Intermittent stickiness synchronization

TL;DR

This paper investigates the phenomenon of Intermittent Stickiness Synchronization in high-dimensional Hamiltonian systems using FTLEs, revealing universal decay behaviors and rare long-term full synchronization events in mixed phase space.

Contribution

It introduces a novel analysis of ISS using FTLEs, demonstrating universal decay exponents and the rarity of long-term full synchronization in complex systems.

Findings

Full SS occurs when all FTLEs tend to zero over long times.

Partial SS depends on system parameters and phase space dimension.

Full SS can appear after very long times due to slow decay of correlations.

Abstract

This work uses the statistical properties of Finite-Time Lyapunov Exponents (FTLEs) to investigate the Intermittent Stickiness Synchronization (ISS) observed in the mixed phase space of high-dimensional Hamiltonian systems. Full Stickiness Synchronization (SS) occurs when all FTLEs from a chaotic trajectory tend to zero for arbitrary long time-windows. This behavior is a consequence of the sticky motion close to regular structures which live in the high-dimensional phase space and affects all unstable directions proportionally by the same amount, generating a kind of collective motion. Partial SS occurs when at least one FTLE approaches to zero. Thus, distinct degrees of partial SS may occur, depending on the values of nonlinearity and coupling parameters, on the dimension of the phase space and on the number of positive FTLEs. Through filtering procedures used to precisely characterize the sticky motion, we are able to compute the algebraic decay exponents of the ISS and to obtain remarkable evidence about the existence of a universal behavior related to the decay of time correlations encoded in such exponents. In addition, we show that even though the probability to find full SS is small compared to partial SSs, the full SS may appear for very long times due to the slow algebraic decay of time correlations in mixed phase space. In this sense, observations of very late intermittence between chaotic motion and full SSs become rare events.

Tables

Table 1. Table 1: Values of K i subscript 𝐾 𝑖 K_{i} used to couple the standard maps and the thresholds ε i subscript 𝜀 𝑖 \varepsilon_{i} .

Value of $K_i$	$N=2$	$N=3$	$N=4$	$N=5$
$K_1$	0.65	0.65	0.65	0.65
$K_2$	0.60	0.63	0.64	0.64
$K_3$	-	0.60	0.63	0.63
$K_4$	-	-	0.62	0.62
$K_5$	-	-	-	0.61
Threshold
${\epsilon }_1$	0.10	0.10	0.10	0.10
${\epsilon }_2$	0.05	0.08	0.08	0.08
${\epsilon }_3$	-	0.05	0.06	0.07
${\epsilon }_4$	-	-	0.04	0.06
${\epsilon }_5$	-	-	-	0.04

Equations

$$\mathbf{M_{i}}\left(\begin{array}[]{c}p_{i}\\ x_{i}\\ \end{array}\right)=\left(\begin{array}[]{llll}p_{i}+K_{i}\sin(2\pi x_{i})&\hskip 2.84544pt\mathrm{mod}\hskip 5.69046pt1\\ x_{i}+p_{i}+K_{i}\sin(2\pi x_{i})&\hskip 2.84544pt\mathrm{mod}\hskip 5.69046pt1\\ \end{array}\right),$$

$$\mathbf{T_{i}}\left(\begin{array}[]{c}p_{i}\\ x_{i}\\ \end{array}\right)=\left(\begin{array}[]{llll}p_{i}+\sum_{j=1}^{N}\xi_{i,j}\hskip 1.42271pt\sin[2\pi(x_{i}-x_{j})]\\ x_{i}\\ \end{array}\right),$$

$$P_{\mathrm{cum}}({\tau }_M)\equiv \sum _{{\tau }_M^{\prime }={\tau }_M}^{\infty }P({\tau }_M^{\prime }).$$

$$P(S_M)=\frac{1}{\beta }\sum _{t=0}^{t_L}{\delta }_{t\in S_M},$$

Full text

Stickiness, finite-time Lyapunov spectrum, Poincaré recurrences, synchronization.

I Introduction

Synchronization in high-dimensional Hamiltonian systems has been defined as measure synchronization in Refs. Hampton and Zanette (1999); Vincent (2005). In these works the authors use models consisting of two coupled maps. By starting two distinct initial conditions from the uncoupled system, which lead to a regular dynamics, they observe what happens to them by adding a small coupling between the maps. A kind of synchronized (collective) motion appears named measure synchronization. As the synchronization of dissipative chaotic systems Boccaletti et al. (2002); Pecora and Carroll (2015), synchronization in generic Hamiltonian systems is also an interesting issue since such systems present a mixed phase-space dynamics which contains a rich variety of behaviors. However, it is important to mention that the synchronization phenomenon observed in dissipative systems is not possible in Hamiltonian system due to the Liouville’s theorem that prevents the full collapse of the orbits to an invariant manifold, since volume must be preserved in phase space.

For symplectic two-dimensional maps the chaotic component is clearly separated from the regular motion Lichtenberg and Lieberman (1992); Zaslavsky (2008). However in higher dimensions the chaotic trajectory may visits ergodically the whole phase space but, until this happens, it may suffer a dynamical trapping (or sticky motion) Chirikov and Shepelyansky (1984); Artuso (1999) close to quasi-regular structures. The effect of the sticky motion on the chaotic trajectory can be classified in distinct regimes Contopoulos et al. (1978); Malagoli et al. (1986), defined by the spectrum of Finite-Time Local Lyapunov Exponents (FTLLEs). When all FTLLEs are positive, the regime is chaotic, when all are zero, we have an ordered regime. In between we have semiordered regimes. Separating the dynamics in distinct regimes, like a filtering procedure, not only a substantial increase in the characterization of the sticky motion was achieved da Silva et al. (2015a), but allowed to find a synchronized-like state, leading to the Intermittent Stickiness Synchronization (ISS) discussed in the present work. Essentially the ISS is characterized by the intermittent behaviour between the chaotic motion and a kind of transient measure synchronization generated by stickiness. Such synchronized-like states due to stickiness were also detected some years ago and classified as common motion Manchein et al. (2012). A somehow similar analysis allowed to synchronize drive and slave coupled standard maps Mahata et al. (2016). In this case, since the coupling between the two maps is unidirectional, once the synchronized state is reached, it does not change along the simulations. Such behavior changes when the coupling interaction between the maps is bidirectional, as considered in the present work, which uses global (all-to-all) interactions.

Since events with long times are associated to times for which the trajectory was trapped to the nonhyperbolic components of the phase space Artuso (1999); Cristadoro and Ketzmerick (2008); Altmann and Kantz (2007); Shepelyansky (2010); Lange et al. (2014, 2016), in the present work the ISS decay is qualitatively described using the time decay of the ordered regime in the case of coupled maps. To mention, other alternatives approaches using FTLEs Kantz and Grassberger (1987); Falcioni et al. (1991); Konishi and Kaneko (1992); Kaneko and Konishi (1994); Tomsovic and Lakshminarayan (2007); Szezech et al. (2005); Harle and Feudel (2007) can be used to characterize the phase space of conservative systems, with recent applications using large deviation techniques Artuso and Manchein (2009); de Oliveira et al. (2018); Laffargue et al. (2013) and the cumulants Manchein et al. (2012, 2014) of the FTLEs distribution.

Recently, a methodology that uses time-series of local Lyapunov exponents to define the above mentioned regimes of ordered, semiordered, and totally chaotic motion was proposed, making it possible to improve the characterization of stickiness in Hamiltonian with few degrees of freedom da Silva et al. (2015a) and non-Hamiltonian da Silva et al. (2015b) systems.

The present work uses such filtering procedure da Silva et al. (2015a) to check precisely the algebraic decay exponents of the ISS in higher-dimensional Hamiltonian systems. This investigation is motivated by the few amount of numerical studies related to weakly chaotic properties and consequently the long time correlations observed in higher-dimensional mixed phase spaces of Hamiltonian systems (at least for small and moderate number of homogeneously coupled two-dimensional maps). We find that only the full SS obeys a power-law decay, while all other partial SSs decay exponentially. Thus, sticky effects from the semiordered regimes are almost irrelevant for long time ISS decay. Additionally, the algebraic decay exponent of full SS seems to be independent of the (i) number of coupled maps (at least for a moderate number of them), and (ii) the coupling intensities used here. Although it is still under debate whether such an exponent persists in the weak-coupling regime, our investigation corroborates with the claim suggested in de Oliveira et al. (2018) that predicts the existence of generic decay exponent for time correlations $\chi\sim 0.20$ for Hamiltonian systems with few degrees of freedom which is smaller than what says the conjecture proposed in Shepelyansky (2010) to predict the existence of an universal decay of Poincaré recurrences $\gamma\approx 1.30-1.40$ (see also Ding et al. (1990) for earlier work). The corresponding relationship between these algebraic exponents is given by the well-known equation $\chi=\gamma-1$.

The plan of this paper is presented as follows. Section II presents the coupled maps model used for the simulations. In Sec. III the precise definition of ordered, semiordered, and chaotic regimes is given, which leads to the definition of the synchronized-like state, together with some numerical examples. While in Sec. IV the ISS decay is discussed qualitatively, Sec. V summarizes our conclusions.

II The coupled-maps model

Consider the time-discrete composition $\mathbf{T}\circ\mathbf{M}$ of independent one-step iteration of $N$ symplectic $2$-dimensional maps $\mathbf{M}=(M_{1},\ldots,M_{N})$ and a symplectic coupling $\mathbf{T}=(T_{1},\ldots,T_{N})$. This constitutes a $2N$-dimensional Hamiltonian system. For our numerical investigation we use the $2$-dimensional Standard Map (SM):

[TABLE]

and for the coupling

[TABLE]

with $\xi_{i,j}=\xi_{j,i}=\frac{\xi}{\sqrt{N-1}}$ (all-to-all coupling). This system is a typical Hamiltonian benchmark tool with mixed phase space presenting all essential features expected in complex systems. It was studied in Refs. Altmann and Kantz (2007); Altmann (2007) using the Recurrence Time Statistic (RTS) and used in Ref. da Silva et al. (2015a) to propose the classification of Lyapunov regimes for improving stickiness characterization. In all numerical simulations we used nonlinearity parameters corresponding to a mixed phase space, namely $0.60\leq K\leq 0.65$ (see Fig. 1 which is discussed next).

III Definition of regimes and stickiness synchronization

The numerical technique uses the FTLLEs spectrum $\{\lambda_{i=1\ldots N}^{(\omega)}\}$ computed along a chaotic trajectory during a window of size $\omega$, where $\lambda^{(\omega)}_{1}>\lambda^{(\omega)}_{2},\ldots,\lambda^{(\omega)}_{N}>0$, and explores temporal properties in the time series of $\{\lambda_{i}^{(\omega)}\}$ da Silva et al. (2015a). The window size $\omega$ has to be sufficiently small to guarantee a good resolution of the temporal variation of the $\lambda^{(\omega)}_{i}$’s, but sufficiently large in order to have a reliable estimation (see Refs. Kantz and Grassberger (1987); Szezech et al. (2005); Harle and Feudel (2007)). The sharp transitions towards $\lambda_{i}^{(\omega)}\approx 0$ observed earlier motivates the classification in regimes of motion Contopoulos et al. (1978); Malagoli et al. (1986). For a system with $N$ degrees of freedom, the trajectory is in a regime of type $S_{M}$ if it has $M$ local Lyapunov exponents $\lambda^{(\omega)}_{i}>\varepsilon_{i}$, where $\varepsilon_{i}\ll\lambda_{i}^{(\infty)}$ are small thresholds. Thereby, $S_{0}$ and $S_{N}$ are ordered and chaotic regimes respectively. Regimes $S_{i}$ with $0<i<N$ are called semiordered. For the computation of the FTLLEs we use the traditional Benettin’s algorithm Benettin et al. (1980); Wolf et al. (1985), which includes the Gram–Schmidt re-orthonormalization procedure. On average, the FTLLEs are in decreasing order. However, inversions of the order ($\lambda_{i+1}^{(\omega)}>\lambda_{i}^{(\omega)}$) may happen for some times $t$ and we have chosen to impose the order of $\lambda_{i}^{(\omega)}$ for all $t$.

III.1 The uncoupled case: $N=1$

To get a better understanding of the involved complexity in the dynamics and the behavior of the FTLLEs, Fig. 1 displays the phase-space dynamics for a chaotic trajectory for one uncoupled ($\xi=0$) SM together with the corresponding positive FTLLE $\lambda^{(\omega)}_{1}$ (see color bar) for $\omega=100$. In this case the Lyapunov spectrum has only two Lyapunov exponents which, asymptotically, one is positive and the other one negative. Thus, only two regimes are observed: (i) the ordered one, if $\lambda_{1}^{(\omega)}<\varepsilon_{1}$ and, (ii) the chaotic one, if $\lambda_{1}^{(\omega)}>\varepsilon_{1}$. While the upper row in Fig. 1 presents the $K=0.60$ case, the lower row shows results for $K=0.65$. For both cases, the phase space has a large regular island located in the center, surrounded by higher order resonances. In Fig. 1(a) we observe a $6$-order resonance and in Fig. 1(e) a $8$-order resonance [better seen in Figs. 1(b) and 1(f), respectively]. It is well known that additional higher-order resonances (not visible in the scale of these Figures) live around the island. These islands lead to the dynamical trapping which can be stronger, or not, depending on the topological structure of the islands. Such dependency becomes better visible when the positive FTLLE $\lambda^{(\omega)}_{1}$ is calculated for the trajectories. This is shown in colors in Figs. 1(b) and 1(f) with some magnifications (see black boxes) shown respectively in Figs. 1(c), 1(d) and 1(g), 1(h). We observe that, when approaching the island borders, the FTLLE decreases, as specified by the color bars in Figs. 1(d), 1(h). A very complex behavior is evident and only motions very close to the regular islands have smaller FTLLEs. This suggests that these motions close to the regular islands will belong to the ordered motion.

III.2 The coupled case: $N=2$

A nice visualization of the regimes becomes clear when two coupled SM are analyzed in phase space, as shown in Fig. 2. Different colors represent points $\boldsymbol{x}_{t}\in S_{M}$ belonging to regimes $S_{0}$ (blue circles), $S_{1}$ (red points), and $S_{2}$ (green points). These points were computed starting a single trajectory in the chaotic sea of the coupled system and iterating it $10^{7}$ times. Table 1 presents the values of $K_{i}$ used in the simulations and the thresholds $\varepsilon_{i}$ used to define the regimes of motion. Blue circles are the points in phase space ($x_{1},p_{1}$) and ($x_{2},p_{2}$) for which $\lambda^{(\omega)}_{i}<\varepsilon_{i}$, for $i=1,2$. The red color indicates points for which $\lambda^{(\omega)}_{1}>\varepsilon_{1}$ and $\lambda^{(\omega)}_{2}<\varepsilon_{2}$. Green points are used if both FTLLEs are larger than the respective thresholds $\varepsilon_{i}$. We observe in Fig. 2 that by increasing the value of the coupling strength $\xi$, the trajectory penetrates the regular domain from the uncoupled case where there is the island’s hierarchy, inside which only $S_{0}$ and $S_{1}$ regimes occur. Thus, by increasing the coupling force between the maps, the number of points which induce sticky motion increase.

III.3 The stickiness synchronization

From above results it is easy to verify that for the ordered regime the position of coupled maps tend to be very close to the almost regular domains and to each other. This can be checked more precisely by determining, for $N=2$ for example, the distance $|x_{1}-x_{2}|$ as a function of time. This is shown by the gray color in Fig. 3 for two distinct time windows. At a given time, the distance $|x_{1}-x_{2}|$ suddenly approaches zero, leading to an approximated synchronization of the positions of the coupled SMs. Since these positions are not exactly equal, we say to have a synchronized-like state. In both cases the synchronization occurs only for a finite-time window. Surprisingly these time windows match with those times for which the ordered regime $S_{0}$ is present. This can be checked in the same picture, where we plot simultaneously the two positive FTLLEs $\lambda^{(\omega)}_{i}$ as a function of time. Thus, synchronization of the position of the maps coincides with the full synchronization of the FTLLEs. For the regime $S_{1}$ we observe in Fig. 3(b) that the distance $|x_{1}-x_{2}|$ is more away from zero than this distance measured in the regime $S_{0}$, leading to a kind of “weaker” synchronization. In this case we say to have a partial synchronization, since only one FTLLE tends to zero. For $S_{2}$ there is no synchronization.

The relation between the position synchronization of the coupled maps shown above allows us to use the concept of stickiness synchronization. We have checked this relation for all $S_{0}$ regimes along a chaotic trajectory of length $t=10^{9}$. Since all regimes $S_{M}$ with $M<N$ are transient, and there is an intermittent transition between these regimes, we say to have the ISS. Our results for higher-dimensions can also be interpreted as the synchronization of FTLLEs. It is in fact a consequence of the synchronization of local expansion/contraction rates along all unstable/stable direction manifolds.

IV Qualitative description of ISS

Numerical techniques used to characterize the sticky motion can now be used to describe the qualitative behavior of the ISS decay in time. For this we use the consecutive time $\tau_{M}$ spent by a trajectory in the same regime $S_{M}$ da Silva et al. (2015a). During a trajectory of length $t_{L}$ we collected a series of $\tau_{M}$ and important results are obtained by analyzing the cumulative distribution of $\tau_{M}$, defined as:

[TABLE]

Applying this alternative procedure to obtain the $P_{\mathrm{cum}}(\tau_{M})$, we are able to estimate the decay exponent for the recurrence times. This technique is much more appropriated to estimate such an exponent when compared to the former one, based on the cumulative distribution of Poincaré recurrence times [or (RTS)], since it remains unclear how to estimate the time scale over which a higher-dimensional system reaches its asymptotic regime under the process of weak Arnold diffusion Lichtenberg and Lieberman (1992). As this technique is based on a filtering process, we can select the events related to longer correlations and then obtain decay curves with several decades. In addition, there is no more dependence on the choice of recurrence set to obtain the RTS Sala et al. (2016).

IV.1 The uncoupled case: $N=1$

To apply the filtering method we have to specify the threshold $\varepsilon$ and the time window $\omega$. Figures 4(a) and 4(b) compare the cumulative distribution $P_{\mathrm{cum}}(\tau_{M})$ for the regime $S_{0}$, obtained using $\omega=100$ and two values of $\varepsilon$, with the RTS $P_{\mathrm{cum}}(\tau)$, both quantities calculated for uncoupled SMs with two different values of $K$, specified in Fig. 4. For the determination of the RTS (plotted in gray) we have: (i) used a recurrence region in the chaotic component of the phase space delimited by $0<x<1$ and $0.45<p<-0.45$, and (ii) collected the lapses of time $\tau$ spent outside of the recurrence region by a trajectory started inside of such predefined box. Straight lines in Fig. 4 are consequences of the sticky motion. We realize that while the usual RTS presents some oscillations as a function of $\tau$, leading to difficulties in the precise decay exponent, the filtering method tends to decrease such oscillations, mainly if the threshold $\varepsilon=0.07$ is used. These results show that, for practical implementations, the thresholds can be defined as $\varepsilon_{i}\approx 0.10\langle\lambda_{i}^{(\omega)}\rangle$, where $\langle\ldots\rangle$ denotes the average over $t$, where $t=1,\ldots,t_{L}$.

It is important to define how sensitive these results are in relation to the time window $\omega$ used to calculate the FTLLEs. For this, we compare $P_{\mathrm{cum}}(\tau_{0})$ obtained using $\omega=100$ and $\varepsilon=0.07$ [blue curves in Figs. 4(a) and 4(b)] with $P_{\mathrm{cum}}(\tau_{0})$ for $\omega=50$ and $\omega=200$, keeping the threshold $\varepsilon=0.07$. Figures 4(c) and 4(d) show this comparison for the cases $K=0.60$ and $K=0.65$, respectively, and the results demonstrate that even though the choice of $\omega$ may affect quantitatively the cumulative distributions $P_{\text{cum}}(\tau_{M})$, our conclusions about the algebraic decay obtained for the regime $S_{0}$ are not changed by oscillations around the chosen value $\omega=100$ da Silva et al. (2015a).

IV.2 The coupled cases: $N=2,3,4,5$

We start determining the cumulative distribution $P_{\mathrm{cum}}(\tau_{M})$ for the $N=2$ case for which we have regimes $S_{0},S_{1}$ and $S_{2}$. This is shown in Fig. 5(a) for coupling $\xi=10^{-3}$ and using values of $K_{i}$ and $\varepsilon_{i}$ according to the Table 1. It shows that the only power-law decay occurs for the $S_{0}$ regime. Thus, while full SS occurs for $S_{0}$ regimes with a power-law decay of the $P_{\mathrm{cum}}(\tau_{M})$, all other regimes have a chaotic component leading to an exponential decay. This indicates that only full FTLLEs synchronized states tend to occur for consecutive very long times, even though with small probability.

Looking at the distributions $P_{\mathrm{cum}}(\tau_{M})$ in Fig. 5(a), we observe for the semiordered regime $M=1$ an exponential tail after an initial power-law decay with scaling $\beta\approx 0.5$ Altmann and Kantz (2007). When the full SS takes place ($M=0$), $P_{\mathrm{cum}}(\tau_{0})\propto\tau_{0}^{-\gamma}$, with $\gamma=1.16$. As shown in Fig. 5(b), this scaling is compatible with the result obtained using RTS. However, the cumulative distribution $P_{\mathrm{cum}}(\tau_{0})$ provides a better characterization of algebraic decay (over several orders of magnitudes), which is essential when dealing with high-dimensional systems (which may contain different pre-asymptotic regimes) and for an accurate estimation of the stickiness exponent $\gamma$. In Fig. 5(c) we show that $P_{\mathrm{cum}}(\tau_{0})$ for the coupled case remains (qualitatively) the same for different values of $\omega$ around $\omega=100$.

Another very interesting quantity to be studied is the residence time $P(S_{M})$ in each regime as a function of the coupling strength, defined by

[TABLE]

where $\beta=t_{L}/\omega$ is the factor of normalization. In Eq. (4), $\delta_{t\in S_{M}}=1$ if in time $t$ the trajectory is in the regime $S_{M}$ and $\delta_{t\in S_{M}}=0$ otherwise. The $P(S_{M})$ is shown in Fig. 6(a) for $N=2$, in Fig. 6(b) for $N=3$, in Fig. 6(c) for $N=4$ and in Fig. 6(d) for $N=5$. For smaller couplings ($\xi\lesssim 3\times 10^{-2}$) the residence time decreases with $M$, namely $P(S_{N})>P(S_{N-1})>\ldots>P(S_{M})>\ldots>P(S_{1})>P(S_{0})$. This means that the probability to find the ordered regimes ($M=0$) is much smaller when compared to semiordered regimes ($M=1$) and so on. For larger couplings $\xi>10^{-1}$, the probability to find order and semiordered regimes has roughly the same values and tend to decrease until zero. Only the probabilities of fully chaotic regimes $S_{N}$ remain for larger values $\xi$.

From Figs. 5 and 6 we conclude that even though the probability to find the full SS is small compared to the partial SS, it can occur for very long times due to the power-decay found for $P_{\mathrm{cum}}(\tau_{0})$. In addition we mention that, in distinction to usual synchronization found in dissipative systems, here the ISS tends to decrease for larger coupling between the maps Heagy et al. (1995).

IV.3 Characterizing the decay of synchronization: the ordered regime

The next step is to precisely quantify the ISS decay for distinct values of coupling $\xi$ between $N=2,3,4$ and $5$ SMs. For this we used only the regime $S_{0}$, which is directly related with the full synchronization between the positions $x_{i}$. As demonstrated in Figs. 4(a), 4(b) and 5(b), the decay of $P_{\mathrm{cum}}(\tau_{0})$ provides an amazing characterization of the sticky motion and allows obtaining accurately the exponent $\gamma$, so that the RTS analysis becomes needless. The results of this study are shown in Fig. 7(a) for $N=2$ and in Fig. 7(b) for $N=3$, using distinct values of $\xi$, as specified in the Figure. The black dotted line is the average over all couplings of each case and fitting this curve we obtain a power-law decay with well defined exponent $\gamma\approx 1.19$, observed for 6 decades. For $\xi=10^{-2}$, long-term trappings tend to disappear. It is worth to mention that the disappearance of the long-term sticky motion manifest itself in the increasing lack of data for $S_{0}$ as the coupling increases.

To finish we would like to present results for $N=4$ and $N=5$. Figure 8 displays the cumulative distribution $P_{\mathrm{cum}}(\tau_{0})$ for the regime $S_{0}$ and for distinct coupling values, specified in the Figure. We observe that for values of $\xi\leq 10^{-4}$ the exponent approaches $\gamma\approx 1.19$ for almost $6$ decades in Fig. 8(a), and $\gamma\approx 1.22$ in Fig. 8(b), values obtained fitting the black dotted line that is the average over all couplings. The amount of long-term sticky motion decreases for $\xi>10^{-4}$ in both cases. Again, this manifests itself in the increasing lack of data for $S_{0}$. However, since we still have at least three decades of power-law behavior, it can still be characterized as sticky motion leading to the full SS.

V Conclusions

This work analysis qualitatively the Intermittent Stickiness Synchronization (ISS) decay in high-dimensional generic Hamiltonian systems. Such synchronization is generated by the regular structures on the chaotic trajectory, and can also be interpreted as the synchronization of FTLLEs. It is a synchronization of local expansion/contraction rates along all unstable/stable direction manifolds. We connect the intermittent motion between ordered, semiordered and chaotic dynamical regimes with, respectively, the full, partial, and absence of synchronization generated by stickiness. By using the cumulative distribution of the consecutive times $\tau_{M}$ spent in each regime $S_{M}$, we demonstrate the ability of the recent proposed filtering procedure da Silva et al. (2015a) to precisely characterize the ISS decay generated by the sticky motion. We also show that even though the residence time in the full SS state is small compared to the residence times in the partial SS states, it may occur for consecutive very long times due to the power-decay of the $P_{\mathrm{cum}}(\tau_{0})$.

In addition, our numerical results demonstrate that the algebraic decay exponent tends to $\gamma\approx 1.20$ for higher-dimensional systems. This is in completely agreement with the estimated decay exponent of time correlations $\chi\approx 0.20$ (both exponents are related by the well-know relationship $\chi=\gamma-1$) obtained in de Oliveira et al. (2018) for $N=2,3$ symplectic maps interacting through a nearest-neighbor coupling scheme. The estimated decay exponents in these two works were obtained through three different approaches and are somehow smaller than recent estimates Shepelyansky (2010) (such observations suggest an universality conjecture, at least for a moderate number of coupled Hamiltonian maps).

Future investigations can analyze a possible relation between the ISS observed here and the hydrodynamic modes found in many body systems Dellago et al. (1996). They present slow, long-wavelength behavior in the tangent space dynamics. Besides, the properties of the covariant Lyapunov vectors Ginelli et al. (2007); Norwood and et al. (2013); Beims and Gallas (2016, 2018) at the full SS might also be promising from the theoretical point of view and applications.

Acknowledgements.

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES)

• Finance Code 001. C. M. and M. W. B. thank CNPq (Brazil) for financial support. The authors also acknowledge computational support from Prof. C. M. de Carvalho at LFTC-DFis-UFPR (Brazil).