International Journal of Bifurcation and Chaos

Modern Science is finding new methods of looking at biological, physical or social phenomena. Traditional methods of quantification are no longer sufficient and new approaches are emerging. These approaches make it apparent that the phenomena the observer is looking at are not classifiable by conventional methods. These phenomena are complex. A complex system, as Chua's oscillator, is a nonlinear configuration whose dynamical behavior is chaotic. Chua's oscillator equations allow to define the basic behavior of a dynamical system and to detect the changes in the qualitative behavior of a system when bifurcation occurs, as parameters are varied. The typical set of behavior of a dynamical system can be detailed as equilibrium points, limit cycles, strange attractors. The concepts, methods and paradigms of Dynamical Systems Theory can be applied to understand human behavior. Human behavior is emergent and behavior patterns emerge thanks to the way the parts or the processes are coordinated among themselves. In fact, the listening process in humans is complex and it develops over time as well. Sound and music can be both inside and outside humans. This tutorial concerns the translation of Chua's oscillators into music, in order to find a new way of understanding complexity by using music. By building up many computational models which allow the translation of some quantitative features of Chua's oscillator into sound and music, we have created many acoustical and musical compositions, which in turn present the characteristics of dynamical systems from a perceptual point of view. We have found interesting relationships between dynamical systems behavior and their musical translation since, in the process of listening, human subjects perceive many of the structures as possible to perceive in the behavior of Chua's oscillator. In other words, human cognitive abilities can analyze the large and complicated patterns produced by Chua's systems translated into music, achieving the cognitive economy and the coordination and synthesis of countless data at our disposal that occur in the perception of dynamic events in the real world. Music can be considered the semantics of dynamical systems, which gives us a powerful method for interpreting complexity.

A new neighborhood selection method is presented for both deterministic and probabilistic cellular automata models. The detection criteria are built explicitly on the corresponding contribution which is made to the value of each updated cell from each detected cell in the evolution. Theoretical analysis and numerical simulations demonstrate the effectiveness of this new method.

Glass networks have been proposed as a model framework for gene regulation, chemical kinetics and neural networks. Their main distinguishing feature is that although the network variables evolve continuously in time, interactions between them depend discontinuously on their sign (i.e. above or below a threshold). While this is a simplification, it has tremendous analytic advantages if the approximation is reasonable in an application. This study explores and classifies bifurcations in Glass networks, and relates them to bifurcations of smooth systems. These bifurcations can often not be studied with traditional bifurcation theory, as the vector fields are discontinuous. However, the theory that has been developed for periodic orbits of Glass networks allows a natural classification for bifurcations of periodic orbits. Some of these are shown to correspond to smooth-system bifurcations, others are shown to fit into the framework of "C-bifurcations" or "border-collision bifurcations" and others are shown to allow truly ambiguous behavior, for which Filippov's theory for discontinuous vector fields is an appropriate tool. Routes to chaos are also explored, and it is demonstrated that period-doubling cascades do not occur. However, sudden transitions to chaos, which are common in Glass networks, can result in a limiting sense from compression to a point of a period-doubling cascade in corresponding networks with sigmoidal interactions as the sigmoid's gain is increased. Other phenomena such as intermittency and multistability are also discussed.

This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.

This paper continues the study of the global dynamic properties specific to maps of the plane characterized by the presence of a denominator that vanishes in a one-dimensional submanifold. After two previous papers by the same authors, where the definitions of new kinds of singularities, called focal points and prefocal sets, are given, as well as the particular structures of the basins and the global bifurcations related to the presence of such singularities, this third paper is devoted to the analysis of nonsimple focal points, and the bifurcations associated with them. We prove the existence of a one-to-one relation between the points of a prefocal curve and arcs through the focal point having all the same tangent but different curvatures. In the case of nonsimple focal points, such a relation replaces the one-to-one correspondence between the slopes of arcs through a focal point and the points along the associated prefocal curve that have been proved and extensively discussed in the previous papers. Moreover, when dealing with noninvertible maps, other kinds of relations can be obtained in the presence of nonsimple focal points or prefocal curves, and some of them are associated with qualitative changes of the critical sets, i.e. with the structure of the Riemann foliation of the plane.

Illuminative deformation patterns of a honeycomb structure are presented. A representative volume element of a honeycomb structure consisting of 2 × 2 hexagonal cells is modeled to be a -equivariant system. The bifurcation mechanism and an exhaustive list of possible bifurcated patterns are obtained by group-theoretic bifurcation theory. A flower mode of the honeycomb is shown to have the same symmetry as the so-called anti-hexagon in the Rayleigh–Bénard convection. A numerical bifurcation analysis is conducted on an elastic in-plane honeycomb structure consisting of 2×2 cells to produce beautiful wallpapers of bifurcating deformation patterns and, in turn, to highlight the achievement of the paper. New deformation patterns of a honeycomb structure have been found and classified in a systematic manner. Knowledge of the symmetries of the bifurcating solutions has turned out to be vital in the successful numerical tracing of the bifurcated paths. This paper paves the way for the introduction of the results hitherto obtained for flow patterns in fluid dynamics into the study of patterns on materials.

Based on the classical Lotka–Volterra predator–prey system, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the system can be permanent. We observe that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Numerical results show that the system we considered has more complex dynamics including period-doubling bifurcation, symmetry-breaking bifurcation, period-halving bifurcation, quasi-periodic oscillation, chaos and nonunique dynamics, meaning that several attractors coexist. Finally, a pest–predator stage-structured model for the pest concerning this kind of impulsive control strategy is proposed, and we also show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some threshold.

In this paper, we are concerned with the chaotic behavior of a class of control systems described by partial differential equations. By means of finite dimensional output feedback control, it is shown that the flow on the global attractor is topologically equivalent to that of a finite dimensional difference system. In addition, an example is given to illustrate that the chaotic behavior of the flow on the global attractor can be determined by computing the Lyapunov exponent of an associated finite dimensional difference system.

This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg–de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth-order Boussinesq equation is considered for illustrations.

In this paper, we consider a linear wave equation on an interval with a van der Pol nonlinear boundary condition at one end and an energy-pumping condition at the other end. We study the dynamical behavior of the Riemann invariants (u,v) of the wave equation in terms of the growth rates of the total variations of the snapshots on the spatial interval. Our main contributions here are the detection of rapid fluctuations of the snapshots of u and v in the long run. The results here sharpen those in the earlier works of [Chen et al., 2001] and [Huang, 2003].

In this paper, we develop a simple linear feedback controller, which employs only one of the states of the system, to stabilize the modified Chua's circuit to an invariant set which consists of its nontrivial equilibria. Moreover, we show for the first time that the closed loop modified Chua's circuit satisfies set stability which can be considered as a generalization of common Lyapunov stability of an equilibrium point. Simulation results are presented to verify our method.

In this paper, some global exponential stability criteria for the equilibrium point of discrete-time recurrent neural networks with variable delay are presented by using the linear matrix inequality (LMI) approach. The neural networks considered are assumed to have asymmetric weighting matrices throughout this paper. On the other hand, by applying matrix decomposition, the model is embedded into a cooperative one, the latter possesses important order-preserving properties which are basic to our analysis. A sufficient condition is obtained ensuring the componentwise exponential stability of the system with specific performances such as decay rate and trajectory bounds.

In this paper, a scheme of a bidirectional hyperchaotic communication system is proposed. The hyperchaotic communication system including the transmitter and the receiver is composed of a pair of Chua circuit with coupling characteristics respectively. From the viewpoint of communication security, directly adopting the bidirectional communication system would provide a so-called "theoretical security". This is caused by the coupling relationship between the transmitter and the receiver. With the technique of frequency spectrum, the theoretical security of bidirectional communication systems can be proven. In other words, even if an intruder knows the parameter values of the system, he could not steal information from the system. As a result, it is not necessary for us to use the additional encryption. On the other hand, to achieve the synchronization of bidirectional communications, the method which applies a suitable Lyapunov function and the property of positive definite of a matrix is proposed to design the feedback controllers. Therefore, the message masked by chaotic signal from the transmitter can be perfectly recovered in the receiver. Finally, the simulation results can verify that the proposed method and the scheme of the bidirectional communication are favorable.

In this paper, we propose a method to research lag synchronization of the identical master-slave chaotic Lur'e systems via replacing variables control with time delay. By means of absolute stability theory, we prove two types of sufficient conditions for the lag synchronization: Lur'e criterion and frequency domain criterion. Based on the criteria, we suggest an optimization scheme to design the control variables. Applying the scheme to general Chua's circuits, we obtain the parameter ranges in which the master-slave Chua's circuits laggingly synchronize or not by varied single-variable control. Finally, we cite the examples by illustration of the results.

A negative conductance forced LCR circuit exhibiting strong chaos via the torus breakdown route as well as period-doubling route is described. The strong chaoticity is evidenced by the high value of the largest Lyapunov exponent and statistical studies. The dual nature of this circuit exhibiting the rich dynamics of both the Murali–Lakshmanan–Chua (MLC) circuit [Murali et al., 1994] and the circuit due to Inaba and Mori [1991] is also explored. The performance of the circuit is investigated by means of laboratory experiments, PSpice circuit simulation, numerical integration of appropriate mathematical model and explicit analytical studies, which all agree well with each other.

This paper investigates the weaknesses of cryptosystems that use observer based synchronized chaotic systems. It is shown that known plaintext and chosen plaintext attacks can successfully be launched against such cryptosystems to recover the system parameters and subsequently eavesdrop on the message transmission. The methods employed rely only on the basic mathematical relations that exist between the output sequence and the message sequence of the transmitter system and require very less computations.

Some techniques based on delay coordinates embedding for detecting unstable periodic orbits from chaotic time series work successfully in several applications but frequently yield false periodic orbits. We present a simple method for excluding false ones from candidates of periodic orbits obtained from chaotic time series. We demonstrate our method in numerical simulations for the Ikeda map and experiments for periodically forced, single and coupled pendula.

We consider the nonlinear bistable dynamic system that is the archetypal system giving way to the phenomenon of stochastic resonance for noise-improved signal processing. Independently of a strict stochastic resonance effect, we use this bistable system as a nonlinear filter for a detection task on a binary signal. We expose a methodology to tune the nonlinear filter at its best performance that minimizes its probability of detection error. The optimally tuned nonlinear filter is then compared to the ideal matched filter, which is the optimal filter for the detection with Gaussian noise. We show that the performance of the nonlinear filter, although (expectedly) not as good, comes close to that of the ideal matched filter operating in its strict nominal conditions. We next examine several possible departures, quite plausible in practical operation, from the nominal conditions of the ideal matched filter. We demonstrate that in such degraded conditions, the nonlinear filter can catch up and surpass the performance of the matched filter. This reveals a robustness superiority of the nonlinear filter, compared to the matched filter operating outside its strict nominal conditions.

A method for controlling onto saddle-type fixed points developed by Yagasaki and Uozumi is extended so as to make the capture region many times larger than that of the original method.

This paper presents a simple numerical scheme for estimating the attraction region of a fixed point in one-dimensional discrete-time chaotic systems controlled by the delayed-feedback method. This scheme employs the well-known linear matrix inequality approach. A systematic procedure for estimating the region is provided, and numerical examples are used to validate the results.