Leveraging temporal features of the divergence measure to detect chaos in conservative systems

Abstract

The recurrence-based divergence quantifier ($DIV$), traditionally applied to dissipative systems, is shown here to be an effective finite-time chaos indicator for conservative dynamics. We benchmark its performances against the well-established fast Lyapunov indicator (FLI), focusing on the standard map, a canonical model of Hamiltonian chaos. Through extensive numerical simulations on moderately long orbits, we find strong agreement between $DIV$ and FLI, supporting the reported correlation between the divergence of recurrences and positive Lyapunov exponents. Additionally, our study sheds more light into asymptotic time properties of $DIV$ by revealing distinct power laws for regular and chaotic trajectories, both in the original and reconstructed phase spaces. In particular, for regular dynamics, $DIV$ decays in average with the time $N$ as $1/N$, mirroring the decay rate of the maximal Lyapunov exponent. For chaotic trajectories, $DIV$ decreases at a much slower rate in average, close to $1/\sqrt{N}$. This scaling insight opens new avenues for characterizing chaos from time series. Our numerical results thus demonstrate $DIV$ to be a computationally viable and theoretically rich tool for chaos detection in conservative systems.

Figures (13)

• Figure 1: (Left) The phase portrait of the standard map at $K=1$ for $N=500$ iterations. (Right) Recurrence plots associated to four representative trajectories shown in color in the phase space; red - quasi-periodic orbit $(\theta_0,p_0)=(3.5,1.0),$ blue - orbit trapped in a secondary resonance $(\theta_0,p_0)=(3.85,1.7),$ green - large scale chaotic orbit $(\theta_0,p_0)=(3.3,1.8),$ black - sticky orbit $(\theta_0,p_0)=(3.14,2.215).$ The different textures in the RPs serve as a basis for recurrence quantification analysis. In particular, we focus here on the divergence metric $DIV$, associated to the non-trivial longest diagonal line in a given RP.
• Figure 2: The measure $DIV$ as a function of the time series length, $N.$ The ensemble of 200 trajectories is separated according to the threshold values into regular (blue) and chaotic (red) parts, 152 and 48 members, respectively. The fit (dashed lines) to the ensemble averages (gray solids) shows different power-law exponents for the two kinds of motion. The green curve will be discussed thereafter.
• Figure 3: Same as in Fig. \ref{['fig:fig2']} for varying nonlinearity parameters $K$ ranging dynamical regimes dominated by stability ($K=0.6$) and chaoticity ($K=4$).
• Figure 4: Evolution of the power laws exponent $\gamma$ in regular or chaotic cases as a function of $K$ computed at $N=500$. The separation between regular and chaotic orbits is well reflected by the exponent. For regular components, the exponent is always smaller than $\gamma_{\textrm{reg.}} \le 1$, whilst for chaotic ensembles, $\gamma_{\textrm{cht.}}$ is larger and tends to align to $\gamma_{\textrm{cht.}}=-1/2$ for very chaotic components.
• Figure 5: (Left) Phase space of the chaotic orbit with initial condition marked in red extracted from Fig. \ref{['fig:fig2']}. The orbit is chaotic with a low divergence value (green curve). Two portions of the orbit are colored (in blue and green) in accordance with the $2$ most prominent sticky events encountered during its evolution. (Right, from top to bottom) Time evolution of the angle $\theta$, the FLI and the $DIV$. Sticky events (localised variations of $\theta$) correspond to plateau in the FLI and sharp decay in the $DIV$ measure, as visually materialised with the blue and green boxes serving as guides. The sharp decreases of $DIV$ during the sticky events explain the low final value of $DIV$, despite being chaotic.