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This work shows that chaotic signals with different power spectrum are robust to linear superposition, meaning that the superposition preserves Ergodic quantities (Lyapunov exponents) and the information content of the source signals, even after being transmitted over non-ideal physical medium. This wonderful property that chaotic signals have allows me to propose a novel communication system based on chaos, where information composed from and to multiple users each operating with different base frequencies and that is carried by chaotic wavesignals can be fully preserved after transmission in the open air wireless physical medium, and it can be trivially decoded with low probability of errors. This work tackles with great detail how chaotic signals and their information content are affected when travelling through medium that presents the non-ideal properties of multipath propagation, noise and chaotic interference (linear superposition), and how this impacts on the proposed communication system. Physical media with other non-ideal properties (dispersion and interference with periodic signals) are also discussed.

In previous work, the $k$-logistic map [Machicao and Bruno, Chaos, vol. 27, 053116 (2017)] was introduced as a transformation operating in the $k$ less significant digits of the Logistic map. It exploited the map's pseudo-randomness character that is present in its less significant digits. In this work, we comprehensively analyze the dynamical and ergodic aspects of this transformation, show its applicability to generic chaotic maps or sets, and its potential impact on enabling the creation of a cryptosystem that is fast, light and reliable.

Information Theory concepts and methodologies conform the background of how communication systems are studied and understood. They are mainly focused on the source-channel-receiver problem and on the asymptotic limits of accuracy and communication rates, which are the classical problems studied by Shannon. However, the impact of Information Theory on networks (acting as the channel) is just starting. Here, we present an approach to understand how information flows in any connected complex network. Our approach is based on defining linear conservative flows that travel through the network from source to receiver. This framework allows us to have an analytical description of the problem and also linking the topological invariants of the network, such as the node degree, with the information flow. In particular, our approach is able to deal with information transmission in modular networks (networks containing community structures) or multiplex networks (networks with multiple layers), which are nowadays of paramount importance., 5 pages, 3 figures, proceedings paper

The electric power system is one of the cornerstones of modern society. One of its most serious malfunctions is the blackout, a catastrophic event that may disrupt a substantial portion of the system, playing havoc to human life and causing great economic losses. Thus, understanding the mechanisms leading to blackouts and creating a reliable and resilient power grid has been a major issue, attracting the attention of scientists, engineers and stakeholders. In this paper, we study the blackout problem in power grids by considering a practical phase-oscillator model. This model allows one to simultaneously consider different types of power sources (e.g., traditional AC power plants and renewable power sources connected by DC/AC inverters) and different types of loads (e.g., consumers connected to distribution networks and consumers directly connected to power plants). We propose two new control strategies based on our model, one for traditional power grids, and another one for smart grids. The control strategies show the efficient function of the fast-response energy storage systems in preventing and predicting blackouts in smart grids. This work provides innovative ideas which help us to build up a robuster and more economic smart power system.

Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the systems parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a ghost of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.

Abrupt changes of behaviour in complex networks can be triggered by a single node. This work describes the dynamical fundamentals of how the behaviour of one node affects the whole network formed by coupled phase-oscillators with heterogeneous coupling strengths. The synchronisation of phase-oscillators is independent of the distribution of the natural frequencies, weakly depends on the network size, but highly depends on only one key oscillator whose ratio between its natural frequency in a rotating frame and its coupling strength is maximum. This result is based on a novel method to calculate the critical coupling strength with which the phase-oscillators emerge into frequency synchronisation. In addition, we put forward an analytical method to approximately calculate the phase-angles for the synchronous oscillators.

Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behaviour, such as frequency synchronisation (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase-angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, without imposing any restriction on the distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behaviour of networks.

The brain of mammals are divided into different cortical areas that are anatomically connected forming larger networks which perform cognitive tasks. The cat cerebral cortex is composed of 65 areas organised into the visual, auditory, somatosensory-motor and frontolimbic cognitive regions. We have built a network of networks, in which networks are connected among themselves according to the connections observed in the cat cortical areas aiming to study how inputs drive the synchronous behaviour in this cat brain-like network. We show that without external perturbations it is possible to observe high level of bursting synchronisation between neurons within almost all areas, except for the auditory area. Bursting synchronisation appears between neurons in the auditory region when an external perturbation is applied in another cognitive area. This is a clear evidence that pattern formation and collective behaviour in the brain might be a process mediated by other brain areas under stimulation.

We have obtained a high resolution parameter space of an experimental Chua's circuit and shown that the topology of the chaotic and periodic regions present not only expected features previously observed from high resolution numerical simulations of idealised Chua's circuit, but also novel unexpected features. Unmatched feedback resistances cause the formation of at least two competing spirals with consequent disrupted or malformed shrimps. We have also confirmed experimentally that the period-adding bifurcation route is formed by periodic regions whose size decrease exponentially with their period, and consequently, periodic behaviour with higher period is unlikely to be observed. The higher-resolution span of parameters was possible by the use of a newly designed potentiometer that could be potentially used in other electronic equipments to reveal hidden behaviours. To have such resistances we developed in series arrays of resistors short-circuited by relays as discrete potentiometers with 1024 steps, and resolutions of 0.100 $\Omega$ for $r_L$ in series with the inductor, and 0.200 $\Omega$ for R connecting the two capacitors.

In this work, we present novel general analytical solutions for the currents that are developed in the edges of network-like circuits when some nodes of the network act as sources/sinks of DC or AC current. We assume that Ohm's law is valid at every edge and that charge at every node is conserved (with the exception of the source/sink nodes). The resistive, capacitive, and/or inductive properties of the lines in the circuit define a complex network structure with given impedances for each edge. Our solution for the currents at each edge is derived in terms of the eigenvalues and eigenvectors of the Laplacian matrix of the network defined from the impedances. This derivation also allows us to compute the equivalent impedance between any two nodes of the circuit and relate it to currents in a closed circuit which has a single voltage generator instead of many input/output source/sink nodes. Contrary to solving Kirchhoff's equations, our derivation allows to easily calculate the redistribution of currents that occurs when the location of sources and sinks changes within the network. Finally, we show that our solutions are identical to the ones found from Circuit Theory node analysis., 9 pages, 6 figures, 3 figure captions

The phenomenon of oral tolerance refers to a local and systemic state of tolerance, induced in the gut associated lymphoid tissues, after its exposure to innocuous antigens, such as food proteins. While recent findings shed light in the cellular and molecular basis of oral tolerance, the network of interactions between the components mediating oral tolerance has not been investigated yet. Our work brings a complex systems theory approach, aiming to identify the contribution of each element in an oral tolerance network. We also propose a model that allows dynamical plus topological quantifying which must encompass functional responses as the local host involved on the oral tolerance. To keep track of reality of our model, we test knockout (KO) of immunological components (i. e. silencing a vertex) and see how it diverges when the system is topologically health. The results from these simulated KO's are then compared to real molecular knock-outs. To infer from these processing we apply a new implementation of a random walk algorithm for directed graphs, which ultimately generate statistical quantities provided by the dynamical behaviour of the simulated KO's. It was observed that the a specifics KO caused the greatest impact on network standard flux. In a brief analysis, the results obtained correspond to biological data. Our model addresses both topological proprieties and dynamical relations. The construction of a qualitative dynamic model for oral tolerance could reflect empirical observations, through the standard flux results and relative error based on individual knockout., 21 pages, 2 figures, 2 tables

The intriguing nature of classical Homeric narratives has always fascinated the occidental culture contributing to philosophy, history, mythology and straight forwardly to literature. However what would be so intriguing about Homer's narratives' At a first gaze we shall recognize the very literal appeal and aesthetic pleasure presented on every page across Homer's chants in Odyssey and rhapsodies in Iliad. Secondly we may perceive a biased aspect of its stories contents, varying from real-historical to fictional-mythological. To encompass this glance, there are some new archeological finding that supports historicity of some events described within Iliad, and consequently to Odyssey. Considering these observations and using complex network theory concepts, we managed to built and analyze a social network gathered across the classical epic, Odyssey of Homer. Longing for further understanding, topological quantities were collected in order to classify its social network qualitatively into real or fictional. It turns out that most of the found properties belong to real social networks besides assortativity and giant component's size. In order to test the network's possibilities to be real, we removed some mythological members that could imprint a fictional aspect on the network. Carrying on this maneuver the modified social network resulted on assortative mixing and reduction of the giant component, as expected for real social networks. Overall we observe that Odyssey might be an amalgam of fictional elements plus real based human relations, which corroborates other author's findings for Iliad and archeological evidences.

We show that the probability distribution function that best fits the distribution of return times between two consecutive visits of a chaotic trajectory to finite size regions in phase space deviates from the exponential statistics by a small power-law term, a term that represents the deterministic manifestation of the dynamics, which can be easily experimentally detected and theoretically estimated. We also provide simpler and faster ways to calculate the positive Lyapunov exponents and the short-term correlation function by either realizing observations of higher probable returns or by calculating the eigenvalues of only one very especial unstable periodic orbit of low-period. Finally, we discuss how our approaches can be used to treat data coming from complex systems., subm. for publication. Accepted fpr publication in Chaos