Estimation of diffusion time with the Shannon entropy approach

The present work revisits and improves the Shannon entropy approach when applied to the estimation of an instability timescale for chaotic diffusion in multidimensional Hamiltonian systems. This formulation has already been proved efficient in deriving the diffusion timescale in 4D symplectic maps and planetary systems, when the diffusion proceeds along the chaotic layers of the resonance's web. Herein the technique is used to estimate the diffusion rate in the Arnold model, i.e., of the motion along the homoclinic tangle of the so-called guiding resonance for several values of the perturbation parameter such that the overlap of resonances is almost negligible. Thus differently from the previous studies, the focus is fixed on deriving a local timescale related to the speed of an Arnold diffusion-like process. The comparison of the current estimates with determinations of the diffusion time obtained by straightforward numerical integration of the equations of motion reveals a quite good agreement.

Figure 1

MEGNO contour plot revealing the actual resonance web of the Arnold Hamiltonian (19) for $\varepsilon =0.25,\mu =0.010$ on the section ${\vartheta }_1=\pi ,{\vartheta }_2=0$.

Figure 2

An initial ensemble indicated as a green point is followed onto the MEGNO contour plot for $\epsilon =0.25,\mu =0.010$; white and light gray denote stable motion, and black indicates strong chaotic dynamics. The concomitant trajectories for the initial ensemble that intersect the section $|{\vartheta }_1-\pi |+|{\vartheta }_2|\le 0.02$ are depicted in red.

Figure 3

Left: Evolution of the mean value $\lambda \left(t\right)=N_s\left(t\right)/q_0\left(t\right)$ for both values of $n_p$. Middle: Evolution of $I_2$ for $n_p=800$. Right: Distribution of the $I_2$ values binned in 150 intervals.

Figure 4

Evolution of the normalized entropies $\widehat{S}\left(t\right),{\widehat{S}}_0\left(t\right)$ and $D_S\left(t\right)$, for $\epsilon =0.25,\mu =0.010$ and two different values of $n_p$.

Figure 5

Evolution of ${\mathrm{var}}_s$ and ${\mathrm{var}}_e$ for $\epsilon =0.25,\mu =0.010,n_p=400$. Left: Fit of the form ${\mathrm{var}}_e\left(t\right)=D{t}^b$ is included in blue with $b\approx 0.68$ and $D\approx {10}^{-5}$. Right: Four different linear fits to ${\mathrm{var}}_e$ are performed in the ranges $[0,8\times {10}^5]$ in black, $(8\times {10}^5,2\times {10}^6]$ in sky blue, $(2\times {10}^6,3\times {10}^6]$ in magenta, and $[3\times {10}^6,4\times {10}^6]$ in green.

Figure 6

Left: Diffusion and instability time in blue and red, respectively, in logarithmic scale for $\varepsilon =0.25,0.0005\le \mu \le 0.08$. Right: Zoom for $0.0005\le \mu \le 0.02$.