Introduction to Focus Issue: Recent advances in modeling complex systems: Theory and applications

Complex systems are present in a variety of research areas corresponding to different disciplines within natural sciences, social sciences, engineering, as well as mathematical and physical sciences. They are often composed of large numbers of interconnected and interacting units, whose behavior is usually difficult to predict precisely due to particular relationships and interactions between their components. As a consequence, these systems possess some remarkable properties, such as nonlinearity, emergence, spontaneous order, adaptation, and their feedback interplay. There are many examples of applications of complex systems in fields such as transportation, communication and social networks, ecological and biochemical systems, computational, engineering (mechanical and electrical), and social and economic systems, besides many examples from basic sciences. Modeling of complex systems comprises a broad category of research problems including those derived from chaotic dynamics, stochastic phenomena, fractional dynamics, and complex networks methods. It constitutes a rich and fruitful field of research in physics, mathematics, biology, engineering, and social sciences. The present Issue provides a collection of papers focused on recent advances in the modeling of complex systems, with contributions from leading scientists working in nonlinear dynamics, chaotic dynamics, and complex systems including fractional dynamics and networks. The main goal is to provide an overview of cutting-edge research in different topics, including both fundamental and applied research in this flourishing area of science. The main purpose of this introduction to the Focus Issue is to review the contributions gathered within the issue. Needless to say, some contributions have a rather fundamental character while others give more attention to applications. In any case, four main groups of contributions can be distinguished: chaotic and fractional dynamics with applications; complex oscillations; complex traffic dynamics; and complex networks and applications. The first group comprises eight papers, the second and the third group consist of three and two papers, respectively, and the last group is represented by five papers. Hence, we provide an overview showing recent advances in this active and rich research field with emphasis put on novel ideas and inspiration for further original results in this active field of scientific endeavor.

Chaotic dynamics constitutes a very fruitful interdisciplinary field of both theoretical and applied research. Numerous methods of analysis are employed to observe new phenomena in complex systems.

In the paper Curry–Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems,¹ Mugnaine et al. study transitions from attractor on torus to chaotic motion in dissipative nontwist systems. This is in the context of dissipative systems obtained by adding a dissipation term in a given Hamiltonian system. The applications are manifold in various topics within nonlinear dynamical systems. Transitions to chaos in dissipative twist systems are well established; for instance, the Curry–Yorke route describes the transition from quasi-periodic attractor on torus to chaos, passing by a chaotic banded attractor. The authors consider the dissipative standard nontwist map, which is a nonconservative version of the standard nontwist map. They observe the same transition to chaos that happens in twist systems. It is known as a soft one, where the quasi-periodic attractor becomes wrinkled and then chaotic through the Curry–Yorke route. The authors study the nature of the orbits for different sets of parameters and observe that quasi-periodic, periodic, and chaotic behavior are possible in the system. They note that these behaviors can coexist in phase space, exhibiting multistability, and different coexistence scenarios have been analyzed by using the concepts of basin entropy and boundary basin entropy.

The evolution of the probability density of ensembles of iterates of the logistic map that advance toward and finally remain at attractors of representative dynamical regimes has been studied in the paper Logistic map trajectory distributions: Renormalization-group, entropy and criticality at the transition to chaos by Diaz-Ruelas et al.² The mirror families of superstable attractors along the period-doubling cascade and of chaotic-band attractors along the inverse band-splitting cascade are considered. Furthermore, the authors examine a common aperiodic accumulation point of the attractors. The iteration time progress of the densities of trajectories is determined via the action of the Frobenius–Perron (FP) operator. In contrast to the study of individual orbits, the analysis of ensembles of positions offers a viewpoint from which the nonlinear dynamical features of this iconic model can be better characterized in statistical–mechanical terms. The scaling of the densities along the considered families of attractors conforms to a renormalization-group (RG) structure, while their entropies are observed to attain extrema at the fixed points of the RG flows. Additionally, this entropy as a function of the map control parameter displays the characteristic features of an equation of state of a thermal system undergoing a second-order phase transition.

One paradigmatic system in nonlinear and chaotic dynamics is the Lorenz system. Much work has been done to characterize and utilize it. We include here the paper On infinite homoclinic orbits induced by unstable periodic orbits in the Lorenz system by Guo and Luo.³ The authors present an analytical study of the infinite homoclinic orbits induced by unstable periodic orbits on bifurcation trees through period-doubling cascades. Each unstable periodic orbit ends at its corresponding homoclinic orbit. Traditional computational methods cannot obtain homoclinic orbits from the corresponding unstable periodic orbits. This is because unstable periodic orbits in the Lorenz system cannot be achieved in numerical simulations. Herein, the stable and unstable periodic motions to chaos on the period-doubling cascaded bifurcation trees are determined through a discrete mapping method. The corresponding homoclinic orbits induced by the unstable periodic orbits are predicted analytically. The authors provide period-doubling bifurcation trees of different periodic orbits as well as the homoclinic orbits relative to the unstable periodic orbits, illustrating the results. A relevant result of this study is how to determine infinite homoclinic orbits through unstable periodic orbits in three-dimensional or higher-dimensional nonlinear systems.

Routes to chaos constitute a relevant topic of chaotic dynamics. Among the well-known routes to chaos, we can mention intermittency, and it is on this important topic that del Rio and Elaskar⁴ devoted their work Type III intermittency without characteristic relation. As it is well known, chaotic intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD) that maps trajectories from the chaotic region into the laminar one. Traditionally, the RPD is taken as a constant. This hypothesis is behind the classically reported characteristic relations, a tool describing how the mean value of the laminar length goes to infinity as the control parameter goes to zero. Recently, a generalized nonuniform RPD has been observed in a wide class of 1D maps, and hence the intermittency theory, as well as the characteristic relations, has been generalized. However, the RPD and the characteristic relations observed in some experimental Poincaré maps still cannot be well explained in the actual intermittency framework. The authors extend the previous analytical results to deal with the mentioned class of maps. They find that in the mentioned maps, there is no well-defined RPD in the sense that its shape drastically changes depending on a small variation of the parameter of the map. Consequently, the characteristic relation conventionally associated with every type of intermittency is not well defined and, in general, cannot be determined experimentally. The authors illustrate the results with a 1D map and develop the analytical expressions for every RPD and its characteristic relations. Moreover, they find a characteristic relation going to a constant value, instead of increasing to infinity, finding a good agreement with the numerical simulations.

Among many interesting applications of chaotic dynamics, one can mention the memristor. Since its discovery, numerous studies have been devoted to the analysis of its chaotic dynamics. In particular, Gu et al.⁵ in the paper A novel non-equilibrium memristor-based system with multi-wing attractors and multiple transient transitions propose a novel memristor-based chaotic system without equilibrium points based on the pure mathematical model of the memristor. By selecting different parameters and initial conditions, the system shows extremely diverse forms of wing-like attractors, such as period-1 to period-12 wings, chaotic single wing (CSW), and chaotic double wing (CDW). It is shown that the attractor basins with three different sets of parameters are interwoven in a complex manner within a relatively large (but not entire) initial phase plane. This means that small perturbations in the initial conditions in the mixing region will lead to the production of hidden extreme multistability. At the same time, these sieve-shaped basins are confirmed by the uncertainty exponent. Additionally, in the case of fixed parameters, when different initial values are chosen, the system exhibits a variety of coexisting transient transition behaviors. These 14 were also where the same state transition from period 18 to period 18 was first discovered. The above dynamical behaviors are analyzed in detail through time-domain waveforms, phase diagrams, attraction basin, bifurcation diagrams, and Lyapunov exponent spectrum (LES). Eventually, the circuit implementation based on the digital signal processor (DSP) is used to verify the numerical simulation and theoretical analysis.

One important field of applications of chaotic dynamics is mechanical engineering. In the paper On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness,⁶ Yakovleva et al. present a mathematical model of nonlinear vibration of a beam nanostructure with low shear stiffness and subjected to uniformly distributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elastic material obey Hooke’s law and are governed by the kinematic third-order approximation (Sheremetev–Pelekh–Reddy model). The von Kármán geometric nonlinear relation between deformations and displacements is taken into account. In order to describe the size-dependent coefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations (PDEs) as well as the initial and boundary conditions. A solution to the dynamical problem is found with the use of the second-order finite difference method (FDM), and next, the Runge–Kutta method of orders from two to eight as well as the Newmark method. Investigations of the nonlinear nanobeam vibrations are carried out with the help of signals (time histories) and phase portraits as well as through Fourier and wavelet-based analyses. The strength of the nanobeam chaotic vibration is quantified through the Lyapunov exponents computed based on the Sano-Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling procedure enabled them to achieve reliable results. In particular, chaotic and hyper-chaotic vibrations of the studied nanobeam are detected, and the results are authentic, reliable, and accurate.

Another aspect of this topic is studied by Krysko-jr et al.⁷ in the paper Chaotic vibrations of size-dependent flexible rectangular plates. The authors propose a mathematical model to describe nonlinear vibrations of size-dependent rectangular plates. The plates are treated as the Cosserat continuum with bounded rotations of their particles (pseudo-continuum). The governing PDEs and boundary/initial conditions are obtained using the von Kármán geometric relations, and they are yielded by the energetic Hamilton’s principle. The derived mixed-form partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) and algebraic equations (AEs) using: (i) the Galerkin–Krylov–Bogoliubov Method (GKBM) in higher approximations and then solved with the help of a combination of the second- and fourth-order Runge–Kutta methods, (ii) the finite difference method (FDM), and the (iii) Newmark method. The convergence of FDM vs the interval of the space coordinate grids and of GKBM vs the number of employed terms of the approximating function are investigated. The latter approach enabled them to achieve reliable results by taking the account of almost infinite-degree-of-freedom approximation to the regular and chaotic dynamics of the studied plates. The problem of stability loss of the size-dependent plates under harmonic load is also tackled.

Fractional dynamics also constitutes an interdisciplinary field of a study aiming at investigating the behavior of dynamical systems that are described by differentiation of fractional orders, using methods of fractional calculus. Fractional-order systems are dynamical systems that can be described by fractional differential equations containing derivatives of noninteger order. They are used to model nonlocal systems and systems with power-law memory in science and engineering.

In the paper Fractional-calculus analysis of the transmission dynamics of the dengue infection⁸ by Srivastava et al., a novel approach in dengue modeling with the asymptomatic carrier and reinfection via the fractional derivative is suggested to deeply interrogate the comprehensive transmission phenomena of dengue infection. The proposed system of dengue infection is represented in Liouville–Caputo fractional framework and investigated for basic properties, that is, uniqueness, positivity, and boundedness of the solution. Authors use the next-generation technique in order to determine the basic reproduction number $R0$ for the suggested model of dengue infection. Moreover, they conduct sensitivity test of $R0$ through PRCC (Partial Rank Correlation Coefficient) technique to know the contribution of input factors on the output of $R0$⁠. They also show that the infection-free equilibrium of dengue dynamics is globally asymptomatically stable for $R0<1$ and unstable in other circumstances. The system of dengue infection is then structured in the Atangana–Baleanu framework to represent the dynamics of dengue with the nonsingular and nonlocal kernel. The existence and uniqueness of the solution of the Atangana–Baleanu fractional system are interrogated through fixed point theory. Finally, the authors present a novel numerical technique for the solution of their fractional-order system in Atangana–Baleanu framework. They obtain numerical results for different values of fractional-order and input factors to highlight the consequences of fractional-order and input parameters on the system. On the basis of the analysis, they predict the most critical parameters in the system for the elimination of dengue infection.

Oscillations and waves are ubiquitous in nature, and nonlinear and complex oscillations are of particular interest in the description of numerous physical and natural phenomena. Under the label of complex oscillations, we refer here to Canard oscillations or noise-induced oscillations.

In the paper Canard oscillations in the randomly forced suspension flows,⁹ Bashkirtseva and Ryashko study complex Canard-type oscillatory regimes in stochastically forced flows of suspensions. They use the nonlinear dynamical model with an N-shaped rheological curve. Amplitude and frequency characteristics of self-oscillations in the zone of the Canard explosion are studied in dependence on the stiffness of this N-shaped function. A constructive role of random noise in the formation of complex oscillatory regimes is investigated. A phenomenon of the noise-induced splitting of stochastic cycles is discovered and studied both numerically and analytically by the stochastic sensitivity technique. Supersensitive Canard cycles are described and their role in noise-induced transitions from order to chaos is discussed.

Furthermore, in Noise-induced complex oscillatory dynamics in the Zeldovich–Semenov model of a continuous stirred tank reactor,¹⁰ Ryashko studies the noise-induced variability of thermochemical processes in continuous stirred tank reactor based on the Zeldovich–Semenov dynamical model. For the deterministic variant of this model, mono- and bistability parametric zones, as well as local and global bifurcations, are determined. Noise-induced transitions between coexisting attractors (equilibria and cycles) and stochastic excitement with spike oscillations are investigated by direct numerical simulation and an analytical approach based on the stochastic sensitivity technique. For the stochastic model, the phenomenon of coherence resonance is discovered and studied.

Finally, Abdulwahed and Verhulst provide a paper on Recurrent canards producing relaxation oscillations.¹¹ For three three-dimensional chaotic systems (Sprott NE1, NE8, and NE9) with linear and quadratic terms only, and one parameter but without equilibria, they consider second-order asymptotic approximations in the case that the parameter is small and near the origin of phase space. The calculation leads to the existence and approximation of periodic solutions with neutral stability for systems NE1, NE9, and asymptotic stability for system NE8. Extending to a larger neighborhood in phase space, the authors find a new type of relaxation oscillations with pulse behavior that can be understood by identifying hidden canards. The relaxation dynamics co-exists with invariant tori and chaos in the systems.

Traffic dynamics constitute a particular case of complex phenomena with some advances in the last few years by using techniques and methods derived from complex systems studies. In particular, we consider here some cases of interest related to chaotic dynamics and complex systems approaches.

In Simulating the city traffic complexity induced by traffic light periods,¹² Carrasco et al. revisit the global traffic light optimization problem through a cellular automata model, which enables us to address the relationship between the traffic lights and the car routing. They conclude that both aspects are not separable. Their results show that a good routing strategy weakens the importance of the traffic light period for mid densities, thus limiting the parameter space where such optimization is relevant. This is confirmed by analyzing the travel time normalized by the shortest path between origin and destination. As an unforeseen result, the authors report what seems to be a power-law distribution for such quantity, indicating that the travel time distribution slowly decreases for long travel times. The power-law exponent depends on the density; the traffic light period; and the routing strategy, which in this case is parameterized by the tendency of agents to abandon a route if it becomes stagnant. These results could have relevant consequences on how to improve the overall traffic efficiency in a particular city, thus providing insight into useful measurements, which are often counter-intuitive that may be valuable to traffic controllers that operate through traffic light periods and phases.

Moreover, in Chaotic semi-express buses in a loop,¹³ Saw et al. analyze an interesting example of urban mobility. Urban mobility involves many interacting components: buses, cars, commuters, pedestrians, trains, etc., making it a very complex system to study. Even a bus system responsible for delivering commuters from their origins to their destinations in a loop service already exhibits very complicated dynamics. Here, the dynamics of a simplified version of such a bus loop system consisting of two buses serving three bus stops is investigated. Specifically, the authors consider a configuration of one bus operating as a normal bus, which picks up passengers from bus stops $A$ and $B$⁠, and then delivers them to bus stop $C$⁠, while the second bus acts as an express bus, which picks up passengers only from bus stop $B$ and then delivers them to bus stop $C$⁠. The two buses are like asymmetric agents coupled to bus stop $B$ as they interact via picking up passengers from this common bus stop. Intriguingly, this semi-express bus configuration is more efficient and has a lower average waiting time for buses, compared to a configuration of two normal buses or a configuration of two express buses. The authors reckon that the efficiency arises from the chaotic dynamics exhibited in the semi-express system, where the tendency toward anti-bunching is greater than that toward bunching, in contradistinction to the regular bunching behavior of two normal buses or the independent periodic behavior of two non-interacting express buses.

Complex networks and applications is a very fruitful research field with many applications. In this last part, we describe a series of papers dealing with this research area.

The modern view of network resilience and epidemic spreading has been shaped by percolation tools from statistical physics, where nodes and edges are removed or immunized randomly from a large-scale network. Shang provides the contribution Immunization of networks with limited knowledge and temporary immunity.¹⁴ In this paper, the author produces a theoretical framework for studying targeted immunization in networks, where only nodes can be observed at a time with the most connected one among them being immunized and the immunity it has acquired may be lost subject to a decay probability $ρ$⁠. They examine analytically the percolation properties as well as scaling laws, which uncover distinctive characters for Erdős–Rényi and power-law networks in the two dimensions of $n$ and $ρ$⁠. The author studies both the case of fixed immunity loss rate and an asymptotic total loss scenario, paving the way to further understand temporary immunity in complex percolation processes with limited knowledge.

In Size matters: Effects of the size of heterogeneity on the wave re-entry and spiral wave formation in an excitable media,¹⁵ Rajagopal et al. deal with neuron networks. Network performance of neurons plays a vital role in determining the behavior of many physiological systems. In this paper, the authors discuss the wave propagation phenomenon in a network of neurons considering obstacles in the network. Numerous studies have shown the disastrous effects caused by the heterogeneity induced by the obstacles but the studies have been mainly focused on the orientation effects. Hence, Rajagopal et al. are interested in investigating the effects of both size and orientation of the obstacles in the wave re-entry and spiral wave formation in the network. For this analysis, they consider two types of neuron models and a pancreatic beta cell model. In the first neuron model, they use the well-known differential equation-based neuron models and in the second type, they use the hybrid neuron models with resetting phenomenon. They show that the size of the obstacle decides the spiral wave formation in the network and horizontally placed obstacles will have a lesser impact on the wave re-entry than the vertically placed obstacles.

Identification of complex networks from limited and noise-contaminated data is an important yet challenging task, which has attracted researchers from different disciplines recently. In Complex networks identification using Bayesian model with independent Laplace prior,¹⁶ Zhang et al. analyze the underlying feature of a complex network identification problem and translate it into a sparse linear programming problem. Then, a general framework based on the Bayesian model with independent Laplace prior is proposed to guarantee the sparseness and accuracy of identification results after analyzing the influence of different prior distributions. At the same time, a three-stage hierarchical method is designed to resolve the puzzle that the Laplace distribution is not conjugated to the normal distribution. Last, the variational Bayesian is introduced to improve the efficiency of the network reconstruction task. The high accuracy and robust properties of the proposed method are verified by conducting both general synthetic networks and real networks identification task based on the evolutionary game dynamic. Compared with the other five classical algorithms, the numerical experiments indicate that the proposed model can outperform these methods in both accuracy and robustness.

In Disintegrating spatial networks based on region centrality,¹⁷ Wang et al. find an optimal strategy at a minimum cost to efficiently disintegrate a harmful network into isolated components. The problem considered is important and interesting, with applications, in particular, to anti-terrorism measures and epidemic control. The paper focuses on optimal disintegration strategies for spatial networks, aiming to find an appropriate set of nodes or links whose removal would result in maximal network fragmentation. The authors refer to the sum of the degree of nodes and the number of links in a specific region as region centrality. This metric provides a comprehensive account of both topological properties and geographic structure. Numerical experiments on both synthetic and real-world networks demonstrate that the strategy is significantly superior to conventional methods in terms of both effectiveness and efficiency. Moreover, their strategy tends to cover those nodes close to the average degree of the network rather than concentrating on nodes with higher centrality.

Finally, in Active learning and relevance vector machine in efficient estimate for basin stability of large-scale dynamic networks,¹⁸ Che and Cheng analyze the interconnectivity between constituent nodes giving rise to the cascading failure in most dynamic networks, such as traffic jam in transportation network and large-scale blackout in power grid system. Basin stability (BS) has recently garnered tremendous traction to quantify the reliability of such dynamical systems. In the power grid network, it quantifies the capability of the grid to regain the synchronous state after being perturbated. It is noted that the detection of the most vulnerable node or generator with the lowest BS or reliability is critical toward the optimal decision making on maintenance. However, the conventional estimation of BS relies on the Monte Carlo (MC) method to separate the stable and unstable dynamics originated from the perturbation, which incurs an immense computational cost, particularly for large-scale networks. As BS estimate is in essence a classification problem, the authors investigate relevance vector machine (RVM) and active learning to locate the boundary of stable dynamics or the basin of attraction in an efficient manner. This novel approach eschews a large number of sampling points in MC method and reduces over $95%$ of the simulation cost in the assessment of the reliability of the power grid network.

We have summarized here different topics covered in this Focus Issue. The contributions gathered can be grouped into four categories. The first one is a collection of papers dealing with chaotic dynamics and fractional dynamics and applications, where papers discuss the coexistence of attractors in dissipative nontwist systems; logistic map trajectory distributions; computation of infinite homoclinic orbits in the Lorenz System; type III intermittency without characteristic relation; a novel non-equilibrium memristor-based system; chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness; chaotic vibrations of size-dependent flexible rectangular plates; and fractional-calculus analysis of the transmission dynamics of the dengue infection.

The second and third groups of papers deal with complex oscillations and complex traffic dynamics. The second one comprises two topics, canard oscillations and noise-induced complex oscillations, while the third one focuses on simulating the city traffic complexity induced by traffic light periods and chaotic semi-express buses in a loop.

Finally, the fourth group encompasses complex networks and applications. The contributions deal with immunization of networks; effects of the size of heterogeneity on spiral wave formation in an excitable media; identification of complex networks with Bayesian methods; spatial networks; and further methods for basin stability of large-scale dynamic networks.

Undoubtedly, the contents of this Focus Issue provide a rich perspective of some recent advances in the modeling of complex systems. In our opinion, the topics and approaches presented here are interesting and actual and will contribute to the further development of novel methods as well as will stimulate researchers to search for solutions to future scientific problems. The methods and techniques shared by different problems contemplated here under the umbrella of complex systems dynamics provide new perspectives for future research endeavors.

M.A.F.S. acknowledges financial support by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF, EU) under Project No. PID2019-105554GB-I00.