[en] We analyse the probability densities of daily rainfall amounts at a variety of locations on Earth. The observed distributions of the amount of rainfall fit well to a q-exponential distribution with exponent q close to $q\approx <!--\; ???\; -->1.3$. We discuss possible reasons for the emergence of this power law. In contrast, the waiting time distribution between rainy days is observed to follow a near-exponential distribution. A careful investigation shows that a q-exponential with $q\approx <!--\; ???\; -->1.05$ yields the best fit of the data. A Poisson process where the rate fluctuates slightly in a superstatistical way is discussed as a possible model for this. We discuss the extreme value statistics for extreme daily rainfall, which can potentially lead to flooding. This is described by Fréchet distributions as the corresponding distributions of the amount of daily rainfall decay with a power law. Looking at extreme event statistics of waiting times between rainy days (leading to droughts for very long dry periods) we obtain from the observed near-exponential decay of waiting times extreme event statistics close to Gumbel distributions. We discuss superstatistical dynamical systems as simple models in this context. (paper)

[en] Highlights: • We study the synchronization of networks having an uncountable number of systems. • We assume the couplings between the nodes arise from a two-by-two hierarchical organization. • Global synchronization takes place when such a network is fully coupled, if the parameters are close to 1/2. • We study the possibility of synchronization in case of an infinity of broken links inside the hierarchical structure. • We show local synchronization occurs if the broken links are well placed and the parameters are very close to 1/2. We study the synchronization of massively connected dynamical systems for which the interactions come from the succession of couplings forming a global hierarchical coupling process. Motivations of this work come from the growing necessity of understanding properties of complex systems that often exhibit a hierarchical structure. Starting with a set of $2^n$ systems, the couplings we consider represent a two-by-two matching process that gather them in larger and larger groups of systems, providing to the whole set a structure in $n$ stages, corresponding to $n$ scales of hierarchy. This leads us naturally to the synchronization of a Cantor set of systems, indexed by ${\{0,1\}}^{\mathbb{N}}$, using the closed-open sets defined by $n$-tuples of $0$ and $1$ that permit us to make the link with the finite previous situation of $2^n$ systems: we obtain a global synchronization result generalizing this case. In the same context, we deal with this question when some defects appear in the hierarchy, that is to say when some couplings among certain systems do not happen at a given stage of the hierarchy. We prove we can accept an infinite number of broken links inside the hierarchy while keeping a local synchronization, under the condition that these defects are present at the $N$ smallest scales of the hierarchy (for a fixed integer $N$) and they be enough spaced out in those scales.

[en] We consider a class of nonideal oscillating (by Sommerfeld and Kononenko) dynamical systems and establish the existence of two types of hyperchaotic attractors in these systems. The scenarios of transitions from regular to chaotic ones attractors and the scenarios of transitions between chaotic attractors of different types are described.

[en] We present new approaches for solving constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose stationary solution is the solution to the time-independent nonlinear Schrödinger equation. Constraints are often considered by projection onto the constraint set, here we include them explicitly into the dynamical system. We show the applicability and efficiency of the methods on examples of relevance in modern physics applications. (paper)

[en] In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.

[en] We analyse the existence of limiting laws of rare events for dynamically generated stochastic processes. We consider two-dimensional dynamical systems and observable functions maximised on Cantor dust sets and prove the existence of distributional limits for the partial maxima. We show how the extremal index, measuring the degree of clustering of rare events, is linked to the compatibility between the dynamics and the fractal structure of the maximal sets. (paper)

[en] We consider implications of dynamical Borel–Cantelli lemmas for rates of growth of Birkhoff sums of non-integrable observables $\mathrm{\phi <!--\; ??\; -->}(x)=d(x,q){}^{-<!--\; ???\; -->k}$, k  >  0, on ergodic dynamical systems $(T,X,\mathrm{\mu <!--\; ??\; -->})$ where $\mathrm{\mu <!--\; ??\; -->}(X)=1$. Some general results are given as well as some more concrete examples involving non-uniformly expanding maps, intermittent type maps as well as uniformly hyperbolic systems. (paper)

[en] Let G be a group equipped with a cyclic order such that G be a cyclically ordered group. Under the given order and a particular condition on G, the positive cone P(G) of G is a cyclically ordered semigroup. We induced the representation of a C ^*-algebra A in a Hilbert space such that we obtain a covariant representation of an action of P(G) by automorphism of A. C ^*-dynamical system (A, α, P(G)), where α is an action of P(G) by automorphism of A. (paper)

[en] Highlights: • We introduce a new mechanism to create clustering of rare events for dynamical systems. • The mechanism consists in using multiple correlated maxima belonging to the same orbit. • We study the effect of the mechanism on the Extremal Index and on Rare Events Point processes. • We observe that the multiple correlated maxima may create different clustering patterns. • We study the problem of competition between different domains of attraction for extremes. We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The novelty is that we will consider observables achieving a global maximum value (possible infinite) at multiple points with special emphasis for the case where these maximal points are correlated or bound by belonging to the same orbit of a certain chosen point. These multiple correlated maxima can be seen as a new mechanism creating clustering of extreme observations, i.e., the occurrence of several extreme observations concentrated in the time frame. We recall that clustering was intimately connected with periodicity when the maximum was achieved at a single point. We will study this mechanism for creating clustering and will address the existence of limiting Extreme Value Laws, the repercussions on the value of the Extremal Index, the impact on the limit of Rare Events Points Processes, the influence on clustering patterns and the competition of domains of attraction. We also consider briefly and for comparison purposes multiple uncorrelated maxima. The systems considered include expanding maps of the interval such as Rychlik maps but also maps with an indifferent fixed point such as Manneville–Pomeau maps.

[en] Highlights: • Entropy of low-significance bits in digital measurements of chaos is examined. • Low-significance bits yield a two-symbol partition with a corrugated structure. • Corrugation at low-significance bits better approximates a generating partition. • Entropy rate estimation using lower-significance bits requires longer block lengths. • Considering only short block lengths can overestimate entropy of physical system. We examine the entropy of low-significance bits in analog-to-digital measurements of chaotic dynamical systems. We find the partition of measurement space corresponding to low-significance bits has a corrugated structure. Using simulated measurements of a map and experimental data from a circuit, we identify two consequences of this corrugated partition. First, entropy rates for sequences of low-significance bits more closely approach the metric entropy of the chaotic system, because the corrugated partition better approximates a generating partition. Second, accurate estimation of the entropy rate using low-significance bits requires long block lengths as the corrugated partition introduces more long-term correlation, and using only short block lengths overestimates the entropy rate. This second phenomenon may explain recent reports of experimental systems producing binary sequences that pass statistical tests of randomness at rates that may be significantly beyond the metric entropy rate of the physical source.