Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem

We study chaotic orbits of conservative low--dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of $N$ iterates in the large $N$ limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long--lasting quasi--stationary states (QSS) are found, whose pdfs appear to converge to $q$--Gaussians associated with nonextensive statistical mechanics. More generally, however, as $N$ increases, the pdfs describe a sequence of QSS that pass from a $q$--Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.Comment: 15 pages, 14 figures, accepted for publication as a Regular Paper in the International Journal of Bifurcation and Chaos, on Jun 21, 201

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-$\beta$) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within "small size" phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to $t\approx10^6$) by a $q$-Gaussian ($1<q<3$) distribution and tend to a Gaussian ($q=1$) for longer times, as the orbits eventually enter into "large size" chaotic domains. However, in agreement with other studies, we find in certain cases that the $q$-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called "crossover" function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these $q$-Gaussian distributions to identify two energy regimes of "weak chaos"-one where the system melts and one where it transforms from liquid to a gas state-by observing where the $q$-index of the distribution increases significantly above the $q=1$ value of strong chaos.Comment: 32 pages, 13 figures, Submitted for publication to Physica

Weak chaos detection in the Fermi-Pasta-Ulam-α\alpha system using qq-Gaussian statistics

We study numerically statistical distributions of sums of orbit coordinates, viewed as independent random variables in the spirit of the Central Limit Theorem, in weakly chaotic regimes associated with the excitation of the first ($k=1$) and last ($k=N$) linear normal modes of the Fermi-Pasta-Ulam-$\alpha$ system under fixed boundary conditions. We show that at low energies ($E=0.19$), when the $k=1$ linear mode is excited, chaotic diffusion occurs characterized by distributions that are well approximated for long times ($t>10^9$) by a $q$-Gaussian Quasi-Stationary State (QSS) with $q\approx1.4$. On the other hand, when the $k=N$ mode is excited at the same energy, diffusive phenomena are \textit{absent} and the motion is quasi-periodic. In fact, as the energy increases to $E=0.3$, the distributions in the former case pass through \textit{shorter} $q$-Gaussian states and tend rapidly to a Gaussian (i.e. $q\rightarrow 1$) where equipartition sets in, while in the latter we need to reach to E=4 to see a \textit{sudden transition} to Gaussian statistics, without any passage through an intermediate QSS. This may be explained by different energy localization properties and recurrence phenomena in the two cases, supporting the view that when the energy is placed in the first mode weak chaos and "sticky" dynamics lead to a more gradual process of energy sharing, while strong chaos and equipartition appear abruptly when only the last mode is initially excited.Comment: 12 pages, 3 figures, submitted for publication to International Journal of Bifurcation and Chaos. In honor of Prof. Tassos Bountis' 60th birthda

Transport phenomena and anomalous diffusion in conservative systems of low dimension

[eng] Apart from this introductory chapter, the contents of the thesis is splitted among four more chapters. Chapters 2, 3 and 4 deal with the planar case, while chapter 5 deals with the 3D volume preserving case. More specifically, - In Chap. 2 we start by considering conservative quadratic Hénon maps (both orientation preserving and orientation reversing cases). First, we study the main features of the domain of stability of these two maps, mainly from the point of view of the area that they occupy, and how it does evolve as parameters change. To be as exhaustive as possible, we review the theory that allows to explain what one can observe in the phase space of these maps. We finish the chapter by considering the Chirikov standard map (1.13) in the 2-torus T2 for large values of the parameter, k ≥ 1. The most prominent sources of regular area in this setting are accelerator modes that appear periodically in k, and scaled somehow. We give numerical evidence of such a scaling, and guided by the experimental evidence, we derive limit representations for the dynamics in some compact set containing these islands, which turn out to be conjugated to the orientation preserving quadratic Hénon map or conjugated to the square of the orientation reversing quadratic Hénon map. Some of these islands are the accelerator modes we checked that appeared in Sect. 1.3. This motivates the following chapter. - Chap. 3 is devoted to study the role of these islands of stability that ’jump’ when the standard map is considered in the cylinder. The stability domain of these islands is determined and studied independently from the standard map Mk in Chap. 2, and is recovered in some regions in the phase space of Mk under suitable scalings. We focus in two main observables: the squared mean displacement of the action under iteration of Mk and the trapping time statistics. We study them both in an adequate range of the parameters, where we can see the effect of considering more and more iterations and the fact that we change parameters and the size of the gaps of a Cantorus change. We provide evidence of the fact that the trapping time statistics behave as the superposition of the effect of two distinctive phenomena: the one of the stickiness, detected as power-law statistics, and the one of the outermost Cantorus, detected as bumps. These bumps change their position in the time axis accordingly to the change of the size of the largest gap in the Cantorus. First, assuming that the stickiness effect gives rise to power law statistics with a certain value of the exponent, and under some other mild conditions (that also are suggested by the simulations), we are able to give a lower bound on the growth of the mean squared displacement of the actions. This is the way these two phenomena are related to each other in this context. Then, the fact that we can identify the source of the bumps as being due to the effect of the outermost Cantorus, motivates the topic of the next chapter: studying this effect by its own in a proper context. - In Chap. 4 we return to the Chirikov standard map, but for values of the parameter close to the destruction of the last RIC, that is, for value of the parameter close but larger than k(G) and approaching it from above. In this setting, we study escape rates across this Cantorus, and we deal with this problem from two different points of view. First, as k decreases to k(G). In this setting, it is known that the mean escape ratio across the Cantorus, that we will denote as hNki, behaves essentially as (k−kG)−B,B ≈ 3. The Greene-MacKay renormalisation theory, and the interpretation of DeltaW as an area justify that, in fact, hNki (k − kG)B should eventually be periodic in a suitable logarithmic scale, as k → kG. In this chapter we give the first evidence of the shape of this periodic behaviour, and perform a numerical study of a region surrounding the Cantorus that allows to give a first (partial) explanation of it. Second, we consider a problem related to the previous topic but for each fixed value of k: the probability that an orbit crosses the Cantorus in a prescribed time. We explain how to compute these statistics, and we show that in logarithmic scale in the number of iterates, as k → kG, they seem to behave the same way, but shifted in this log-scale in time. - Finally, Chap. 5 is devoted to study the stickiness problem in the 3D volume preserving setting. To do so, a map inspired in the Standard map is constructed following the scheme in Sect. 1.3. This map depends on various parameters, one of them, say ε, being a distance-to-integrable one. The map is considered in such a way that: 1. Invariant tori subsist until moderate values of ε, and 2. At integer values of the parameter the origin becomes an accelerator mode, and that exactly at integer values it undergoes a Hopf-Saddle-Node bifurcation, giving rise to a stability bubble. The normal form of the unfolding of this bifurcation justifies that, in fact, there are just two relevant parameters (since it is a co-dimension 2 bifurcation). An analysis inspired in that of Chap. 3 is performed by fixing one of them. Also in this case one can observe a power law decay of the trapping time statistics, but with slightly different values of the exponent in different ranges of the number of iterates. Preliminary results of more massive simulations seem to indicate that the effect decreases as the number of iterates increases

Statistics of a Family of Piecewise Linear Maps

We study statistical properties of the truncated flat spot map $f_t(x)$. In particular, we investigate whether for large $n$, the deviations $\sum_{i=0}^{n-1} \left(f_t^i(x_0)-\frac 12\right)$ upon rescaling satisfy a $Q$-Gaussian distribution if $x_0$ and $t$ are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if $f_t$ is the rotation by $t$, then it has been shown that in this case the rescaled deviations are distributed as a $Q$-Gaussian with $Q=2$ (a Cauchy distribution). This is the only case where a non-trivial (i.e. $Q\neq 1$) $Q$-Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, $\lim_n S_n/n$ converges to a random variable with a curious distribution which is clearly not a $Q$-Gaussian or any other standard smooth distribution

Cooperative surmounting of bottlenecks

The physics of activated escape of objects out of a metastable state plays a key role in diverse scientific areas involving chemical kinetics, diffusion and dislocation motion in solids, nucleation, electrical transport, motion of flux lines superconductors, charge density waves, and transport processes of macromolecules, to name but a few. The underlying activated processes present the multidimensional extension of the Kramers problem of a single Brownian particle. In comparison to the latter case, however, the dynamics ensuing from the interactions of many coupled units can lead to intriguing novel phenomena that are not present when only a single degree of freedom is involved. In this review we report on a variety of such phenomena that are exhibited by systems consisting of chains of interacting units in the presence of potential barriers. In the first part we consider recent developments in the case of a deterministic dynamics driving cooperative escape processes of coupled nonlinear units out of metastable states. The ability of chains of coupled units to undergo spontaneous conformational transitions can lead to a self-organised escape. The mechanism at work is that the energies of the units become re-arranged, while keeping the total energy conserved, in forming localised energy modes that in turn trigger the cooperative escape. We present scenarios of significantly enhanced noise-free escape rates if compared to the noise-assisted case. The second part deals with the collective directed transport of systems of interacting particles overcoming energetic barriers in periodic potential landscapes. Escape processes in both time-homogeneous and time-dependent driven systems are considered for the emergence of directed motion. It is shown that ballistic channels immersed in the associated high-dimensional phase space are the source for the directed long-range transport

Predictability: a way to characterize Complexity

Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing different kind of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to finite-time and finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system. The characterization of irregular behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports. Related information at this http://axtnt2.phys.uniroma1.i