Diffusion and Lyapunov timescales in the Arnold model

In the present work, we focus on two dynamical timescales in the Arnold Hamiltonian model: the Lyapunov time and the diffusion time when the system is confined to the stochastic layer of its dominant resonance (guiding resonance). Following Chirikov's formulation, the model is revisited, and a discussion about the main assumptions behind the analytical estimates for the diffusion rate is given. On the other hand, and in line with Chirikov's ideas, theoretical estimations of the Lyapunov time are derived. Later on, three series of numerical experiments are presented for various sets of values of the model parameters, where both timescales are computed. Comparisons between the analytical estimates and the numerical determinations are provided whenever the parameters are not too large, and those cases are in fact in agreement. In particular, the case in which both parameters are equal is numerically investigated. Relationships between the diffusion time and the Lyapunov time are established, like an exponential law or a power law in the case of large values of the parameters.

Figure 1

The Arnold web given by (10) with $\varepsilon =0.25$ and just for $|m_1|+|m_2|+|m_3|<6$, where ${\omega }_1={\omega }_p$ for $|I_1|\le 2\sqrt{\varepsilon }(h_1<0)$ while ${\omega }_1=2{\omega }_p$ for $|I_1|>2\sqrt{\varepsilon }(h_1>0)$.

Figure 2

Left: Critical value of $\mu$ (in logarithmic scale) against $\varepsilon$ according to (19) for the overlap of the first-order resonances ${\omega }_1\left(I_1\right)=0$ and ${\omega }_1\left(I_1\right)=\pm 1$. Right: MEGNO contour plot for $\varepsilon =0.25,\mu =0.10$ showing that for these values of the parameters, the overlap occurs. In the MEGNO contour plot, black indicates $\langle Y\rangle >65$, gray denotes $2.1<\langle Y\rangle <65$, while white means $\langle Y\rangle <2.1$. The blue dot indicates the location of a given initial ensemble.

Figure 3

Observed diffusion during a time interval $4\times {10}^6$ for $\varepsilon =0.25,\mu =0.025$ of a small ensemble (depicted in blue) of $n_p=100$ initial conditions in the stochastic layer of the guiding resonance where ${\vartheta }_1\left(0\right)=\pi ,{\vartheta }_2\left(0\right)=0,I_1\left(0\right)=2\sqrt{\varepsilon },I_2\left(0\right)=0.01\sqrt{3}$. The plot is a projection of the full motion on the 3D section ${\mathcal{S}}_{{\vartheta }_2=0}=\{(I_1,{\vartheta }_1,I_2):|{\vartheta }_2|<2\times {10}^{-4}\}.$

Figure 4

Left: Recurrence time, $T_{\mathrm{r}}$, over the section ${\mathcal{S}}_{{\vartheta }_1=\pi }^{{\vartheta }_2=0}$ as a function of $\mu$ just for $\varepsilon =0.25, 0.01\le \mu \le 0.25$. Right: An initial ensemble (indicated as a blue circle) is followed onto the MEGNO contour plot for $\epsilon =0.25,\mu =0.025$; white and light gray denote stable motion, while black indicates strongly chaotic dynamics. The concomitant trajectories for the initial ensemble that intersect the 2D section ${\mathcal{S}}_{{\vartheta }_1=\pi }^{{\vartheta }_2=0}$ are depicted in red.

Figure 5

Left: Evolution of $sin{\vartheta }_2$ for $\varepsilon =0.25,\mu =0.025$; ${\vartheta }_1\left(0\right)=\pi ,{\vartheta }_2\left(0\right)=0,I_1\left(0\right)=2\sqrt{\varepsilon },I_2\left(0\right)=0.01\sqrt{3}$. Right: Numerical determination of $T_{\mathrm{D}}$ on ${\mathcal{S}}_{{\vartheta }_1=\pi }^{{\vartheta }_2=0}$ and its corresponding theoretical value as $d^2{D}^{-1}$ with $d^2\approx 0.49$, the mean-square distance associated with the wandering of $I_2$ (see the text for details).

Figure 6

MEGNO contour plot for $\varepsilon =\mu =0.12$, where black indicates $\langle Y\rangle >35$, gray indicates $2.1<\langle Y\rangle <35$, and white indicates $\langle Y\rangle <2.1$. The evolution of the initial ensemble is depicted in red only for $|I_2-I_2\left(0\right)|\le 0.5$.

Figure 7

$T_{\mathrm{D}}$ computed on ${\mathcal{S}}_{{\vartheta }_1=\pi }^{{\vartheta }_2=0}$ and $T_{\mathrm{L}}$ vs $\mu$; different colors indicate the values of $\varepsilon$. The black full lines in the plots correspond to particular numerical fits (see the text for details).

Figure 8

Left: $T_{\mathrm{L}}$ for $\varepsilon =0.10,0.15$ computed up to $t={10}^6$ for $n_p=50,\xi ={10}^{-7}$ and the same initial conditions as in Fig. 7 (right). $T_{\mathrm{L}}$ up to $t={10}^5$ for the same values of $\varepsilon$ are also included for comparison. Center: similar to the figure at the left but just for $\varepsilon =0.10$ and taking $n_p=1$ instead of $n_p=50$. Also $T_{\mathrm{L}}$ for $n_p=50$ and $t={10}^5$–${10}^6$ are included. The theoretical expected value given in (23) multiplied by 1.45 is depicted in black. Right: Distribution of $T_{\mathrm{L}}$ (in logarithmic scale) at $\varepsilon =0.10$ and $0.001\le \mu \le 0.05$ for the $n_p=50$ initial conditions in the ensemble for each value of $\mu$.

Figure 9

MEGNO contour plot in $(I_1,I_2)$ space for $\varepsilon =0.10,\mu =0.015$ and the evolution of an ensemble of $n_p=50$ initial conditions at the separatrix of the guiding resonance and ${\omega }_2=0.01\sqrt{3}$ on ${\mathcal{S}}_{{\vartheta }_1=\pi }^{{\vartheta }_2=0}$. Black and gray denote strong and mild chaos, respectively, while white indicates stable motion.

Figure 10

${10}^{-4}{T}_{\mathrm{D}}$ (in logarithmic scale) vs $T_{\mathrm{L}}$ for $\varepsilon =0.15,0.20,0.25$ and $22<T_{\mathrm{L}}<40$. The green line corresponds to the fit (26) for all the data with ${\gamma }_1\approx 0.18$ and $C_1$ taken as the half-value obtained by LSF since its uncertainty is large, $C_1\approx 0.06$.

Figure 11

$T_{\mathrm{D}}$ (in units of ${10}^4$) and $T_{\mathrm{L}}$ as a function of $\mu =\varepsilon$ and $T_{\mathrm{D}}$ against $T_{\mathrm{L}}$. The green curve in the three plots represents a LSF of the data, while the black one in the central panel corresponds to the theoretical estimation of $T_{\mathrm{L}}$ by means of (23) (see the text for details).

Figure 12

Left: $T_{\mathrm{D}}$ (in units of ${10}^4$) and $T_{\mathrm{L}}$ as a function of $\mu$, where different colors indicate the corresponding $\varepsilon$ values. The black curves represent a LSF of the data for particular values of $\varepsilon$ (see the text for details).

Figure 13

Left: $T_{\mathrm{D}}$ (in units of ${10}^4$) vs $T_{\mathrm{L}}$; different colors indicate the corresponding $\varepsilon$ values. The black curve in the figure on the left is the fit for $\varepsilon =0.35$, and that on the right encompasses the whole data. The two numerical fits are included, one with $C_3\approx 0.874,{\gamma }_3\approx 0.46$ in blue, while the black one is for $C_3\approx 0.346,{\gamma }_3\approx 0.824$ (see the text for details).