The exciting field of complexity, chaos, and non-linear science has experienced impressive growth in recent decades. Not only classic problems in statistical mechanics, fluids, pattern formation, nonlinear optics, chaos, and dynamical systems have received a lot of attention but also new interdisciplinary fields have emerged, including complex networks, social dynamics, theoretical neuroscience and brain dynamics, biomathematics, and data science, among others. This Focus Issue of Chaos, “Nonlinear dynamics of non-equilibrium complex systems,” features an interesting selection of papers by leading researchers working in the field of non-equilibrium statistical mechanics and nonlinear physics and covers a wide range of relevant topics in nonlinear science.

The research in non-equilibrium statistical mechanics and nonlinear physics is a scientific approach to the investigation of how relationships between parts give rise to the collective behaviors of a system and how the system interacts and forms relationships with its environment. The topics cover information theory, statistical mechanics, and nonlinear dynamics and represent organized but unpredictable behaviors of natural systems that are considered fundamentally complex. Such research is used to model processes in biology, economics, physics, chemistry, computational neuroscience, complex networks, and material science, among others. Thus, we hope that the present issue will be able to provide a scientific basis and guidelines for future developments in the area.

The contributions to this focus issue are a selection of most interesting works presented at XIX MEDYFINOL 2016 that took place in Valdivia, Chile, from 5 to 9 December 2016. MEDYFINOL is a traditional and regular meeting of the statistical and nonlinear physics community in Latin America. The purposes of the MEDYFINOL meetings are to keep updated the scientific community on the new developments and tendencies in the statistical mechanics and nonlinear dynamical fields, to incentive collaborative international science programs, and to identify and discuss the most relevant advances in the area.

The current issue gathers the contributions of leading scientists that delineate the state of the art in areas strongly influenced by non-equilibrium statistical mechanics and nonlinear dynamical systems. The authors have made special efforts in writing 20 high-level papers with an introduction general enough so as to reach a wide audience and introduce people outside their fields to their research.

In the first contribution, Zabaleta and Arizmendi (2018) propose a quantum minority game that exploits the properties of quantum superposition and entanglement at the same time. This is a toy model of users of a communication network competing for scarce spectrum resources. The level of expectation of the players is defined as a decision parameter. When all players have low expectation level, quantum improves and complexity could not be necessary because the theoretically highest Nash equilibrium cannot be obtained. On the other hand, the highest Nash equilibrium is reached when some players have high expectation, and all the players apply the quantum strategy Q. Finally, classic equilibrium, which implies low expectations, is reached in fewer amounts of iterations. This is an important result because the number of iterations is related to the time needed to reach equilibrium and consequently to the energy consumption in spectrum allocation systems.

In the next contribution, the article by Peralta et al. (2018) considers a well-established model of social behavior based on a mechanism of imitation. The main idea is to understand under which rules a society, or a group of people, can reach consensus on a given topic, considering (i) the influence of the network of interactions amongst the people, (ii) the eventual desire to depart from the group main behavior, and (iii) the reluctance to change opinion despite the majority consensus. The effect of these three main ingredients, studied with the tools of statistical mechanics and critical phenomena, leads to a very rich phenomenology with different coexisting phases (corresponding to consensus and disagreement), tricritical points, catastrophes, etc.

Nowadays, numerous studies manifest that noise has rather counterintuitive and constructive effects on the behavior of a dynamical system. Wu et al. (2018) investigate here a quantitative bistable two-dimensional model of gene expression dynamics describing the competence development in the Bacillus subtilis under the influence of Lévy as well as Brownian motions. This article provides new insights into the prevention and therapy of complex disease system with a normal state, a pre-disease state, and a disease state. Further, it sheds light on quantifying random influences for complex systems from biology, to geophysics and to statistical physics.

The next article deals with birdsong production, which requires delicate and fast activation of several muscles in order to control the configuration of the syrinx (the avian vocal organ) and the respiratory system. In particular, the syringealis ventralis muscle has been reported to be involved in the control of the tension of the vibrating labia and thus to affect the frequency modulation of the generated sound. Recently, a biophysically inspired dynamical system has allowed researchers to carry out explicitly this translation of electrical activity into frequency modulation of birdsong. In the work by Doppler et al. (2018), it is shown that the information carried out by the syringealis ventralis muscle can also be used to determine whether the songbird is phonating at a given time. In awake animals, this work reports a quantification of the similarity between synthetic songs (generated with the song production model driven by syringealis ventralis activity) and the actual birdsong (produced as the muscle activity was being recorded). This is particularly interesting since it has been shown that male zebra finches (Taeniopygia guttata) exhibit spontaneous song-like activity in the syringealis ventralis muscle during the night. In this way, the results of this work allow one to transduce song like activity during sleep into synthetic sounds.

Electroencephalography (EEG) signals depict the electrical activity that takes place at the surface of the brain and provide an important tool for understanding a variety of cognitive processes. The EEG is the product of synchronized activity of the brain, and variations in EEG oscillation patterns reflect the underlying changes in neuronal synchrony. In a contribution, Baravalle et al. (2018) provide a quantification of the degree of complexity of different brain rhythmic oscillations of the EEG signals, by means of information theoretical approach. Their aim is to characterize the complexity of the EEG rhythmic oscillation bands when the subjects perform a visuomotor or imagined cognitive tasks (imagined movement), providing a causal mapping of the dynamical rhythmic activities of the brain as a measure of attentional investment. The obtained results enhance the functional role of gamma oscillations in the formation of neural representations of perception during visuomotor action and imagery activities.

The article by Legnani et al. (2018) analyzes the evolution of an ischaemic process using causal Information Theory descriptors. Ischaemic wall behavior under this condition was analyzed through wall thickness and ventricular pressure variations, acquired during an obstructive flow maneuver performed on left coronary arteries of surgically instrumented animals. Basically, the induction of ischaemia consisted on the temporary occlusion of left circumflex coronary artery (which supplies blood to the posterior left ventricular wall) for a few seconds. Normal perfusion of the wall was then reestablished while the anterior ventricular wall remained adequately perfused during the entire maneuver. The obtained results show that system dynamics could be effectively described by entropy-complexity loops, in both abnormally and well perfused walls. These results could contribute to make an objective indicator of the recovery heart tissues after an ischaemic process, in a way to quantify the restoration of myocardial behavior after the supply of oxygen to the ventricular wall was suppressed for a brief period.

Causality quantifies how past events in a variable influences the future events of another variable. It is central to understand and model natural and man-made systems. There are several approaches to quantify causality. The article by Bianco-Martinez and da Silva Baptista (2018) deals with the approaches that measure causality as a consequence of the direction of the flow of information between variables. So far, causality has been accessed by considering the probability of “temporal events” happening in the variables being observed. This work extends the notion of causality to the probability of “spatial events,” representing the state of the variables at a given time within a specific spatial resolution. Causality has thus space and time signatures: two temporally related events are also intrinsically related in space. The authors have shown that if a system X causes an effect in a system Y, then not only information flows from X to Y but also longer-time or higher-resolution observations in Y can be used to predict the past states of the system X. Moreover, we have proposed a quantity, the CaMI, that explores the space-time features of causality, and one which allows for a simple, experimental appealing and less computational demanding approach, but rigorous, quantification of causality.

Real-world systems in physics, chemistry, biology, economy, etc. are typically described by a large number of equations, involving many variables, and therefore, their dynamical evolution occurs in a high dimensional phase space. One of the most exciting discoveries in the field of dynamical systems in the last decades is that, in spite of their high dimensionality, these systems can be described by low-dimensional attractors, which can be reconstructed even if one can only observe one variable, during a finite time interval, with finite resolution and with large measurement noise. Examples of such high dimensional systems are one-dimensional spatially extended systems (1D SESs) and time delayed systems (TDSs). In a space-time representation, these systems show similar phenomena (e.g., wave propagation, pattern formation, defects and dislocations, turbulence, etc.). Quintero-Quiroz et al. (2018) study the state space reconstruction of bistable 1D SESs and TDSs from the time series of one scalar “observed” variable and show that, with appropriated definitions of the three pseudo-variables, the dynamics of these systems can be reconstructed in a three-dimensional pseudo space, where the evolution is governed by the same polynomial potential.

Bandt and Pompe introduced in 2002 [Phys. Rev. Lett. 88, 174102 (2002)] a successfully symbolic encoding scheme, based on the ordinal relation between the amplitude of neighboring values of a given data sequence, from which the permutation entropy can be evaluated. Equalities in the analyzed sequence, for example, repeated equal values, deserve special attention and treatment as was shown recently by Zunino and co-workers [Phys. Lett. A 381, 1883 (2017)]. A significant number of equal values can give rise to false conclusions regarding the underlying temporal structures in practical contexts. In the present contribution, Travesaro et al. (2018) propose computational affordable solutions for the problem of tied observations in time series. Such observations may lead to wrong conclusions when using Causal Information Theory descriptors. The authors compare several strategies and show the ones more apt to solve this problem.

In recent years, there has been a development of a new type of financial asset called cryptocurrency. This synthetic asset has an almost unique feature that it is traded 24/7 in unregulated trading platforms. The huge amount of money traded daily in such a market and the singularity of the asset nature deserves a deep study on prices dynamics and comparison among different cryptocurrencies. Bariviera et al.(2018) analyze 12 of the most important ones, using a powerful statistical tool based on Information Theory, namely, the complexity-entropy causality plane. The importance of this study lies in the extended coverage of the cryptoworld, accounting for more than 90% of the total daily turnover. It allows discriminating these cryptocurrencies into different dynamical regimes. Whereas most of the cryptocurrencies follow a similar pattern, there are two currencies (ETC and ETH) that exhibit a more persistent stochastic dynamics, and two other currencies (DASH and XEM) whose behavior is closer to a random walk. Consequently, similar financial assets, using blockchain technology, are differentiated by market participants.

The article by Criado et al. (2018) deals with the comparison between two approaches to the concept of personalized edge PageRank of a directed and weighted network, one of them obtained via the classical PageRank algorithm and the other one obtained through the line graph associated with this network. From the definition of the line graph of a directed and weighted network, the equivalence between both approaches is shown. This result makes it possible to calculate the edge's PageRank of a directed and weighted network from the PageRank of its nodes, with the consequent computational advantages. The usefulness of this result in the area of cybersecurity and intentional cyber-risk is related to the computation of edge accessibility, one of the three basic parameters underpinning intentional risk (along with anonymity and value). Also, by means of some simulations on a real subway network, how to use a personalization vector suitable for biasing the PageRank is shown with the aim of determining the segments of the subway network with the highest number of passengers depending on the time zone under consideration.

Lange et al. (2018) investigate the dynamics of water flow, given as time series of river runoff from long-term measurement stations (up to 85 years of data) in Brazil. The time series are analyzed using “Horizontal Visibility Graphs” (HVG). In this method, time series are represented as a network: each value of the time series is a node of the network, and two nodes are linked to each other if they can “see” at each other in the horizontal direction (no higher values are in between them), i.e., analogous to horizontal visibility in a landscape. Properties of the network provide insight into the temporal structure of the river runoff; in particular, it can be determined to which extent river runoff resembles certain types of random processes. The authors demonstrate that the analysis has to be carried out with great care in order to avoid misinterpretations and wrong conclusions. In particular, they show the consequences of the presence of identical values in the time series, of different versions of taking out the seasonal trend, and of the finite length of the series. For the latter, they use computer-generated data from random processes, where analytical results for infinite length are known. If thoroughly applied, the HVG are tools for the analysis of time series providing insights into the dynamics and a presentation of their behavior not easily obtained otherwise.

The next contribution by Descalzi and Brand (2018) deals with collisions of non-explosive dissipative solitons. The interactions of stable spatially localized solutions in dissipative driven systems have attracted a considerable amount of attention. Systems for which such interactions have been studied experimentally include binary fluid convection, catalytic oxidation of CO on surfaces, as well as the biologically inspired mutants of Dictyostelium discoideum. Here, they investigate the interaction of stationary and oscillatory dissipative solitons for two coupled cubic-quintic complex Ginzburg-Landau equations for counter-propagating waves. They analyze the case of a stabilizing as well as a destabilizing cubic cross-coupling between the counter-propagating dissipative solitons. The three types of interacting localized solutions investigated are stationary, oscillatory with one frequency, and oscillatory with two frequencies. They show that there are a large number of different outcomes as a result of these collisions including stationary as well as oscillatory bound states and compound states with one and two frequencies. The two most remarkable results are (a) the occurrence of bound states and compound states of exploding dissipative solitons as an outcome of the collisions of stationary and oscillatory pulses and (b) spatio-temporal disorder due to the creation, interaction, and annihilation of dissipative solitons for colliding oscillatory dissipative solitons as initial conditions.

The article by Cisternas et al. (2018) deals with the occurrence of normal and anomalous diffusion of dissipative solitons. Solitons are localized waves, which appear in several branches of science such as hydrodynamics, optics, and surface reactions. Dissipative solitons show a variety of interesting dynamical behaviors, for instance, explosions, which are transient asymmetric enlargements of the soliton, which lead to a spatial displacement of the center of mass of the soliton. A sequence of such explosions can cause—despite its deterministic nature—an apparently random motion of the soliton, which is reminiscent of a diffusion process. The latter can be normal or anomalous depending on the specific parameters of the model, in this case the cubic-quintic complex Ginzburg-Landau equation. This motion can be characterized by quantities such as the distribution of generalized diffusivities, which has already been used successfully for a deeper analysis of anomalous diffusion processes. With the help of this tool and other statistical quantities, it is found that the anomalous diffusion of solitons can be described by a subdiffusive continuous time random walk showing weak ergodicity breaking, which implies the non-equivalence of ensemble and time averages. These results are very important for experiments where explosions of solitons have already been observed.

The critical collapse, which is usually accompanied by instability, is a critical problem in the general nonlinear-wave theory, especially for optical wave's propagation in nonlinear optics and evolution of matter waves in atomic Bose-Einstein condensates. In particular, such collapse appears in self-focusing media with two-dimensional cubic nonlinearity and with one-dimensional (1D) quintic term. Linear lattices (spatially periodic linear potentials) and nonlinear lattices (spatially periodic modulations of the local nonlinearity strength), as well as their combination have been introduced against the critical collapse for the stabilization of solitons. The use of nonlinear lattices to stabilize 1D solitons is proved to be a challenging issue since the relevant stability area is in a very narrow parameter region. The article by Shi et al. (2018) deals with such an issue on the suppression of critical collapse for solitons by combining 1D quintic nonlinear lattices and saturation of the quintic nonlinearity. They demonstrate the existence of spatially symmetric fundamental solitons and excited states in the form of localized antisymmetric (dipole) modes. Stability areas for all the soliton species are identified by numerical and analytical methods.

The article by Escaff et al. (2018) deals with interacting self-propelled particles, which have the potential to exhibit a number of self-coordinated motions. Nature offers many examples surprising for their beauty, such as flocking birds or swarming fish. The keys to understanding the emergence of such collective behaviors are two: the motion of the self-propelled entities themselves and the interaction that leads to the coordination. In this work, the authors present a mathematical model for the sort of self-propelled particles that under appropriate conditions are capable of collective motions. This model deepens our understanding of the emergence of collective motion in terms of the theoretical framework provided by non-equilibrium statistical mechanics and nonlinear physics.

Cortes-Dominguez and Burguete (2018) present experiments where different nonlinear dynamics have been obtained on a conducting fluid. The destabilizing mechanism has its origin on a time-dependent magnetic field. This field produces a beating of the fluid generating different patterns. They compare the effect of various geometries (disc-shaped and toroid drops) and relate this problem with a recent description of adverse symptoms (vertigoes and dizziness) on MRI devices which could be generated by the interaction of the magnetic field with the electrolyte present in the inner ear.

The article by Mancini et al. (2018) deals with synchronization and control of complex dynamical systems. Mathematical models based in ODE equations with their parameters adjusted to fit different states in fluid dynamics are very useful tools to help in the design of these systems (i.e., number and location of sensors and actuators). In previous works, they presented complete synchronization obtained by coupling two equal hyperchaotic systems of Takens-Bogdanov equations (T-B), and their control by very small amplitude harmonic signals and tuning the frequency. In this work, they extend those results to other two different systems of hyperchaotic equations derived from the Lorenz 3D model. Complete synchronization, as in the T-B system, and the results of control with a small amplitude signal (less than 1%) by tuning their frequency are shown.

Alvarez-Martinez et al. (2018) study the probability distribution of symbolic strings of different lengths generated by the symbolic coding of unimodal maps taken from two families which include both the tent and logistic map as particular cases. When arranged in a decreasing order, these rank-ordered distributions follow a generalized discrete beta distribution which is determined by two exponents: exponent a, which is shown to be associated to long-range correlations, and exponent b, related to disordered behavior. By examining the variation of these exponents with respect to the parameters of the studied unimodal families, they establish a correspondence between the difference Δ = (ab) > 0 and the prevalence of intermittency over chaos. When Δ < 0 and chaos dominates, they show the existence of a first order phase transition in a thermodynamic model whose interaction potential is directly derived from the probability distribution of symbolic strings in the corresponding unimodal mapping. With these findings, some meaning is provided to the exponents, which contributes to the understanding of the ubiquity of the beta distribution found in natural phenomena, social behavior, and the arts.

Finally, the article by Rameshwar et al. (2018) deals with the thermal convection of binary liquids in a porous medium. Since this system exhibits stationary and oscillatory instabilities leading to transitions from supercritical to subcritical dynamics the authors have derived corresponding cubic-quintic Ginzburg-Landau equations. Some analytical solutions like fronts and localized structures are also examined for the stationary case.

We thank all the authors for their very interesting contributions, and we are grateful for the invaluable help of the referees who critically evaluated the papers that now form the present Focus Issue of Chaos. We also appreciate the professionalism of the Chaos editorial staff, especially Kristen Overstreet and Matthew Kershis, as well as editor-in-chief Jürgen Kurths, all of whom provided unfailing support through every step of the process.

1.
Alvarez-Martinez
,
R.
,
Cocho
,
G.
, and
Martinez-Mekler
,
G.
, “
Rank ordered beta distributions of nonlinear map symbolic dynamics families with a first-order transition between dynamical regimes
,”
Chaos
28
, 075515 (
2018
).
2.
Baravalle
,
R.
,
Rosso
,
O. A.
, and
Montani
,
F.
, “
Rhythmic activities of the brain: quantifying the high complexity of beta and gamma oscillations during visuomotor tasks
,”
Chaos
28
, 075513 (
2018
).
3.
Bariviera
,
A. F.
,
Zunino
,
L.
, and
Rosso
,
O. A.
, “
An analysis of high frequency cryptocurrencies prices dynamics using permutation-information-theory quantifiers
,”
Chaos
28
, 075511 (
2018
).
4.
Bianco-Martinez
,
E.
and
da Silva Baptista
,
M.
, “
Space-time nature of causality
,”
Chaos
28
, 075509 (
2018
).
5.
Cisternas
,
J. E.
,
Albers
,
T.
, and
Radons
,
G.
, “
Normal and anomalous random walks of 2-d solitons
,”
Chaos
28
, 075505 (
2018
).
6.
Cortes-Dominguez
,
I.
and
Burguete
,
J.
, “
Instabilities triggered in different conducting fluid geometries due to slowly time-dependent magnetic fields
,”
Chaos
28
, 075514 (
2018
).
7.
Criado
,
R.
,
Moral
,
S.
,
Perez
,
A.
, and
Romance
,
M.
, “
On the edges’ PageRank and line graphs
,”
Chaos
28
, 075503 (
2018
).
8.
Descalzi
,
O.
and
Brand
,
H.
, “
Collisions of non-explosive dissipative solitons can induce explosions
,”
Chaos
28
, 075508 (
2018
).
9.
Doppler
,
J. F.
,
Bush
,
A.
,
Amador
,
A.
,
Goller
,
F.
, and
Mindlin
,
G. B.
, “
Gating related activity in a syringeal muscle allows the reconstruction of zebra finches songs
,”
Chaos
28
, 075517 (
2018
).
10.
Escaff
,
D.
,
Toral
,
R.
,
Van den Broeck
,
C.
, and
Lindenberg
,
K.
, “
A continuous-time persistent random walk model for flocking
,”
Chaos
28
, 075507 (
2018
).
11.
Lange
,
H.
,
Sippel
,
S.
, and
Rosso
,
O. A.
, “
Nonlinear dynamics of streamflow from brazilian rivers elucidated by Horizontal Visibility Graphs
,”
Chaos
28
, 075520 (
2018
).
12.
Legnani
,
W. E.
,
Traversaro
,
F.
,
Redelico
,
F. O.
,
Cymberknop
,
L.
,
Armentano
,
R.
, and
Rosso
,
O. A.
, “
Analysis of ischaemic crisis using the informational causal entropy-complexity plane
,”
Chaos
28
, 075309 (
2018
).
13.
Mancini
,
H. L.
,
Becheikh
,
R.
, and
Vidal
,
G.
, “
Control and synchronization of hyperchaotic states in mathematical models of Bénard-Marangoni convective experiments
,”
Chaos
28
, 075519 (
2018
).
14.
Peralta
,
A. F.
,
Carro
,
A.
,
San Miguel
,
M.
, and
Toral
,
R.
, “
Analytical and numerical study of the non-linear noisy voter model on complex networks
,”
Chaos
28
, 075516 (
2018
).
15.
Quintero-Quiroz
,
C.
,
Torrent
,
M. C.
, and
Masoller
,
C.
, “
State space reconstruction of spatially extended systems and of time delayed systems from the time series of a scalar variable
,”
Chaos
28
, 075504 (
2018
).
16.
Rameshwar
,
Y.
,
Anuradha
,
V.
,
Srinivas
,
G.
,
Pérez
,
L.
,
Laroze
,
D.
, and
Pleiner
,
H.
, “
Nonlinear convection of binary liquids in a porous medium
,”
Chaos
28
, 075512 (
2018
).
17.
Shi
,
J.
,
Zeng
,
J.
, and
Malomed
,
B.
, “
Suppression of the critical collapse for one-dimensional solitons by saturable quintic nonlinear lattices
,”
Chaos
28
, 075501 (
2018
).
18.
Travesaro
,
F.
,
Redelico
,
F. O.
,
Risk
,
M. R.
,
Frery
,
A. C.
, and
Rosso
,
O. A.
, “
Bandt-Pompe symbolization dynamics for time series with tied values: a data-driven approach
,”
Chaos
28
, 075502 (
2018
).
19.
Wu
,
F.
,
Chen
,
X.
,
Zheng
,
Y.
,
Duan
,
J.
,
Kurths
,
J.
, and
Li
,
X.
, “
Levy noise induced transition and enhanced stability in a gene regulatory network
,”
Chaos
28
, 075510 (
2018
).
20.
Zabaleta
,
O. G.
and
Arizmendi
,
C. M.
, “
Evolutionary quantum minority game: a wireless network application
,”
Chaos
28
, 075506 (
2018
).