Table of contents

The XI Latin American Workshop on Nonlinear Phenomena (LAWNP) has been held in Búzios-RJ, Brazil, from 5–9 October 2009. This international conference is one in a series that have gathered biennially, over the past 21 years, physicists and other scientists who direct their work towards several aspects of nonlinear phenomena and complex systems. The main purpose of LAWNP meetings is to create a friendly and motivating environment, such that researchers from Latin America and from other parts of the globe can discuss not only their own latest results but also the trends and perspectives in this very interdisciplinary field of investigation. Hence, it constitutes a forum for promoting scientific collaboration and fomenting the emergence of new ideas, helping to advance the field.

The XI edition (LAWNP'09) has gathered more than 230 scientists and students (most from Latin America), covering all of the world (27 different countries from North and South America, Asia, Europe, and Oceania). In total there were 18 plenary lectures, 80 parallel talks, and 140 poster contributions. A stimulating round-table discussion also took place devoted to the present and future of the Latin American Institutions in Complex Phenomena (a summary can be found at http://lawnp09.fis.puc-rio.br, in the Round-Table report link). The 2009 workshop was devoted to a wide scope of themes and points of view, pursuing to include the latest trends and developments in the science of nonlinearity.

In this way, we have a great pleasure in publishing this Proceedings volume based on the high quality scientific works presented at LAWNP'09, covering already established methods as well as new approaches, discussing both theoretical and practical aspects, and addressing paradigmatic systems and also completely new problems, in nonlinearity and complexity. In fact, the present volume may be a very valuable reference for those interested in an overview on how nonlinear interactions can affect different phenomena in nature, addressing: classical and quantum chaos; instability and bifurcation; cooperative behavior; self-organization; pattern formation and synchronization; far-from-equilibrium and fluctuation dynamics; nonlinearity in fluid, plasmas, granular media, optics, and wave propagation; turbulence onset; and complexity in natural and social systems.

The success of the conference was possible thanks to the financial support from many agencies, especially the Brazilian agencies Capes and CNPq, and the international agencies, Binational Itaupú, ICTP-Trieste, and CAIS-Albuquerque. Equally very important was the support by the organizer's institutions PUC-Rio de Janeiro and UFPR-Curitiba. We also must thank Journal of Physics: Conference Series, for believing in the success and scientific quality of the conference, and to the journal staff, specially Anete Ashton, for the kind and prompt help during the whole production process of this publication.

Finally, and most important, we acknowledge all the participants of the LAWNP'09, whose interest and enthusiasm in advancing the science of nonlinearity constitutes the true moto making the present Proceedings a very valuable scientific contribution.

Celia Anteneodo (PUC-Rio, Brazil) and Marcos G E da Luz (UFPR-Curitiba, Brazil) Conference Chairs

Some of the conference participants.

This issue was supported by CAPES (Agency for Evaluation and Support of Graduate Studies Programs), Brazilian govern entity devoted to the formation of human resources. CA would like to thank CAPES for financial support.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

Chaotic dynamical systems with certain phase space symmetries may exhibit riddled basins of attraction, which can be viewed as extreme fractal structures in the sense that, regardless of how small is the uncertainty in the determination of an initial condition, we cannot decrease the fraction of such points that are certain to converge to a given attractor. We investigate a mechanical system exhibiting riddled basins of attraction: a particle under a two-dimensional potential with friction and time-periodic forcing. The verification of riddling is made by checking its mathematical requirements through computation of finite-time Lyapunov exponents as well as by scaling laws describing the fine structure of basin filaments densely intertwined in phase space.

We consider a one-dimensional harmonic oscillator with a random frequency, focusing on both the standard and the generalized Lyapunov exponents, λ and λ* respectively. We discuss the numerical difficulties that arise in the numerical calculation of λ* in the case of strong intermittency. When the frequency corresponds to a Ornstein-Uhlenbeck process, we compute analytically λ* by using a cumulant expansion including up to the fourth order. Connections with the problem of finding an analytical estimate for the largest Lyapunov exponent of a many-body system with smooth interactions are discussed.

This paper gives an overview of the simple yet fundamental bouncing ball problem, which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity. The dynamics is modeled on the basis of a discrete map of difference equations, which numerically solved fully reveals a rich variety of nonlinear behaviors, encompassing irregular non-periodic orbits, subharmonic and chaotic motions, chattering mechanisms, and also unbounded non-periodic orbits.

The effect of rounded edges in the rectangular billiard is analyzed via the escape times statistic. Such realistic and sometimes intrinsic edges are shown to generate autosimilar structures and power law decays for the escape times statistic. As the rounding effects increase, stickiness and long lived trajectories are observed. Rounding edges of around 0.01% → 1% from the whole billiard size generate the sticky motion.

In the present work, we quantify the fraction of trajectories that reach a specific region of the phase space when we vary a control parameter using two symplectic maps: one non-twist and another one twist. The two maps were studied with and without a robust torus. We compare the obtained patterns and we identify the effect of the robust torus on the dynamical transport. We show that the effect of meandering-like barriers loses importance in blocking the radial transport when the robust torus is present.

The stability of periodic orbits under translation and also dilatation (or contraction) of coordinates is analysed using a q-Jacobian. Specifically we analyse the stability of period-1 and period-2 orbits of the quadratic map and show that the dilatation changes the parameter values for which bifurcations take place. The dilatation (|q| > 1) or contraction (|q| < 1) can be used to stabilize (destabilize) the unstable (stable) periodic orbits. In addition, dilatation effects on the stability depend only on the period of the orbit, and not on the location of orbital points in phase-space. This is shown to be true for any period-n orbit of the quadratic map. As a possible practical application, we suggest that the dilatation (contraction) considered here could represent temperature variations in physical systems.

In this contribution the weak stability boundary algorithmic definition was numerically accomplished with the inclusion of lunar and earth collisional sets and a subclassification of the unstable set. Then, the associated sets to WSB definition were analyzed and characterized according to relevant dynamical properties in order to clarify their applicability in earth-moon transfer orbit design. The obtained stable, unstable, and collisional sets are defined as a function of the osculating ellipse eccentricity for prograde and retrogade initial conditions. The stable sets, candidates to ballistic capture transfers, are subclassified according to chosen specific criteria, namely, the Jacobi constant intervals defined by distinct classes of Hill regions, the location of the final state after a complete cycle with respect to the Hill sphere, the permanence in the lunar sphere of influence in a full cycle around the moon, and exit basins for retrograde evolution. By the first time, with this investigation, elucidative criteria based on three-body problem elements are employed to identify initial condition subsets with required properties to design ballistic capture transfers.

We investigate the propagation of kinks in inhomogeneous media. We show that the extended character of the kink, the internal mode instabilities and the phenomenon of disappearance of the translational mode can affect the kink motion in the presence of space-dependent external perturbations. We apply the results to the analysis of kink ratchets and the propagation of kinks driven by wave fields.

It is shown that the synchronization behavior of a system of chaotic maps subject to either an external forcing or a coupling function of their internal variables can be inferred from the behavior of a single element in the system, which can be seen as a single drive-response map. From the conditions for stable synchronization in this single driven-map model with minimal ingredients, we find minimal conditions for the emergence of complete and generalized chaos synchronization in both driven and autonomous associated systems. Our results show that the presence of a common drive or a coupling function for all times is not indispensable for reaching synchronization in a system of chaotic oscillators, nor is the simultaneous sharing of a field, either external or endogenous, by all the elements. In the case of an autonomous system, the coupling function does not need to depend on all the internal variables for achieving synchronization and its functional form is not crucial for generalized synchronization. What becomes essential for reaching synchronization in an extended system is the sharing of some minimal information by its elements, on the average, over long times, independently of the nature (external or internal) of its source.

The phenomenon of synchronization occurring in coupled chaotic maps on a directed random network is studied. The network is characterized by the average degree of its nodes and the fraction of directed links. It is found that the required coupling strength so that the chaotic synchronization emerges is smaller when the fraction of directed links is increased. In addition, the system undergoes a transition from an asynchronous phase to a synchronous one at some critical values of its parameters. The critical boundary separating the synchronous from the asynchronous regime is calculated on the parameter space of the system, given by the coupling strength and the fraction of directed links of the network. The phase transition between the two regimes is of second order for all values of the fraction of directed links, and the critical exponent depends of it.

We construct classical stochastic mass transport processes for stationary states which are chosen to factorize over pairs of sites of an undirected, connected, but otherwise arbitrary graph. For the special topology of a ring we derive static properties such as the critical point of the transition between the liquid and the condensed phase, the shape of the condensate and its scaling with the system size. It turns out that the shape is not universal, but determined by the interplay of local and ultralocal interactions. In two dimensions the effect of anisotropic interactions of hopping rates can be treated analytically, since the partition function allows a dimensional reduction to an effective one-dimensional zero-range process. Here we predict the onset, shape and scaling of the condensate on a square lattice. We indicate further extensions in the outlook.

In this work we study a growing network model with chaotic dynamical units that evolves using a local adaptive rewiring algorithm. Using numerical simulations we show that the model allows for the emergence of hierarchical networks. First, we show that the networks that emerge with the algorithm present a wide degree distribution that can be fitted by a power law function, and thus are scale-free networks. Using the LaNet-vi visualization tool we present a graphical representation that reveals a central core formed only by hubs, and also show the presence of a preferential attachment mechanism. In order to present a quantitative analysis of the hierarchical structure we analyze the clustering coefficient. In particular, we show that as the network grows the clustering becomes independent of system size, and also presents a power law decay as a function of the degree. Finally, we compare our results with a similar version of the model that has continuous non-linear phase oscillators as dynamical units. The results show that local interactions play a fundamental role in the emergence of hierarchical networks.

Synchronous Digital Hierarchy (SDH) is the standard technology for information transmission in broadband optical networks. Unlike systems with unplanned growth, such as those of natural origin or the Internet network, the SDH systems are strictly planned as rings, meshes, stars or tree-branches structures designed to connect different equipments. In spite of that, we have found that the SDH network operated by Telefónica in Spain shares remarkable topological properties with other real complex networks as a product of its evolution since 1992. In fact, we have found power-law scaling in the degree distribution (N·P(k) = k^−γ) and small-world networks properties. The complexity found in SDH systems was reproduced by two models of complex networks, one of them considers real planning directives that take into account geographical and technological variables and the other one is based in the compatibility among SDH equipments.

One of the challenges in obtaining long lasting magnetic confinement of fusion plasmas in tokamaks is to control electrostatic turbulence near the vessel wall. A necessary step towards achieving this goal is to characterize the turbulence level and so as to quantify its effect on the transport of energy and particles of the plasma. In this paper we present experimental results on the characterization of electrostatic turbulence in Tokamak Chauffage Alfven Bresilien (TCABR), operating in the Institute of Physics of University of São Paulo, Brazil. In particular, we investigate the effect of certain magnetic field fluctuations, due to magnetohydrodynamical (MHD) instabilities activity, on the spectral properties of electrostatic turbulence at plasma edge. In some TCABR discharges we observe that this MHD activity may increase spontaneously, following changes in the edge safety factor, or after changes in the radial electric field achieved by electrode biasing. During the high MHD activity, the magnetic oscillations and the plasma edge electrostatic turbulence present several common linear spectral features with a noticeable dominant peak in the same frequency. In this article, dynamical analyses were applied to find other alterations on turbulence characteristics due to the MHD activity and turbulence enhancement. A recurrence quantification analysis shows that the turbulence determinism radial profile is substantially changed, becoming more radially uniform, during the high MHD activity. Moreover, the bicoherence spectra of these two kinds of fluctuations are similar and present high bicoherence levels associated with the MHD frequency. In contrast with the bicoherence spectral changes, that are radially localized at the plasma edge, the turbulence recurrence is broadly altered at the plasma edge and the scrape-off layer.

The interaction between a turbulent fountain and its stratified environment was studied. A heavy fluid, cold water, was injected vertically upwards into a linearly stratified medium. The round heavy-fluid jet reaches a maximum height before it begins to fall due to the effect of gravity. Because of the effects of friction and mixing, the vertical momentum and density of the jet fluid decrease as it submerges to an intermediate height of zero buoyancy. At this point, the jet fluid spreads as a horizontal front, intruding into the stratified environment. The degree of fluctuation in the proximity of the injection point was studied under both unrestricted- and restricted-flow configurations at the injection, using two differently sized stainless-steel woven-wire screens at the injection port as flow-restricting means. Using visualization and velocimetry techniques, both maximum and spreading heights were found to decrease with increasing turbulence at the point of injection.

A model for propagation of arches on cubic lattices, to simulate the internal mobility of grains in a dense granular system under shear is proposed. In this model, the role of the arches in granular transportation presents a non-linear dependence on the local values of the stress components that can be modeled geometrically. In particular, we study a modified Couette flow and were able to reproduce qualitatively the experimental results found in the literature.

We have observed some features of the coexistence of foams and granular materials in Hele-Shaw cells. The most part of the liquid and granular material stays at the bottom of the cell, with only a small quantity of the mixture resting on the froth. The fractal dimensions of the final states of the foams are close to the values obtained from the Random Apollonian Packing model. The disperse structure of the granular material affects the probability distribution of number of sides of the foam bubbles. The nearest neighbor distances between the peaks of the sand piles at the bottom of the cell are close to a lognormal distribution.

The Blume-Capel model with infinite-range interactions presents analytical solutions in both canonical and microcanonical ensembles and therefore, its phase diagram is known in both ensembles. This model exhibits nonequivalent solutions and the microcanonical thermodynamical features present peculiar behaviors like nonconcave entropy, negative specific heat, and a jump in the thermodynamical temperature. Examples of nonequivalent ensembles are in general related to systems with long-range interactions that undergo canonical first-order phase transitions. Recently, the extended gaussian ensemble (EGE) solution was obtained for this model. The gaussian ensemble and its extended version can be considered as a regularization of the microcanonical ensemble. They are known to play the role of an interpolating ensemble between the microcanonical and the canonical ones. Here, we explicitly show how the microcanonical energy equilibrium states related to the metastable and unstable canonical solutions for the Blume-Capel model are recovered from EGE, which presents a concave "extended" entropy as a function of energy.

Bootstrap (or k-core) Percolation with k = 3 is studied numerically on two-dimensional lattices with long-range links whose lengths r_ij are distributed according to P(r) ~ r^−α. By varying the decay exponent α the topology of these networks can be made to range from two-dimensional short-range networks to ∞-dimensional random graphs. The 3-core transition is found to be of first-order character with a divergent correlation length for α < 2.75 and of second order for larger α. Whenever the transition is first-order an associated critical corona is found to exist. The correlation length exponent ν defined from the corona correlation length above the first order transition is estimated as v ≈ 1/2 for α = 0, and only shows a weak α dependence for α ≤ 2.50. The second-order transition at large α is found to be in the universality class of two-dimensional Percolation.

In this work we analyze the Hamiltonian Mean Field model, which consists of a set of N interacting rotators that move around an unitary circle. This model presents a very complex dynamical behavior and, in particular, the canonical predictions do not necessarily coincide with the temporal averages obtained along numerical simulations. Recently, a topological explanation for the existence of these anomalies has been introduced. In this work we further analyze this hypothesis by considering the role of the energy of the system in the dynamics of the model.

Quasi two dimensional systems with competing interactions usually display complex patterns in the relevant order parameter. In many cases these patterns are analog to liquid-crystal phases, showing smectic, nematic or hexatic order. We show that order parameters suitable for the characterization of these phases in systems with nearly isotropic competing interactions emerge naturally from an analysis of a Landau model. We describe with some detail the nematic case, which characterizes orientational order of striped domain walls. The Landau model presents an isotropic-nematic transition of the Kosterlitz-Thouless type. Although for the perfectly isotropic model long range nematic order is absent in infinite systems, we show that in real systems of finite size nematic order of domain walls can be observed.

Geometric properties of polymixtures after a sudden quench in temperature are studied through the q-states Potts model on a square lattice, and their evolution with Monte Carlo simulations with non-conserved order parameter. We analyse the distribution of hull enclosed areas for different initial conditions and compare with exact and numerical findings for the q = 2 (Ising) case.

The problem of Laplacian growth is considered within the Loewner-equation framework. A new method of deriving the Loewner equation for a large class of growth problems in the half-plane is presented. The method is based on the Schwarz-Christoffel transformation between the so-called 'mathematical planes' at two infinitesimally separated times. Our method not only reproduces the correct Loewner evolution for the case of slit-like fingers but also can be extended to treat more general growth problems. In particular, the Loewner equation for the case of a bubble growing into the half-plane is presented.

The nonlinear response and sliding friction behavior of a phase-field crystal model for driven adsorbed atomic layers is determined numerically. The model describes the layer as a continuous density field coupled to the pinning potential of the substrate and under an external driving force. Dynamical equations which take into account both thermal fluctuations and inertial effects are used for numerical simulations of commensurate and incommensurate layers. At low temperatures, the velocity response of an initially commensurate layer shows hysteresis with dynamical melting and freezing transitions at different critical forces. The main features of the sliding friction behavior are similar to the results obtained previously from molecular dynamics simulations of particle models. However, the dynamical transitions correspond to nucleations of stripes rather than closed domains.

We show that the two complementary parts of the dynamics associated to the Feigenbaum attractor, inside and towards the attractor, form together a q-deformed statistical-mechanical structure. A time-dependent partition function produced by summing distances between neighboring positions of the attractor leads to a q-entropy that measures the ratio of ensemble trajectories still away at a given time from the attractor (and the repellor). The values of the q-indexes are given by the attractor's universal constants, while the thermodynamic framework is closely related to that first developed for multifractals.

We analyze the stochastic resonance response in an extended system, considering different transport/coupling mechanisms: diffusion, KPZ, and also include the possibility of a non-local interaction. Our aim, since these mechanisms correspond to different forms of coupling of resonant units leading to an extended system, is to obtain information about the way to optimize the system's response to weak signals. To reach such a goal, we exploit the knowledge of the so called "non-equilibrium potential" for the above indicated situations.

The non-linear coupling between lateral diffusions on an interface and the geometry of this interface is studied numerically on a simple 1D model which presents the same kind of non-linearities as fully 2D ones. It is found that small-scale geometry fluctuations influence the diffusion at all scales, including scales much larger than those of the fluctuations. At large enough scales, the diffusion can be approximately described by introducing an effective or renormalized diffusion coefficient. Several theoretical estimates for this coefficient are introduced and compared to the value derived from numerical simulations.

In this short note we review a recently found formulation of two-dimensional causal quantum gravity defined through Causal Dynamical Triangulations and stochastic quantization. This procedure enables one to extract the nonperturbative quantum Hamiltonian of the random surface model including the sum over topologies. Interestingly, the generally fictitious stochastic time corresponds to proper time on the geometries.

The influence of nonlinear effects in stochastic equations of motion with both additive and multiplicative noises is studied. Non-Markovian stochastic dynamics are compared with their corresponding Markovian (local approximations). Non-Markovian effects are implemented through Ornstein-Uhlenbeck and exponential damped harmonic dissipative kernels.

We illustrate, with reference to the so-called Fano factor, that dramatic differences between the two most popular phase-space distributions (Wigner's and Husimi's) become evident at the level of the harmonic oscillator treatment.

We review and analyze the main features of fractal sets, as well as argue that the geometric property called fractality is not a well defined concept in the literature. As an example, we consider the mode locking phenomenon exhibited by the Sine Circle Map, concerning those sets in parameter space called Arnold Tongues and Devil's Staircase (more exactly, their complements). It is shown that well-known results for the fat fractal exponent of the ergodic region turn out to be valid only in a tiny region of parameter space where ρ ∊ [0,a], a → 0, being ρ the winding number. A careful geometric analysis shows that a misleading simplification has led to a loss of information that hindered relevant conclusions. We propose an alternative, broader and more rigorous approach in that it selects a generic interval from the whole domain which excludes the mentioned restricted region. Our results reveal that the measure shows a different dependence on tongue widths, so we argue that such a discrepancy is only possible if we assume that the set in question does not satisfy a scale-free property. Since there is no invariant exponent for the complementary set of Arnold Tongues (and consequently for that of Devil's Staircase), we conclude that the exponent found by Ecke et al.[4] does not characterize those sets as true fractals. We also consider those sets as an example that statistical criteria are insufficient to characterize any set as being a fractal; they should always follow an analysis of the topological process responsible for generating the set.

In this work we are interested in the concept of market efficiency and its relationship with the algorithmic complexity theory. We employ a methodology based on the Lempel-Ziv index to analyze the relative efficiency of high-frequency data coming from the Brazilian stock market.

Within the context of agent-based Monte-Carlo simulations, we study the well-known majority-vote model (MVM) with noise applied to tax evasion on simple square lattices, Voronoi-Delaunay random lattices, Barabasi-Albert networks, and Erdös-Rényi random graphs. In the order to analyse and to control the fluctuations for tax evasion in the economics model proposed by Zaklan, MVM is applied in the neighborhod of the noise critical q_c to evolve the Zaklan model. The Zaklan model had been studied recently using the equilibrium Ising model. Here we show that the Zaklan model is robust because this can be studied using equilibrium dynamics of Ising model also through the nonequilibrium MVM and on various topologies cited above giving the same behavior regardless of dynamic or topology used here.

Climate can be described by statistical analysis of mean values of atmospheric variables over a period. It is possible to detect correlations in climatological time series and to classify its behavior. In this work the Hurst exponent, which can characterize correlation and persistence in time series, is obtained by using the Detrended Fluctuation Analysis (DFA) method. Data series of temperature, precipitation, humidity, solar radiation, wind speed, maximum squall, atmospheric pressure and randomic series are studied. Furthermore, the multifractality of such series is analized applying the Multifractal Detrended Fluctuation Analysis (MF-DFA) method. The results indicate presence of correlation (persistent character) in all climatological series and multifractality as well. A larger set of data, and longer, could provide better results indicating the universality of the exponents.

Ecosystems are complex systems which can respond to gradual changes of their conditions by a sudden shift to a contrasting regime or alternative stable state (ASS). Predicting such critical points before they are reached is extremely difficult and providing early warnings is fundamental to design management protocols for ecosystems. Here we study different spatial versions of popular ecological models which are known to exhibit ASS. The spatial heterogeneity is introduced by a local parameter varying from cell to cell in a regular lattice. Transport of biomass among cells occurs by simple diffusion. We investigate whether different quantities from statistical mechanics -like the variance, the two-point correlation function and the patchiness-may serve as early warnings of catastrophic phase transitions between the ASS. In particular, we find that the patch-size distribution follows a power law when the system is close to the catastrophic transition. We also provide links between spatial and temporal indicators and analyze how the interplay between diffusion and spatial heterogeneity may affect the earliness of each of the observables. Finally, we comment on similarities and differences between these catastrophic shifts and paradigmatic thermodynamic phase transitions like the liquid-vapor change of state for a fluid like water.

Epilepsy is commonly associated with synchronous activity in the form of spikes and also in developed seizures. Desynchronised activity seems to play an important role also in the seizure process, favouring the initiation of seizures. The aim of the present work is to explore synchronization activity in the inner areas in the temporal lobe of epileptic patients by a novel approach. Two temporal lobe epilepsy (TLE) patients' records have been analyzed through a cluster analysis. Electrical activity in the inner part of the temporal has been recorded by using Foramen Ovale Electrodes (FOE), a semi-invasive technique frequently used in drug resistant epileptic patients. Instead of tracking synchronized activity, we give here special attention to desynchronized activity, mainly those areas which are not included in synchronization clusters. Our results show that electrical activity in the epileptic side behaves in a less cohesive fashion than the contra-lateral side. There exists a clear tendency in the epileptic side to be organized as isolated clusters of electrical activity as compared with the contra-lateral side, which is organized in the form of large clusters of synchronous activity. In particular, we shall give special attention to the cluster desynchronization during the seizures. As we shall show, our results can help in understand several characteristics of the seizures dynamics.

In this work it is analyzed a one-dimensional lattice which is composed by mass-spring systems with one additional Rosen-Morse potential on site. This kind of lattice is used to study thermodynamic properties of DNA, especially its thermal denaturation. On the context of this work, the Rosen-Morse potential simulates hydrogen bonds between double strands of the molecule. From the graphic of the average stretching of base pairs versus temperature it is possible to observe the thermal denaturation of the system. This result shows that it is possible to obtain phase transition with an asymmetric potential without an infinite barrier.

The displacement of particles or probes in the cell cytoplasm as a function of time is characterized by different anomalous diffusion regimes. The transport of large cargoes, such as organelles, vesicles or large proteins, involves the action of ATP-consuming molecular motors. We investigate the motion of pigment organelles driven by myosin-V motors in Xenopus laevis melanocytes using a high spatio-temporal resolution tracking technique. By analyzing the turning angles (ϕ) of the obtained 2D trajectories as a function of the time lag, we determine the critical time of the transition between anticorrelated and directed motion as the time when the turning angles begin to concentrate around ϕ = 0. We relate this transition with the crossover from subdiffusive to superdiffusive behavior observed in a previous work [5]. We also assayed the properties of the trajectories in cells with inhibited myosin activity, and we can compare the results in the presence and absence of active motors.

This work studies through the Floquet theory the stability of breathers generated by the anti-continuous limit. We used the Peyrard-Bishop model for DNA and two kinds of nonlinear potential: the Morse potential and a potential with a hump. The comparison of their stability was done in function of the coupling parameter. We also investigate the dynamic behaviour of the system in stable and unstable regions. Qualitatively, the dynamic of mobile breathers resembles DNA.

We consider a two-dimensional CA model with three possible states for the system individual cells, 0 and ±. As for the dynamical rules, only ± can exert pressure to change the cells actual states. In this way, the 0 state is neutral and in some sense competitively weaker than the other two states. We further assume an inner property, the inertia, which is an intrinsic resistance to changes in the system. We evolve an ensemble of initial configurations for the CA until reaching steady states. By calculating averages over some relevant quantities for the final stationary configurations, we discuss how certain features of the problem, namely, initial states population and degree of aggregation as well as the values of inertia, can determine the different characteristics of the spatio-temporal pattern created by the CA evolution. We finally discuss how our findings may be relevant in the understanding of structures formation due to species competition in biology, specially in the transition regions between different biomes, the so called ecotones.