Since the pioneering work by Vicsek and his collaborators on the motion of self-propelled particles, most of the subsequent studies have focused on the onset of ordered states through a phase transition driven by particle density and noise. Usually, the particles in these systems are placed within periodic boundary conditions and interact via short-range velocity alignment forces. However, when the periodic boundaries are eliminated, letting the particles move in open space, the system is not able to organize into a coherently moving group since even small amounts of noise cause the flock to break apart. While the phase transition has been thoroughly studied, the conditions to keep the flock cohesive in open space are still poorly understood. I this talk I will present an extension of the Vicsek model of collective motion by introducing long-range alignment interactions between the particles. The results show that just a small number of these interactions is enough for the system to build up long lasting ordered states of collective motion in open space and in the presence of noise. This finding was verified for other models in addition to the Vicsek one, suggesting its generality and revealing the importance that long-range interactions can have for the cohesion of the flock.
The spatial distribution of living organisms in heterogeneous environments is a central issue in the dynamics of biological populations. In particular, it is relevant to know how fragmented structures arise and, mainly, if in the long term the population will survive or become extinct. We address these problems for single species populations. The nonlinear Fisher-KPP equation provides a fundamental mathematical description of the spatial distribution at the mesoscopic level, governed by elementary processes (growth, competition for limited resources and random dispersion), and can be generalized in several realistic directions by including for instance: density-dependencies in growth and diffusion rates, selective mobility, fluctuations, all under appropriate boundary conditions. We will discuss the role of these factors, on the survival of the population, as well as on pattern formation and shaping.
The mass media plays a fundamental role in the formation of public opinion, either by defining the topics of discussion or by making an emphasis on certain issues. Directly or indirectly, people get informed by consuming news from the media. Naturally, two questions appear: What are the dynamics of the agenda and how the people become interested in their different topics? These questions cannot be answered without proper quantitative measures of agenda dynamics and public attention. In this work we study the agenda of newspapers in comparison with public interests by performing topic detection over the news. We define Media Agenda as the distribution of topic’s coverage by the newspapers and Public Agenda as the distribution of public interest in the same topic space. We measure agenda diversity as a function of time using the Shannon entropy and differences between agendas using the Jensen–Shannon distance. We found that the Public Agenda is less diverse than the Media Agenda, especially when there is a very attractive topic and the audience naturally focuses only on this one. Using the same methodology we detect coverage bias in newspapers. Finally, it was possible to identify a complex agenda-setting dynamics within a given topic where the least sold newspaper triggered a public debate via a positive feedback mechanism with social networks discussions which install the issue in the Media Agenda.
We consider the nonequilibrium statistical mechanics and thermodynamics of quantum systems. We start by considering open quantum systems evolving with a so-called boundary-driven Lindblad equation. We discuss in simple examples some of its properties and also consider them in the context of quantum thermodynamics [1]. We obtain a thermodynamically consistent description of boundary-driven Lindblad models from the repeated interaction framework for the system and bath dynamics [2]. We discuss some interesting applications of these models [3] and explore differences and similarities of the thermodynamics and statistical mechanics for quantum systems in the repeated interaction framework and under collisional models [4].
References:
[1] F. Barra, Sci. Rep. 5, 14873 (2015).
[2] F. Barra and C. Lledó, Phys. Rev. E 96, 052114 (2017).
[3] F. Barra, Phys. Rev. Lett. 122, 210601, (2019).
[4] j. Ehrich, M. Esposito, F. Barra and J.M.R. Parrondo, Physica A https://doi.org/10.1016/j.physa.2019.122108 (In press, 2019).
We present a method for obtaining the complete resonant structure in spiral galaxy discs, showing that it is well represented by a concentric set of annular density waves, with weak couplings between pairs of them. The results allow us to make predictions about the structure of the bars and the spiral arms, as well as showing how the bars have been progressively braked by the dark matter halos of the galaxies. Comparing our predictions with measurements in over 150 galaxies we show how the non-linear theory of swing amplification for the density waves can be satisfactorily approximated by a linear approximation.
We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite-time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semiordered (or semichaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase space associated to them. Applying our methodology to a chain of coupled standard maps we obtain (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; and (iii) the dependence of the Lyapunov exponents with the coupling strength. Intermittent stickiness synchronization is also discussed in this context.
Physiological time series are typically non-stationary, demanding different approaches to be analyzed in terms of standard methods. Besides environmental conditions, non-stationarities can arise from multiple regulatory mechanisms, for example blood flow and respiration, operating concomitantly and varying over time, with each subsystem presenting its own time scale. Several diseases and conditions, such as myocardial infarction, diabetic neuropathy, and myocardial dysfunction, are related to the reduction in heart rate variability.
We present a non-parametric method which allows to identify changes in the physiological signal and split it into stationary patches, providing local quantities such as mean and variance of the signal in each stationary patch, as well as its duration. We explain the details of the segmentation method and how the outcomes of the segmentation can point to the complexity reduction in heart beating, due to pathological conditions and aging, and we discuss also the detection of sleep apnea through cardiovascular data. The outcomes of segmentation give us access to time characteristics of the signal that were no longer available, making possible a different approach to quantify non-stationarities in physiological time series.
In this work, we present the generation of multiscroll chaotic attractors via heteroclinic orbits based on piecewise linear systems. The collective dynamics of a pair or more coupled systems with different number of scrolls is studied. There are different configuration to get a coupled system via unidirectional coupling or bidirectional coupling. In this work, the master-slave configuration is mainly studied. We investigate the synchrony behavior of different connected networks. Itinerary synchronization is defined in terms of the symbolic dynamics arising by assigning different numbers to the regions where the scrolls are generated. A weaker variant of this notion, \(\epsilon\)-itinerary synchronization is also explored and numerically investigated. Itinerary synchronization is used to detect synchrony behavior when the coupled system presents generalized multistability.
Soaring birds often rely on ascending air currents as they search for prey or migrate across large distances. The landscape of convective currents is rugged and rapidly changing. How soaring birds find and navigate thermals within this complex landscape is unknown. Reinforcement learning provides an appropriate framework to identify an effective navigational strategy as a sequence of decisions taken in response to environmental cues. I will discuss how to use it to train gliders to autonomously navigate atmospheric thermals, in silico and in the field.
Matter under different equilibrium conditions exhibits different aggregation states. Exotic states of matter, such as Bose-Einstein condensates, superfluidity, and superconductivity exist in extreme conditions and low dimensions. These states are of topological nature. In this talk, we show topological states of matter in a liquid crystal cell with temporal modulation injection of energy. A transition from no vortex state to a state in which vortices persist is observed. Based on an amplitude equation, this phenomenon is analyzed. Experimental observation show quite good agreement with experimental observation.
The Fokker–Planck equation is a fundamental partial differential equation in statistical mechanics useful to describe the time evolution of the probability density function in the phase space of a system of particles under the influence of interactions. The equation can be generalized to other observables as well. Now, we discuss a nonlinear Fokker–Planck equation that has been recently developed to apply to problems with nonstandard behaviors. The relevance of this description is the existence of a family of power-law solutions. This type of development has proven to be a valuable tool for the study of diverse complex system in physics, biology and other fields. In this communication, we apply to non-linear diffusion, nonlinear optics and systems with long range interactions.
Fluctuations in the one-port scattering (S) and normalized impedance (z) matrices in polygonal and chaotic time-reversal invariant microwave billiards are experimentally investigated, in several levels of coupling and absorption, at room temperature and at 77 K. The observed distributions of reflection coefficient, phase of the scattering matrix, normalized resistance and reactance exhibit no fingerprint of a given geometry. At low frequencies, the results are consistent with earlier theoretical models by López, Mello and Seligman (S matrix) and by Zheng, Antonsen and Ott (z matrix), who independently predicted that the scattering fluctuations might be the same for the Wigner and Poisson level spacing distributions in the lossless resonator. The uniqueness of the observed scattering statistics at higher absorption levels is discussed with respect to inherent limitations posed by the experimental technique.
Right triangle billiards are well know to be quasi-integrable systems since their configuration space, depending on the acute angles, correspond to genus greater than 1 torus. In the quantum case, an ingenious way to construct such kind of system is by assuming point particles of different masses interacting with each other through elastic collisions and being confined in a 1D box. Notoriously, we cannot solve the Bethe ansatz in this case (of distinct masses particles), what is usually attributed to diffraction effects. However, is there another way to explain this lack of integrability through a semiclassical analysis, based on the system space topology? We show that it is by analyzing a full family of right triangle billiards and the conditions for which the Bethe ansatz fails to solve the problems in terms of classical structures of the dynamics.
Many disciplines are concerned with the links between different levels of observation. This is especially the case for the biology of social or gregarious arthropods (e.g. ants, bees, cockroaches, kissing bugs, woodlice, social fireflies,…) where the relation between the behaviour of the units and the dynamics of the society/group is a key question. Most social arthropods can perform collective decisions when choosing feeding or resting places,…. These decisions can be modulated by environmental factors and by social interactions.
Very often however, the idea that collective behaviors arise from a limited number of simple rules is predominant and that the complexity of individual units is neglected or underestimated. It seems natural therefore to put back the individual complexity into the dynamics of the society.
Based on theoretical and experimental approaches, we will discuss how: (1) the interplay between environmental conditions and the modulation of social interactions between individuals can lead to unexpected outcomes; (2) the individual capabilities and idiosyncrasy affect the global dynamics using cases from the social behaviour of insects.
Quantum chaos and quantum measurement have one constitutive feature in common: They capture information at the smallest scales and amplify it to macroscopic observability. Fundamental bounds on the information content of closed quantum system with finite-dimensional Hilbert space restrict their entropy production to a finite initial time span. Only in open systems where fresh entropy infiltrates from the environment, quantum dynamics (partially) recovers chaotic entropy production.
In quantum measurements, a macroscopic apparatus observes a small quantum system. Typically, notably in spin measurement, their results involve a component of randomness. The analogy with quantum chaos suggests that random outcomes of quantum measurements could, in a similar manner, reveal the entropy generated through the coupling to a macroscopic environment. It is required anyway to explain a crucial feature of quantum measurement, the decoherence that becomes manifest in the collapse of the wavepacket. However, the subsequent step from a set of probabilities to specific individual measurement outcomes (the “second collapse”) still evades a proper understanding in microscopic terms and remains shrouded in concepts such as “quantum randomness”. Could this process be explained by the back action of the macroscopic apparatus on the measured system? While obviously, information on the measured system must reach the apparatus, this would mean that also conversely, information originating in the apparatus would be shared with the object.
To explore this hypothesis in the case of spin measurements, we adopt the microscopic model of the measurement process proposed by Zurek and others and combine it with a unitary approach to decoherence, used in quantum chemistry and quantum optics, with heat baths comprising only a finite number N of modes. We expect the dynamics of the measured spin for growing N to exhibit a transition to a scenario of increasingly abrupt collapses and revivals: episodes of significant spin polarization of increasing length alternating with spin flips, determined by the initial condition of the apparatus. Preliminary analytical and numerical results confirm our expectation. Complementing the quantum model, we study an analogous classical system: A particle, launched from the top of the barrier of a symmetric double-well potential, will fall into either well, depending on random impacts by ambient degrees of freedom to which it couples.
We study a one-dimensional system of off-lattice Brownian particles that interact among themselves through a local velocity-alignment force that does not affect their speed. These conditions restrict the implementation of the aligning forces to a time-based scheme that allows for two different cases to be analyzed: synchronous and asynchronous updates. In the first, velocity-alignment is implemented periodically throughout the whole system while, in the second, probabilistically at every time-step in a Monte Carlo fashion. As the frequency of alignment increases in the synchronous case, or the probability of alignment in the asynchronous one, the system is driven from stationary states close to thermal equilibrium to far-from-equilibrium ones, where the system exhibits spontaneous symmetry breaking and self-organization characterized by long-range order and giant number fluctuations, features typically observed in ordered states of interacting self-propelled particles. Our results show that self-propulsion is not necessary to induce the flocking transition even in one-dimensional systems. Moreover, in the synchronous version of our model, the order parameter shows a regular spiking and resetting activity in the ordered phase, with fluctuations fading with the density, nonetheless, the system is still susceptible to turbulence and transient global disorder typical of these kind of out-of-equilibrium phases. On the other hand, some anomalous statistics for the higher moments of the order parameter become apparent in the disordered phase. Finally, we also analyze the case where self-propulsion is gradually included in the dynamics. In this case, the critical point decreases as self-propulsion becomes stronger in comparison with the cases without it.
Rulkov's discrete-time neuron model is known to display stability phases characterized by trains of pulsing and bursting signals having a multitude of spikes scattered in the control parameter space. By performing a systematic classification of such complex oscillations, we discovered nested sequences of arithmetic progressions among adjacent phases of pulsing and bursting. Such nested progressions can be expressed in terms of simple linear combinations of two "basic" periods. Nestings are robust and can be observed abundantly in distinct control parameter planes that are described in detail.
In this work, we show recent results on pattern formation in a parametric system subject to a heterogeneous forcing. We explore the presence of Rabi-like oscillations in a dissipative (out-of-equilibrium) system. We also present some experimental realizations conducted in a shallow water trough subjected to a parametric force with a tunable localization. New perspectives and future works will be discussed.
It is commonplace that human musical performances differ in various aspects from the corresponding musical scores or their transcription into MIDI-sequences. An important such aspect are so-called microtiming deviations, slight temporal deviations from the exact rhythm, that were claimed e.g. to play a key role for the swing feeling in jazz music. This claim, however, is so far discussed controversially in the musicological literature. In previous work using time series analysis we were able to identify differences between microtiming deviations of rock and jazz music. Here we report the results of an online survey among groups of musicians to whom we presented different versions of jazz music with original and manipulated microtiming deviations in an attempt to better characterize the phenomenon of swing.
*Work in collaboration with G. Datseris, A. Ziereis, T. Albrecht, Y. Hagmayer, C. Nelias, and V. Priesemann
Already in 1926, Schroedinger had pointed out the importance of Gaussian wavepackts in the transition from micro- to macro-mechanics [1]. We discuss the connections of different approximate and numerically exact ways, based on Gaussian wavepackets, to solve the time-dependent Schroedinger equation for a many-particle system. Special emphasis will be laid upon trajectory-guided coherent states that are, e.g., used in semiclassical initial value representations of the propagator [2]. Furthermore, some recent progress in numerical implementations of the Davydov-Ansatz for spin-boson models will be presented [3].
The possibility to combine the recent progress in the two so far disconnected approaches will be elaborated on.
References:
[1] E. Schroedinger, Die Naturwissenschaften 14, 664 (1926)
[2] M. Herman and E. Kluk, Chem. Phys. 91, 27 (1984)
[3] R. Hartmann et al., J. Chem. Phys. 150, 234105 (2019)
Advances in mathematical epidemiology have led to a better understanding of the risks posed by epidemic spreading and strategies to contain disease spread. However, a challenge that has been overlooked is that, as a disease becomes more prevalent, it can limit the availability of the resources needed to effectively treat those who have fallen ill. Here a simple generalized Susceptible-Infected-Susceptible (SIS) model is used to gain insight into the dynamics of an epidemic when the recovery of sick individuals depends on the availability of healing resources that are generated by the healthy population. The epidemics can spiral out of control into explosive spread, if the cost of recovery is above a critical cost. This can occur even when the disease would die out without the resource constraint. The onset of explosive epidemics is very sudden, exhibiting a discontinuous transition under very general assumptions. Analytical expressions can be given for the critical cost and the size of the explosive jump in infection levels in terms the parameters that characterize the spreading process. The model and its results apply beyond epidemics to contagion dynamics that self-induce constraints on recovery, thereby amplifying the spreading process. The spread of an infection is also investigated when a system component can recover only if it remains reachable from a functioning central unit. More precisely, infection spreads from infected to healthy nodes, with the constraint that infected nodes only recover, if they remain connected to a pre-defined central node, through a path that contains only healthy nodes. In this case, clusters of infected nodes will absorb their non-infected interior, because then no path exists between the central node and encapsulated nodes. This gives rise to the simultaneous infection of multiple nodes. Interestingly, the system converges to only one of two stationary states: either the whole population is healthy or it becomes completely infected. This simultaneous cluster infection can give rise to discontinuous jumps of different sizes in the number of failed nodes. Larger jumps emerge at lower infection rates. The network topology has an important effect on the nature of the transition: hysteresis appears for networks with dominating local interactions. The model shows how local spread can abruptly turn uncontrollable, when it disrupts connectivity at a larger spatial scale. We introduce a general mathematical framework to describe and classify a variety of spreading dynamics. Interestingly, some scenarios turn out to exhibit spontaneous, unpredictable breakdown and recovery cascades. To foster the recovery of damaged or infected systems, we also propose a targeted recovery protocol where least- damaged or infected regions recover first. This can lead to spatial confinement of the infection within a well- defined radius.
In order to identify the most relevant environmental parameters that regulate flowering time of bulbous perennial plants, first flowering dates of 329 taxa over 33 years are correlated with monthly and daily mean values of 16 environmental parameters spanning at least one year back from flowering. A machine learning algorithm is deployed to identify the best fitting parameters because the problem is strongly overdetermined for traditional methods. Surprisingly, the best proxy of flowering date fluctuations is the daily snow depth anomaly even for species flowering in October. Moreover, this proxy performs much better than mean soil temperature preceding the flowering, the best monthly explanatory parameter. Our findings support the existence of complicated temperature sensing mechanisms operating on different time scales, which is a prerequisite to precisely observe the length and severity of the winter season and translate e.g., “lack of snow” information to meaningful internal signals related to phenophases.
A secure communication scheme is described based on a chaotic multistable system. Both the emitter and the receiver of the communication system are conformed by two Rössller oscillators with nonlinear coupling. Varying the initial condition of one of the system variable results in different periodic or chaotic regimes. Synchronization between emitter and receiver is reached by a private communication channel, while the information transmission is realized by a public channel adhering a message package in a staggered manner to coexisting chaotic states within the same time series. The changing sequence of the initial condition acts as a dynamical private secrete key, while the parameters of the Rössller oscillators are considered as a static public key. The high security of the proposed communication system is provided by a change in the system parameters faster than synchronization time, so that synchronization attacks are ineffective.
In the 1930s, Philippe Le Corbeiller proposed connecting the mathematical theory of Lyapunov stability with the thermodynamics of engines (understood as devices capable of generating and maintaining a cyclic motion at the expense of an external disequilibrium without any corresponding periodicity). Unfortunately, this had little impact in the scientific community and was not developed very far by Le Corbeiller himself. I will argue that Le Corbeiller's program is still a promising way forward in the theory of non-equilibrium thermodynamics, which until now has, for the most part, failed to capture the detailed dynamics of work extraction by engines. In classical physics, such work extraction requires an active, non-conservative force, something that has been studied almost exclusively in the context of mechanical instabilities. I will treat three separate problems: the generation of waves on the surface of the water by the action of the wind, the hunting oscillation of a train, and the tidal acceleration of the Moon. I will show how the dialogue between dynamical systems theory and thermodynamics simplifies the solutions to these problems while revealing surprising commonalities among them. Finally, I will also argue that a similar approach can throw light on the thermodynamics of non-conservative chaotic systems, including Chua's circuit and the Lorenzian waterwheel.
We investigate the phenomenon of amplitude death [in two scenarios—traveling (TAD) and stationary] in coupled pendula with escapement mechanisms. The possible dynamics of the network is examined in coupling parameters’ plane, and the corresponding examples of attractors are discussed. We analyze the properties of the observed patterns, studying the period of one full cycle of TAD under the influence of system’s parameters, as well as the mechanism of its existence. It is shown, using the energy balance method, that the strict energy transfer between the pendula determines the direction in which the amplitude death travels from one unit to another. The occurrence of TAD is investigated as a result of a simple perturbation procedure, which shows that the transient dynamics on the road from complete synchronization to amplitude death is not straightforward. The pendula behavior during the transient processes is studied, and the influence of parameters and perturbation magnitude on the possible network’s response is described. Finally, we analyze the energy transfer during the transient motion, indicating the potential triggers leading to the desired state. The obtained results suggest that the occurrence of traveling amplitude death is related to the chaotic dynamics and the phenomenon appears as a result of completely random process
The study of pedestrian dynamics has been receiving great attention during the last years. The interest on the subject traverses the several areas of knowledge and researchers from a broad spectra of disciplines are studying those phenomena from complementary points of view. Of particular interest is the dynamics of an evacuation process. During an evacuation, pedestrians need to abandoned an enclosing through one or more available exits, some times narrow enough to generate tragic events. The complex nature of this process allows to trace links with studies in granular matter and game theory among others. In this talk we will focus in the interplay of these two areas to formulate mathematical models and to study and understand experimental data. The novelty introduced in our approach is to consider behavioural aspects related to the decision taking of the pedestrians at the moment of approaching the exit and interacting with other pedestrians. We will show how three different methodologies that include the mentioned behavioural aspects in the description of an evacuation process lead to the same phenomenology, not fully explained by previous works.
The use of mathematical models in Ecology has grown significantly in the last decade. This is due in part to their predictive capacity, but also to their power to order and systematize assumptions and thus contribute to elucidate the behavior of complex biological systems. In fact, the interrelation of factors as diverse as climate, access to resources, predators and human activity, makes it necessary to develop mathematical models that allow predicting the effect of each of them on the species involved, showing possible scenarios of coexistence or extinction in spatially structured populations.
A large number of publications on topics such as predator-prey models, intra- and inter-specific competition, or habitat fragmentation can be found, but more research is still needed on how to integrate all these mechanisms together.
With the purpose of advancing towards the study of trophic web complexity in successive approximations, we started a few years ago the development of metapopulation models of generic predator-prey-competition systems coexisting in environments subjected to disturbances. The use of both, ordinary differential equations and stochastic simulations, allowed us to obtain the average behavior of the relevant variables but also to study the role of fluctuations and spatial correlations.
I will present some recent results obtained with more realistic versions of the models we initially explored. Besides the typical regimes of coexistence and extinction of species, persistent temporal and spatial oscillations appear in some regions of the parameter space. The phenomenon is not present for the more idealized models, suggesting that it can be the source of real ecosystems oscillations.
While topical applications increasingly rely on models of dynamical systems driven by noise, the corresponding theory for random dynamical systems still remains in its infancy. This talk surveys some recent insights into how random dynamical systems may exhibit bifurcations, i.e. qualitative changes in dynamical behaviour under the variation of parameters.
The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by
using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, \(I\), and angle, \(\theta\) and controlled by two control parameters: (i) \(\epsilon\), controlling the nonlinearity of the system, particularly a transition from integrable for \(\epsilon=0\) to non-integrable for \(\epsilon\ne 0\) and; (ii) \(\gamma\) denoting the power of the action in the equation defining the angle. For \(\epsilon\ne 0\) the phase space is mixed and chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures. For the chaotic dynamics far apart from the periodic islands, normal diffusion is observed. The scenario changes significantly when the dynamics passes near stability regions where anomalous diffusion dominates over the dynamics, stickiness is present and a temporary break of ergodicity is observed.
We argue that the applicability of percolation models can be extended from fundamental fluid dynamics to practical aerodynamics relevant for engineering problems and, thus, to a more generally valid concept. As a fundamental phenomenon of fluid mechanics, recent studies suggested laminarturbulent transition belonging to the universality class of directed percolation. Here, the onset of a laminar separation bubble on an airfoil is analyzed in terms of the directed percolation model using particle image velocimetry data. Our findings indicate a clear significance of percolation models in a general flow situation beyond fundamental ones. We show that our results are robust against fluctuations of the parameter, namely, the threshold of turbulence intensity, that maps velocimetry data into binary cells (turbulent or laminar). In particular, this percolation approach enables the precise determination of the transition point of the laminar separation bubble, an important problem in aerodynamics. Finally, we also put our findings into perspective, discussing possible applications of our framework to rotor blades of wind turbines and in computer fluid dynamics approaches to turbulence.
We conceive a new recurrence quantifier (S) for time series based on the concept of information entropy, in which the probabilities are associated with the presence of microstates defined on the recurrence matrix as small binary submatrices. The new methodology to compute the entropy of a time series has advantages compared to the traditional entropies defined in the literature, namely, a good correlation with the maximum Lyapunov exponent of the system and a weak dependence on the vicinity threshold parameter. Furthermore, the new method works adequately even for small segments of data, bringing consistent results for short and long time series. In a case where long time series are available, the new methodology can be employed to obtain high precision results since it does not demand large computational times related to the analysis of the entire time series or recurrence matrices, as is the case of other traditional entropy quantifiers. The method is applied to discrete and continuous systems. We also show that MAX( S) can be used as a parameter-free quantifier of short and long-term memories of stochastic signals. We conclude that the development of a new parameter-free quantifier of stochastic and chaotic time series can open new perspectives to stochastic data and deterministic time series analyses and may find applications in many areas of science.
In the last decade, the southeast region of Brazil has been suffering severe water shortages. Here, we propose to compute the expected drought period length to characterize the drought events in the region of São Paulo. We report the unique properties of the exceptional drought event during the austral summer 2014 by showing the differences and similarities to the very dry season in 2001 and the mild dry seasons in 2006 and 2015. Furthermore, we investigate the correlations of the abnormal precipitation deficit with the ocean and atmospheric patterns.
We will present a survey of results and methods on the mathematical theory of billiards. We will focus on ergodic and statistical properties of classical billiard systems in the plane.
The invariant measure for these systems was studied by Birkhoff in the twenties and the ergodic theory was developed by the soviet school in the sixties - seventies, specially in the seminal works of Jacob Sinai.
Relations with the ergodic hypothesis of Boltzmann and properties of the geodesic flow were well understood from the very beginning.
The survey will include references to my own work on ergodic and statistical properties, and recent results related with decay of correlations and Poisson processes using new methods by, among others, Lai-Sang Young, Carlangelo Liverani, Mark Demers, Dmitry Dolgopyat, Viviane Baladi, Pierre Collet, Hong-Kun Zhang, Ian Melbourne, Pierre Collet, Francoise Pène .
The inference of the underlying network that interconnects the units composing a complex system from observed data is of great interest nowadays. For example, network inference applied to brain fMRI scans or climate data allows for the reconstruction of a functional connectivity network, which has helped scientists to understand better the emerging behaviors in these systems. These networks are usually constructed by analyzing time-series observed at different points and establishing links between them depending on how similar the observations are. However, network inference of real-world systems is still not fully understood. Here, we discuss experimental observations with the goal to reveal the main topological properties that frustrate or facilitate inferring the underlying network from CC measurements. Specifically, we use pulse-coupled dynamical neurons connected as in the Caenorhabditis elegans neural networks as well as in networks with similar randomness and small-worldness. We analyse the effectiveness and robustness of the inference process under different observations and collective dynamics, contrasting the results obtained from using membrane potentials and inter-spike interval time-series. We find that overall, small-worldness favours network inference and degree heterogeneity hinders it. In particular, success rates in C. elegans networks – that combine small-world properties with degree heterogeneity – are closer to success rates in Erdös-Rényi network models rather than those in Watts-Strogatz network models. These results are relevant to understand better the relationship between topological properties and function in different neural networks.
References:
Rubido, N., Martí, A. C., Bianco-Martínez, E., Grebogi, C., Baptista, M. S., & Masoller, C., Exact detection of direct links in networks of interacting dynamical units, New Journal of Physics, 16(9), 093010 (2014).
Rodrigo Garcia, Arturo C. Marti, Nicolás Rubido, and Cecilia Cabeza, Small-worldness favours network inference (preprint, 2019).
In complex systems research, detecting correlations that capture genuine interactions, and anticipating dynamical transitions directly from the observed signals, are challenging tasks with applications across disciplines. In the first part of this talk I will consider networks of oscillators and I will discuss how the analysis of the mutual lags between pairs of oscillators can yield useful information for inferring the system connectivity, and also, for anticipating the transition to synchrony. Synthetic datasets from Kuramoto oscillators and empirical datasets from Rossler-like chaotic electronic circuits will be analyzed. In the second part of the talk I will consider a global empirical climatological dataset (surface air temperature) and I will discuss how the analysis of the instantaneous amplitudes and phases of the seasonal cycles in different geographical regions, computed by using the Hilbert transform, allows to disentangle climatic processes and to track atmospheric waves that propagate across the planet.
Consider a synchronized network where each unit presents only a periodic attractor with a chaotic transient. Depending on the instant that a perturbation is applied, we observe two possible network long-term states: (i) The network neutralizes the perturbation effects and returns to its synchronized configuration. (ii) The perturbation leads the network to an alternative desynchronized state. We show that this time-dependent vulnerability of synchronized state is due to the existence of a fractal set of initial conditions conducing the dynamic to a chaotic set in which trajectories persist for times indefinitely long. We argue that this phenomenon is general and illustrate with a complex network composed of electronic circuits.
Molecular dynamics (MD) studies can be critical to study specific protein-lipid interactions in a given system. Based on statistical thermodynamics, these simulations predict the trajectory of a system based on the forces that act on each of the components. The trajectory is obtained by solving Newton’s law of motion for every component in the system at a given time step. The simulation parameters are called a ‘force field’ and are determined based on experimental and quantum mechanics for bonded, non-bonded, and short/long rage electrostatic interactions.
Using MD, we are interested in understanding the effect of protein binding and aggregation on membrane dynamics in retroviral assembly. Specifically, our studies advance the understanding of key mechanisms in the viral assembly process of HIV-1 and offer potential new perspectives for antiretroviral treatments. Additionally, this work shows the importance of accurate membrane models to study protein dynamics through simulation, and provides insights into relevant interactions between lipidated peripheral proteins on the membrane surface. We built both symmetric and asymmetric membrane models and simulated a single as well as multiple protein units of the membrane targeting domain of Gag, a key protein for HIV assembly at the plasma membrane during. We explored the role of lipid co-localization to the protein binding site and its relationship to the insertion of the lipidated tail of this motif. These observations contribute to our understanding of molecular interactions that prepare a region in the plasma membrane for viral assembly and budding.
We show that polydisperse packings of frictional elastic disks under gravity self-organize, when vibrated vertically with moderate
intensities, onto a rotational state where the average angular velocity \(mrvel_i\) of each disk \(i\) is nonzero when measured over times much longer than the collision time. This previously unobserved phenomenon is studied here by means of experiment and numerical simulation, finding an excellent degree of phenomenological agreement between both methods.
Cellular automata are powerful tools to simulate car traffic. By dividing the streets into cells where individual cars move at discrete time steps, they can simulate streets, neighborhoods, massive transportation systems and even whole cities. They can investigate the effect of an individual behaviour on the system’s performance, and some of their models exhibit anisotropic phase transitions. In contrast, lattice Boltzmann are more adequate to model continuous systems with a set of conservation laws. A hypothetical fluid carries the information from cell to cell and collides by following the Boltzmann’s transport equation, a procedure that is fully parallel and can be easily implemented on graphic cards. Although the method has been successful to simulate fluids, acoustics, diffusion, electrodynamics and even general relativity and non-linear media, its use has been limited by the fact that most of them works on uniform Cartesian grids only. Hereby we show recent results on car traffic cellular automata and on the design of lattice Boltzmann models for generalised curvilinear coordinates that illustrate the power of such discrete models.
We present a study for a laser system consisting of two coupled laser oscillators, each of which shows mixed-mode oscillations and chaos when uncoupled. The type of coupling is via saturable absorbers, which is akin to inhibitory nonlinear coupling in neurons. We have carried out numerical bifurcation analysis and numerical simulations to show that for small enough coupling, well below the synchronization threshold, the onset of certain resonances in a symmetric configuration induce a type of rare events characterized by a very small amplitude. For an asymmetric configuration, we observe extreme rare events (rogue waves), which occur near an in-phase Hopf bifurcation. In both configurations, the rate of these rare events can be tuned by suitably changing physically relevant parameters. We observe similar rare events in other settings composed of these laser oscillators.
Unravelling the packing structure of dense assemblies of semiflexible rings is not only fundamental for the dynamical description of polymers rings, but also key to understand biopackaging, such as observed in circular DNA inside viruses or genome folding. Here we use X-ray tomography to study the geometrical and topological features of disordered packings of rubber bands in a cylindrical container. Assemblies of short bands are found to display a liquid-like disordered structure, with short-range orientational order and a minor influence of the container. On the contrary, as the bands become longer, confinement force folded configurations and the bands interpenetrate and entangle. The degree of entanglement is characterized through minimal surfaces and generalized Vorono\"{\i} diagrams, which allow the identification of bands threadings and near neighbors. Most of the systems are found to display a threading network which percolates the system. Interestingly, for long bands whose diameter doubles the diameter of the container, we found that all bands interpenetrate each other, in a complex fully-entangled structure.
A one dimensional probabilistic cellular automaton with two absorbing states is discussed. Given a quenched random field, the time evolution is deterministic and chaos can be defined. The time evolution depends on two continuous parameters and it makes sense to study phase transitions as a function of these parameters. Synchronization and control are also discussed.
Experiments in semiconductor laser with optical feedback give chaotic power output pulses known for a long time. We shall present how the statistics [1] of these chaotic pulses are affected by the feedback.
[1]. Phys. Rev. A 99, 053828 (2019)
We show the coexistence of coherent and incoherent states, chimera states, in a simple Duffing oscillators chain coupled to nearest neighbors. These intriguing states are observed in the bistability region between a uniform oscillation and a spatiotemporal chaotic state. To characterize the chimera states rigorously, we compute their Lyapunov spectra. Depending on initial conditions, a family of chimera states can appear and disappear, following a snaking-like bifurcation diagram.
General relativity -- itself a nonlinear field theory -- naturally leads to deterministic chaos. For example, the fate of a photon approaching a pair of black holes can be essentially indeterminate, even though it is governed by a deterministic set of equations. Here we explore the intricate structure of the shadow cast by the event horizons of a pair of black holes (BHs). An exciting era for gravitational astronomy is underway. In 2015, the first direct observation of gravitational waves (GWs), by the LIGO/Virgo collaboration, confirmed that binary black holes exist in Nature. The Event Horizon Telescope (EHT) has begun observing nearby galactic centres, and on April 10, 2019, the first picture of a BH shadow at the center of the M87 galaxy was shown. A BH shadow is associated with the set of all photons which, when traced backwards in time from the observer, asymptote towards the event horizon of the BH. In the language of nonlinear dynamics, a BH shadow is an exit basin in an open Hamiltonian dynamical system. Motivated by the GW detections from merging binary BHs, and the future prospects of the EHT, much work has focused on what the shadow of a pair of BHs would look like. The null geodesic equations, which describe the propagation of photons, are non-integrable, and chaotic scattering of photons emerges naturally. One of the hallmarks of chaos is the presence of fractal structures in phase space. In a binary BH system, a photon meets one of three possible fates: it falls into the first BH, the second BH, or it escapes to infinity. Thus, it is natural to define three exit basins. And across the phase space the shadow may exhibit both a regular and a fractal structure. Furthermore, in certain parts of the phase space, the three basins have the more restrictive property of Wada. For the binary BH system, this means that a photon which starts close to a Wada boundary in phase space is uncertain and could end up in one of three final states: the photon could fall into either of the black holes, or escape to spatial infinity. We apply a recently-developed numerical method, the merging method [1] to test for the Wada property in the fractal structures that arise in a binary BH model in general relativity. To our knowledge, this work [2] represents the first demonstration of the Wada property for a general-relativistic system. As well as demonstrating that tools from the field of chaos theory can be used to understand the rich dynamics of scattering processes in general relativity, this work highlights that there exist novel dynamical systems in gravitational physics which can be fruitfully explored by nonlinear dynamicists.
[1] Alvar Daza, Alexandre Wagemakers, Miguel A.F. Sanjuán. Ascertaining when a basin is Wada: the merging method. Scientific Reports 8, 9954 (2018)
[2] Alvar Daza, Jake O. Shipley, Sam R. Dolan and Miguel A. F. Sanjuan. Wada structures in a binary black hole system. Phys. Rev. D 98, 084050 (2018)
From intracellular protein trafficking to large scale motion of animal groups, the physical concepts driving the self-organization of living systems are still largely unraveled. Self-organization of active entities, leading to novel phases and emergent macroscopic properties, recently shed new lights on these complex dynamical processes. Here we show that, under the application of a constant magnetic field, motile magnetotactic bacteria confined in water-in-oil droplets self-assemble into a rotary motor exerting a torque on the external oil phase. A collective motion in the form of a large-scale vortex, reversable by inverting the field direction, builds-up in the droplet with a vorticity perpendicular to the magnetic field. We study this collective organization at different concentrations, magnetic fields and droplets radii and reveal the formation of two torque-generating areas close to the droplet interface. We characterize quantitatively the mechanical energy extractable from this new biological and self-assembled motor. In a second set of experiment, we confine non-magnetic bacteria (E. coli) inside water-in-oil droplets. Here, the bacterial suspension does not develop a global vortex but, rather, the collective dynamics takes the form of small short-lived vertices in a turbulent-like motion. This collective motion of the suspension is able to move the droplet, which performs a persistent random walk. The measured persistence time and diffusion coefficient are of the order of 0.3 s and 0.5 mu m^2/s, respectively, several orders of magnitude larger than for a passive droplet. PIV measurements of the velocity field inside the droplet show that the droplet moves antiparallel to the bacteria, consistent with a rolling and slipping motion. The two examples demonstrate that bacteria can be used to build motors made of motors, that is, microscopic organisms can transfer useful mechanical energy to their confining environment, opening the way to the assembly of mesoscopic motors.
Non-uniform spatial distribution of vegetations in semiarid and arid environments is a rule rather than the exception. These vegetation patterns can be spatially periodic such as gaps, bands often called tiger bush or patches.
In this communication, we focus our analysis on localized patches and gaps. When the level of aridity is increased, the uniform vegetation cover develops localized regions of lower biomass.
These gaps can be either self-organized in a periodic way, or randomly distributed or exhibiting a clustering phenomenon. The appearance of gaps embedded in a uniform vegetation cover behaviour constitutes a warning signal towards desertification. We investigate the combined influence of the facilitative and the competitive interactions between plants, and the role of crow/root allometry, on the formation of gap vegetation structures. We characterize first the formation of the periodic distribution of gaps by drawing their bifurcation diagram. More importantly, we characterize localized and aperiodic distributions of gapped structures in terms of their snaking bifurcation diagram.
Our ability to understand how vegetation manages to survive and propagate through arid and semiarid ecosystems may be useful in the development of future strategies to prevent desertification. We finally discuss a robust phenomena observed in semi-arid ecosystems, by which localized vegetation patches split in a process called self-replication. Localized patches of vegetation are visible in nature at various spatial scales. Even though they have been described in literature, their growth mechanisms remain largely unexplored. Here, we develop an innovative statistical analysis based on real field observations to show that patches may exhibit deformation and splitting. This growth mechanism is opposite to the desertification since it allows to repopulate territories devoid of vegetation. We investigate these aspects by characterizing quantitatively, with a simple mathematical model, a new class of instabilities that lead to the self-replication phenomenon observed.
The issues around the quantization of classically chaotic systems that, under the name of quantum chaos provided a deep insight into the nature of the quantum-classical transition, have now reached the stage of maturity and sophistication where they necessarily collide with some of the deepest problems in the whole Physics. Three of such open questions concern, i) the ultimate relation between the classical and quantum world and the role of nonlinear and dissipative dynamics, ii) the large amount of evidence pointing to chaos in the quantum mechanical description of black holes, and iii) the description of the classical limit of lattice field theories with finite dimensional local Hilbert spaces typical of quantum information theory. In this talk, I will present some of the efforts of the semiclassics community to contribute to this problems by means of extending the incredible success of quantum chaos in particle systems into the realm of quantum fields.
Reaction rates and the mechanism of most electrocatalytic reactions are known to critically depend on the structure of the electrode surface. Examples of structure sensitive electrocatalytic reactions are abundant and include the electro-oxidation of carbon monoxide, formic acid, methanol, etc., on platinum. Even more intricate is the effect of the interfacial structure on the oscillatory dynamics usually observed in those systems. This is somewhat expected because several adsorption and reaction steps are simultaneously active during self-organized potential or current oscillations. Herein we present results of the effect of surface structure on the oscillatory electro-oxidation of methanol and glucose on platinum. The oxidation of methanol was investigated in acidic media on Pt(111), Pt(110), and Pt(100), and stepped surfaces Pt(776), Pt(554), Pt(775), and Pt(332); for glucose, oscillations were studied in alkaline electrolyte on polycrystalline platinum (Ptpoly), Pt(111), Pt(110), and Pt(100). For methanol, very tiny differences in the amount of surface defects were identified by means of the oscillatory pattern. For the electro-oxidation of methanol on stepped surfaces, we observed specificities in the dynamics that were unambiguously assigned to the surface structure. The following features were found according to the specific surface used: period-adding sequences of mixed-mode oscillations; a new type of mixed-mode oscillation; and a particular separation between two types of sequential oscillations. The electro-oxidation of glucose in alkaline media was found to strongly depend on the surface structure, with dramatic differences in the reaction currents, shape of the cyclic voltammogram and also in the onset potential. The potential oscillations were astonishingly stable and of high amplitude on Ptpoly. Pt(111) seems to be the less susceptible to poisoning and no oscillations were found on it. On Pt(110), large amplitude oscillations similarly to that on Ptpoly prevail. The presence of (100) sites brings about a secondary instability and new oscillations emerge. Temporal patterns on Pt(100) set in through small amplitude oscillations which further develop into very rich mixed-mode ones. In this case, higher frequency cycles observed around intermediate potentials and very well defined period-adding sequences appear. Overall, the observed results also indicate that electrochemical oscillations are much more sensitive to the surface structure than conventional electrochemical signatures, c.f. voltammetry, and thus can be used to infer on the evolution of the catalyst as the reaction proceeds. To understand the relationship between the surface structure and the underlying dynamics of the surface chemistry during oscillations is a key challenge and results in this direction will be also discussed.
In this lecture I will examine nature of subtle phenomenon such grazing bifurcations occurring in non-smooth systems. I will start with linear oscillators undergoing impacts with secondary elastic supports, which have been studied experimentally and analytically for near-grazing conditions [1]. We discovered a narrow band of chaos close to the grazing condition and this phenomenon was observed experimentally for a range of system parameters. Through stability analysis, we argue that this abrupt onset to chaos is caused by a dangerous bifurcation in which two unstable period-3 orbits, created at "invisible" grazing collide [2].
The experimentally observed bifurcations are explained theoretically using mapping solutions between locally smooth subspaces. Smooth as well as non-smooth bifurcations are observed, and the resulting bifurcations are often as an interplay between them. In order to understand the observed bifurcation scenarios, a global analysis has been undertaken to investigate the influence of stable and unstable orbits which are born in distant bifurcations but become important at the near-grazing conditions [3]. A good degree of correspondence between the experiment and theory fully justifies the adopted modelling approach.
Similar phenomena were observed for a rotor system with bearing clearances, which was analysed numerically [4] and experimentally [5]. To gain further insight into the system dynamics we have used a path following method to unveil complex bifurcation structures often featuring dangerous co-existing attractors.
References
1. Ing, J., Pavlovskaia, E.E., Wiercigroch, M. and Banerjee, S. 2008 Philosophical Transactions of the Royal Society – Part A 366, 679-704. Experimental study of impact oscillator with one sided elastic constraint.
2. Banerjee, S., Ing, J., Pavlovskaia, E., Wiercigroch, M. and Reddy, R. 2009 Physical Review E 79, 037201. Invisible grazing and dangerous bifurcations in impacting systems.
3. Ing, J., Pavlovskaia, E., Wiercigroch, M. and Banerjee, S. 2010 International Journal of Bifurcation and Chaos 20(11), 3801-3817. Complex dynamics of bilinear oscillator close to grazing.
4. Páez Chávez, J. and Wiercigroch, M. 2013 Communications in Nonlinear Science and Numerical Simulation 18, 2571–2580. Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model.
5. Páez Chávez, J., Vaziri Hamaneh, V. and Wiercigroch, M. 2015 Journal of Sound and Vibration 334, 86-97. Modelling and experimental verification of an asymmetric Jeffcott rotor with radial clearance.