Coupled logistic maps and non-linear differential equations

We study the continuum space-time limit of a periodic one dimensional array of deterministic logistic maps coupled diffusively. First, we analyse this system in connection with a stochastic one dimensional Kardar-Parisi-Zhang (KPZ) equation for confined surface fluctuations. We compare the large-scale and long-time behaviour of space-time correlations in both systems. The dynamic structure factor of the coupled map lattice (CML) of logistic units in its deep chaotic regime and the usual d=1 KPZ equation have a similar temporal stretched exponential relaxation. Conversely, the spatial scaling and, in particular, the size dependence are very different due to the intrinsic confinement of the fluctuations in the CML. We discuss the range of values of the non-linear parameter in the logistic map elements and the elastic coefficient coupling neighbours on the ring for which the connection with the KPZ-like equation holds. In the same spirit, we derive a continuum partial differential equation governing the evolution of the Lyapunov vector and we confirm that its space-time behaviour becomes the one of KPZ. Finally, we briefly discuss the interpretation of the continuum limit of the CML as a Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) non-linear diffusion equation with an additional KPZ non-linearity and the possibility of developing travelling wave configurations.Comment: 23 page

A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking

In this paper, a new chaotic pseudo-random number generator (PRNG) is proposed. It combines the well-known ISAAC and XORshift generators with chaotic iterations. This PRNG possesses important properties of topological chaos and can successfully pass NIST and TestU01 batteries of tests. This makes our generator suitable for information security applications like cryptography. As an illustrative example, an application in the field of watermarking is presented.Comment: 11 pages, 7 figures, In WISM 2010, Int. Conf. on Web Information Systems and Mining, volume 6318 of LNCS, Sanya, China, pages 202--211, October 201

Description of stochastic and chaotic series using visibility graphs

Nonlinear time series analysis is an active field of research that studies the structure of complex signals in order to derive information of the process that generated those series, for understanding, modeling and forecasting purposes. In the last years, some methods mapping time series to network representations have been proposed. The purpose is to investigate on the properties of the series through graph theoretical tools recently developed in the core of the celebrated complex network theory. Among some other methods, the so-called visibility algorithm has received much attention, since it has been shown that series correlations are captured by the algorithm and translated in the associated graph, opening the possibility of building fruitful connections between time series analysis, nonlinear dynamics, and graph theory. Here we use the horizontal visibility algorithm to characterize and distinguish between correlated stochastic, uncorrelated and chaotic processes. We show that in every case the series maps into a graph with exponential degree distribution P (k) ~ exp(-{\lambda}k), where the value of {\lambda} characterizes the specific process. The frontier between chaotic and correlated stochastic processes, {\lambda} = ln(3/2), can be calculated exactly, and some other analytical developments confirm the results provided by extensive numerical simulations and (short) experimental time series

Randomness Quality of CI Chaotic Generators: Applications to Internet Security

Due to the rapid development of the Internet in recent years, the need to find new tools to reinforce trust and security through the Internet has became a major concern. The discovery of new pseudo-random number generators with a strong level of security is thus becoming a hot topic, because numerous cryptosystems and data hiding schemes are directly dependent on the quality of these generators. At the conference Internet`09, we have described a generator based on chaotic iterations, which behaves chaotically as defined by Devaney. In this paper, the proposal is to improve the speed and the security of this generator, to make its use more relevant in the Internet security context. To do so, a comparative study between various generators is carried out and statistical results are given. Finally, an application in the information hiding framework is presented, to give an illustrative example of the use of such a generator in the Internet security field.Comment: 6 pages,6 figures, In INTERNET'2010. The 2nd Int. Conf. on Evolving Internet, Valencia, Spain, pages 125-130, September 2010. IEEE Computer Society Press Note: Best Paper awar

Time Quasilattices in Dissipative Dynamical Systems

We establish the existence of `time quasilattices' as stable trajectories in dissipative dynamical systems. These tilings of the time axis, with two unit cells of different durations, can be generated as cuts through a periodic lattice spanned by two orthogonal directions of time. We show that there are precisely two admissible time quasilattices, which we term the infinite Pell and Clapeyron words, reached by a generalization of the period-doubling cascade. Finite Pell and Clapeyron words of increasing length provide systematic periodic approximations to time quasilattices which can be verified experimentally. The results apply to all systems featuring the universal sequence of periodic windows. We provide examples of discrete-time maps, and periodically-driven continuous-time dynamical systems. We identify quantum many-body systems in which time quasilattices develop rigidity via the interaction of many degrees of freedom, thus constituting dissipative discrete `time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333. Submission to SciPos

Bifurcations in Globally Coupled Map Lattices

The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius--Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The complete bifurcation behaviour of coupled tent maps near the chaotic band merging point is presented. Furthermore the time independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations.Comment: 19 pages, .dvi and postscrip