International Journal of Bifurcation and Chaos

One of the main characteristics of complexity is the emergence of properties due to dynamical processes. This special issue has put together a unique collection of articles written by leading researchers and experts around the globe on recent advances in complex systems and applications, in various fields of science and engineering. It focuses not only on equation-based modeling of eco- or bio-systems analysis but also on the study of eco- or bio-complexity and global emergent properties and self-organization, resulting from various interactions.

The pre-history of chaos in a rationalist context is taken as a point of departure, starting out with ancient China. The related ancient-Greek "unmixing theory" then leads over to two simple formally 2-body Hamiltonian systems exhibiting chaotic behavior. When the two masses involved are unequal, "pseudoattractors" are formed. Deterministic statistical "thermodynamics" with its dissipative behavior arises when the potential is repulsive. Deterministic statistical "cryodynamics" arises when the potential is attractive. The latter class of Newtonian systems is characterized by "antidissipative" behavior. A geometric proof is sketched in the footsteps of Sinai and Bunimovich. Antidissipative behavior is known empirically from Hubble's law which was so far explained in less fundamental terms. Three experimental examples are proposed.

This work clarifies the relationship between network circuit (topology) and behavior (information transmission and synchronization) in active networks, e.g. neural networks. As an application, we show how to determine a network topology that is optimal for information transmission. By optimal, we mean that the network is able to transmit a large amount of information, it possesses a large number of communication channels, and it is robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh–Rose neurons.

In systems theory and science, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems [Aziz-Alaoui & Bertelle, 2009]. In this paper, we use the emergent property of the ultra weak multidimensional coupling of p 1-dimensional dynamical chaotic systems which leads from chaos to randomness.

Generation of random or pseudorandom numbers, nowadays, is a key feature of industrial mathematics. Pseudorandom or chaotic numbers are used in many areas of contemporary technology such as modern communication systems and engineering applications. More and more European or US patents using discrete mappings for this purpose are obtained by researchers of discrete dynamical systems [Petersen & Sorensen, 2007; Ruggiero et al., 2006]. Efficient Chaotic Pseudo Random Number Generators (CPRNG) have been recently introduced. They use the ultra weak multidimensional coupling of p 1-dimensional dynamical systems which preserve the chaotic properties of the continuous models in numerical experiments. Together with chaotic sampling and mixing processes, ultra weak coupling leads to families of (CPRNG) which are noteworthy [Hénaff et al., 2009a, 2009b, 2009c, 2010].

In this paper we improve again these families using a double threshold chaotic sampling instead of a single one.

We analyze numerically the properties of these new families and underline their very high qualities and usefulness as CPRNG when very long series are computed. Moreover, a determining property of such improved CPRNG is the high number of parameters used and the high sensitivity to the parameters value which allows choosing it as cipher-keys. It is why we call these families multiparameter chaotic pseudo-random number generators (M-p CPRNG).

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos.

Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value μ₂ of the map at which there is chaos. We show that often virtually all (i.e. all but finitely many) "regular" periodic orbits at μ₂ are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired — connected to exactly one other cascade, or solitary — connected to exactly one regular periodic orbit at μ₂. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ₂, ⋅). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

We show techniques of analyzing complex dynamics of cellular automata (CA) with chaotic behavior. CA are well-known computational substrates for studying emergent collective behavior, complexity, randomness and interaction between order and chaotic systems. A number of attempts have been made to classify CA functions on their space-time dynamics and to predict the behavior of any given function. Examples include mechanical computation, λ and Z-parameters, mean field theory, differential equations and number conserving features. We aim to classify CA based on their behavior when they act in a historical mode, i.e. as CA with memory. We demonstrate that cell-state transition rules enriched with memory quickly transform a chaotic system converging to a complex global behavior from almost any initial condition. Thus, just in few steps we can select chaotic rules without exhaustive computational experiments or recurring to additional parameters. We provide an analysis of well-known chaotic functions in one-dimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions.

Complex software networks, as a typical kind of man-made complex networks, have attracted more and more attention from various fields of science and engineering over the past ten years. With the dramatic increase of scale and complexity of software systems, it is essential to develop a systematic approach to further investigate the complex software systems by using the theories and methods of complex networks and complex adaptive systems. This paper attempts to briefly review some recent advances in complex software networks and also develop some novel tools to further analyze complex software networks, including modeling, analysis, evolution, measurement, and some potential real-world applications. More precisely, this paper first describes some effective modeling approaches for characterizing various complex software systems. Based on the above theoretical and practical models, this paper introduces some recent advances in analyzing the static and dynamical behaviors of complex software networks. It is then followed by some further discussions on potential real-world applications of complex software networks. Finally, this paper outlooks some future research topics from an engineering point of view.

The aim of this paper is to contribute to the modeling and analysis of complex systems, taking into account the nature of complexity at different stages of the system life-cycle: from its genesis to its evolution. Therefore, some structural aspects of the complexity dynamics are highlighted, leading (i) to implement the morphogenesis of emergent complex network structures, and (ii) to control some synchronization phenomena within complex networks. Specific applications are proposed to illustrate these two aspects, in urban dynamics and in neural networks.

Fast-slow systems are studied usually by "geometrical dissection" [Borisyuk & Rinzel, 2005]. The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems [Dumortier & Roussarie, 1996] and with applications to physiology [Desroches et al., 2008; Toporikova et al., 2008], a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered when the transition points were folds, in this article, we focus on the case where one or several transition points (or "jumps") are instead transcritical. We present several new surprising effects like the "amplification of canards" or the "exceptionally fast recovery" on both (1 + 1)-systems and (2 + 1)-systems associated with tritrophic food chain dynamics. Finally, we also mention their possible relevance to the notion of resilience which has been coined out in ecology [Holling, 1973; Ludwig et al., 1997; Martin, 2004].

We apply recently introduced measures of complexity for the structural quantification of distal tibial bone. For the first time, we are able to investigate the temporal structural alteration of trabecular bone. Based on four patients, we show how the bone may alter due to temporal immobilization.

Classical models of morphogenesis by Murray and Meinhardt and of epidemics by Ross and McKendrick can be revisited in order to consider the colocalizations favoring interaction between morphogens and cells or between pathogens and hosts. The classical epidemic models suppose, for example, that the populations in interaction have a constant size and are spatially fixed during the epidemic waves, but the presently observed pandemics show that the long duration of their spread during months or years imposes to take into account the pathogens, hosts and vectors migration in epidemics, as well as the morphogens and cells diffusion in morphogenesis. That leads naturally to study the occurrence of complex spatio-temporal behaviors in dynamics of population sizes and also to consider preferential zones of interaction, i.e. the zero-diffusion sets, for respectively building anatomic frontiers and confining contagion domains. Three examples of application will be presented, the first proposing a model of Black Death spread in Europe (1348–1350), and the last ones related to two morphogenetic processes, feather morphogenesis in chicken and gastrulation in Drosophila.

We present a classical interspecific competition model. Individuals compete for a resource on a common patch and can go to a refuge. It is assumed that if species would remain on the competition patch, species 1 survives and species 2 would go extinct. Therefore, species 1 is Locally Superior Competitor (LSC) and species 2 Locally Inferior Competitor (LIC). We study the effects of density dependent dispersal from the competition patch to the refuge on the global outcome of competition. We study two cases. The first case considers LSC density dependent dispersal of the LIC trying to escape competition and going to its refuge when the LSC density is large. The second case considers aggressiveness of LIC leading to LIC density dependent dispersal of the LSC. We show that under some conditions, tactic 2 can allow the LIC to survive and even provoke global extinction of the LSC.

Noninvasive ventilation is a common procedure for managing patients having chronic respiratory failure. The success of this ventilatory assistance is often linked with patient's tolerance that is known to be related to the quality of the synchronization between patient's spontaneous breathing cycles and ventilatory cycles delivered by the ventilator. Thirty-four sleep sessions (more than 5000 ventilatory cycles each) were automatically investigated using a specific algorithm processing airflow and pressure time series. Four groups of patients were defined according to the interplay between asynchrony events and leaks. Different mechanisms that depend on sleep stages were thus evidenced. A Shannon entropy was also proposed as a new sleep fragmentation quantification methodology.

This paper considers a sensor fault reconstruction scheme for nonlinear systems, using the sliding mode observer and adaptive observer. The novelty of this contribution is that we consider the disturbances represented in both the state equation and the output equation. Computer simulation experiments are carried out to demonstrate the effectiveness of the proposed design.

We present an analog circuit implementation of the novel partial control method, that is able to sustain chaotic transient dynamics. The electronic circuit simulates the dynamics of the one-dimensional slope-three tent map, for which the trajectories diverge to infinity for nearly all the initial conditions after behaving chaotically for a while. This is due to the existence of a nonattractive chaotic set: a chaotic saddle. The partial control allows one to keep the trajectories close to the chaotic saddle, even if the control applied is smaller than the effect of the applied noise, introduced into the system. Furthermore, we also show here that similar results can be implemented on a circuit that simulates a horseshoe-like map, which is a simple extension of the previous one. This encouraging result validates the theory and opens new perspectives for the application of this technique to systems with higher dimensions and continuous time dynamics.

In this paper, we create a multi-scroll chaotic attractor from Chen system by a nonlinear feedback control. The dynamic behavior of the new chaotic attractor is analyzed. Specially, the Lyapunov spectrum and Lyapunov dimension are calculated and the bifurcation diagram is sketched. Furthermore, via changing the value of the control parameters, we can increase the number of equilibrium points and obtain a family of more complex chaotic attractors with different topological structures. By introducing time delay to the feedback control, we then generalize the multi-scroll attractor to a set of hyperchaotic attractors. Computer simulations are given to illustrate the phase portraits with different system parameters.

In this paper a new concept, spatio-temporal generalized frequency response functions (STGFRF), is introduced for the first time to characterize nonlinear spatio-temporal dynamical systems in the frequency domain. A probing method is developed to calculate the STGFRFs recursively for both continuous and discrete spatio-temporal systems. The algorithm is computationally compact and exposes the explicit relationship between the continuous and discrete models and the elements of the generalized frequency response functions.

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

We analyze the two neuron CNN for the particular parameter range where the system converges to constant outputs. The functional relation between the external inputs and the steady state values of the neuron states is found and proves to be useful to solve classification problems. In fact, an exhaustive classification of the binary input–output relations that can be achieved by a two neuron CNN is established. From this relation, we propose an algorithm relating the CNN parameters and each one of the different classification problems. As an illustration, we attempt to implement the header action of a universal Turing machine and Boolean functions. Our results are compared to the CNN universal cell.

In this paper, we discuss in detail the local stability, Hopf bifurcation and pitchfork bifurcation of an inertial two-neuron system with time delays by applying the second-order approximation approach and analyzing the associated transcendental equation, respectively. Comparison has been made between the two methods. Numerical results have been presented to verify the analytical predictions. Specially, numerical examples show the time-varying control and the impulsive control which can both improve the stability of the system effectively.

A diffusive Brusselator model with delayed feedback control subject to Dirichlet boundary condition is considered. The stability of the unique constant equilibrium and the existence of a family of inhomogeneous periodic solutions are investigated in detail, exhibiting rich spatiotemporal patterns. Moreover, it shows that Turing instability occurs without delay. And under certain conditions, the constant equilibrium switches finite times from stability to instability to stability, and becomes unstable eventually, as the delay crosses through some critical values. Then, the direction and the stability of Hopf bifurcations are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Finally, some numerical simulations are carried out for illustrating the analysis results.

In this paper the models of the transmission line with one DOF, two DOFs and three DOFs are constructed respectively by using Hamilton principle, where the initial location, the geometric nonlinearity caused by deformation and the aerodynamic nonlinearity caused by flow are considered. These simple models are accurate and can be used for the theoretical analysis. In actual engineering, the critical wind speed of galloping has been paid more and more attention. Therefore, in this paper the critical wind speeds are obtained by Lyapunov stability theory and the effects of structural parameters to the critical wind speed are discussed for different models. And the response curves of the system with one DOF and two DOFs (out-of-plane motion) are obtained by using harmonic balance method and multiple scale method, respectively. The bifurcation of the system moving out-of-plane is analyzed by singularity theory. For the systems whose motion is coupled with torsion, the numerical results are given by using the fourth order Runge–Kutta method. Additionally, the effects of the dynamic vibration absorber and detuning pendulum to the anti-galloping of the transmission line are studied for different cases.

Reaction–diffusion (RD) equation behavior has been studied in various fields, including biology, bioengineering and chemistry. Their solution leads to the formation of patterns which are stable in time and unstable in space, especially when RD system parameters are in the Turing space. Such patterns can be changed due to the growth of the domain where the reaction takes place. This article presents RD equations concerning growing domains in 2D and 3D. Several numerical examples have been solved using different geometries to study the effect of growth on pattern formation. The finite element method was used in conjunction with the Newton–Raphson method for the numerical solution to approximate nonlinear partial differential equations. It was found that growth affected Turing pattern formation, thereby generating complex structures in the domain.

We analyze both theoretically and numerically the dynamical behavior of a modification of a nonholonomic Hamiltonian system known as the Heisenberg system. The equations of motion are derived by computing the Lagrange multiplier in terms of the Poisson commutator. The presence of the constraint induces a nontrivial dynamical behavior that has been investigated by plotting the Poincaré section for the reduced system and taking advantage of a conserved quantity and the reversibilities. The dynamics is organized around two Lyapunov families of periodic orbits whose bifurcation behavior has been analyzed with a continuation technique both on the conserved quantity and on the parameters of the problem. The nongeneric branching behavior of the normal modes is theoretically explained by studying the variational equations that reduces to a Hill equation and its well known coexistence property. Invariant tori around the elliptic periodic orbits have been numerically detected but not further analyzed.

The nonlinear stability analysis of boiling water nuclear reactors (BWRs) is conducted with the aid of so-called advanced, well validated, system codes and an advanced reduced order model to build a detailed mathematical understanding of the BWR behavior in the practical relevant parameter space. In the last years, the existence of Hopf-bifurcation points was confirmed by some researchers. In the framework of this paper, a parameter region was analyzed in which the coexistence of different stability states is realized. As a novel result, we found a parameter region in which stable fixed points, unstable limit cycles and stable limit cycles coexist. This system behavior can be explained by a saddle-node bifurcation of cycles (turning point). The existence of this solution type in a BWR system indicates the possibility of large amplitude limit cycle oscillations in the linear stable region.

Dynamically qualitative properties of individual orbits in a new four-dimensional nonlinear circuit are observed on an oscilloscope. Meanwhile, they are also traced numerically with the help of some methods for finding chaos. Comparisons show that the observed results are consistent with the computed ones to a great extent. In addition, the bifurcation, Lyapunvon spectra, fast Lyapunov indicators and small alignment indexes represent almost the same rules of transitivity to chaos on a control parameter. It is found when the parameter has a threshold value from order to chaos, and the chaos gets stronger and stronger, the parameter is smoothly varied from small to large. In particular, the entire set of Lyapunov exponents can lead to another threshold value of the parameter from chaotic to hyperchaotic behaviors.