attractors, $\Lambda=2$ for $\lambda_1 \approx 0$, $\lambda_2 < 0$, $\lambda_3<0$ (orange region) denoting the existence of limit cycles, and $\Lambda=3$ when $\lambda_1, \lambda_2, \lambda_3<0$ (purple region) corresponding to the appearance of stable fixed points. Each color plot is created by considering a set of $2,991 \times 81 = 242,271$ equally spaced grid points on the region $(r, b) = [0, 300] \times [1, 5]$.
• Figure 4: [(a$_1$), (a$_2$), (a$_3$)] $3$D phase space projections of the trajectory with IC $(x,y,z,w) = (3,2,10,1)$ (orange circle points), which asymptotically approaches a stable fixed point attractor of the $4$D Lorenz system \ref{['eq:4DODE']} with $a=35$, $b=8/3$, $c=2$, and $r=-12$. As in the left column panels of Fig. \ref{['fig:3D-01N']}, we use gray color to depict the initial phase of the trajectory’s evolution, and red, green and blue colors to show projections of the orbit in different 2D planes. (b) The time evolution of the four ftLEs of the trajectory. The $\lambda_4$ values have been rescaled for visualization purposes. The horizontal black line indicates $\lambda_j=0$ for comparison. (c) The time evolution of the GALI$_{2}$ (solid blue curve) displays fluctuations around a constant positive value due to the fact that $\lambda_1$ and $\lambda_2$ become practically equal. The GALI$_3$ (solid red curve) and the GALI$_4$ (solid green curve) in the inset, decay to zero following specific exponential laws provided by \ref{['eq:GALI_chaos']} (dashed curves).
• Figure 5: [(a$_1$), (a$_2$), (a$_3$)] $3$D phase space projections of the trajectory with IC $(x,y,z,w) = (3,2,10,1)$ (orange circle points), which asymptotically approaches a stable limit cycle of the $4$D Lorenz system \ref{['eq:4DODE']} with $a=35$, $b=8/3$, $c=2$, and $r=-5$. As in Fig. \ref{['fig:4D-01']}, we use gray color to depict the initial phase of the trajectory’s evolution, and red, green and blue colors to show projections of the orbit in different 2D planes. (b) The time evolution of the four ftLEs of the trajectory. The $\lambda_4$ values have been rescaled for visualization purposes. The horizontal black line (not clearly seen due to the overlap of the $\lambda_1$ values) indicates $\lambda_j=0$ for comparison. (c) The GALI$_{2}$ (solid blue curve), GALI$_3$ (solid red curve) and GALI$_4$ (solid green curve in the inset) decay to zero following specific exponential laws provided by \ref{['eq:GALI_chaos']} (dashed curves).