International Journal of Bifurcation and Chaos

Dramatic advances in the field of complex networks have been witnessed in the past few years. This paper reviews some important results in this direction of rapidly evolving research, with emphasis on the relationship between the dynamics and the topology of complex networks. Basic quantities and typical examples of various complex networks are described; and main network models are introduced, including regular, random, small-world and scale-free models. The robustness of connectivity and the epidemic dynamics in complex networks are also evaluated. To that end, synchronization in various dynamical networks are discussed according to their regular, small-world and scale-free connections.

Beam halo-chaos in high-current accelerators has become a key concerned issue because it can cause excessive radioactivity from the accelerators therefore significantly limits their applications in industry, medicine, and national defense. This article reviews the complexity of accelerator driven clean nuclear power system (ADS) as well as the associate physical mechanism for beam halo-chaos formation in high-intensity proton linear accelerator. Notably, some general engineering methods for chaos control have been developed in recent years, but they are generally unsuccessful for beam halo-chaos suppression due to many technical constraints. In this article, some of these technical problems are addressed. Particles-in-Cell (PIC) simulations are described, for exploring the nature of beam halo-chaos formation. Some efficient nonlinear control methods, including wavelet function feedback control, are reported for beam halo-chaos suppression. PIC simulations show that after control is applied to the initial proton beam with water bag or full Gauss distributions, the beam halo strength factor is quickly reduced to zero, and other statistical physical quantities of beam halo-chaos are also doubly reduced. These performed PIC simulation results demonstrate that the developed methods are very effective for halo-chaos suppression. Potential applications of the beam halo-chaos control methods are finally discussed.

The local activity principle of the Cellular Nonlinear Network (CNN) introduced by Chua [1997] has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. This paper presents some analytical criteria for the local activity of two-port CNN cells with three or four state variables. As a first application, a coupled excitable cell model (ECM) CNN is introduced, which has cells defined by the Chay equations representing ionic events in excitable membranes in terms of a Hodgkin–Huxley type formalism. The bifurcation diagram of the ECM CNN supplies a possible explanation for the mechanism of arrhythmia (from normal to abnormal until stopping) of excitable cells: the cell parameter is changed from an active unstable domain to an edge of chaos. The member potentials along fibers are simulated numerically, where oscillatory patterns, chaotic patterns as well as convergent patterns are observed. As a second application, a smoothed Chua's circuit (SCC) CNN with two ports is presented, whose prototype has been introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN in order to obtain Turing patterns. The bifurcation diagrams of the SCC CNN are the same as those with one port, which have only active unstable domains and edges of chaos. Numerical simulations show that in the active unstable parameter domains, the evolutions of the patterns of the state variables of the SCC CNNs can exhibit divergence, periodicity and chaos, where, in the parameter domains located in the edge of chaos, periodic patterns and divergent patterns are observed. These results demonstrate once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex patterns of CNNs.

Consider the one-dimensional wave equation on a unit interval, where the left-end boundary condition is linear, pumping energy into the system, while the right-end boundary condition is self-regulating of the van der Pol type with a cubic nonlinearity. Then for a certain parameter range it is now known that chaotic vibration occurs. However, if the right-end van der Pol boundary condition contains an extra linear displacement feedback term, then it induces a memory effect and considerable technical difficulty arises as to how to define and determine chaotic vibration of the system. In this paper, we take advantage of the extra margin property of the reflection map and utilize properties of homoclinic orbits coupled with a perturbation approach to show that for a small parameter range, chaotic vibrations occur in the sense of unbounded growth of snapshots of the gradient. The work also has significant implications to the occurrence of chaotic vibration for the wave equation on a 3D annular domain.

Decoding of a sequence of interspike intervals (ISIs) of a neuron model driven by a chaotic stimulus is performed based on the attractor reconstruction method. As stimulus strength increases, both the stimulus estimation error and the prediction error in predicting stimulus crosswise by exploiting ISIs information tend to decrease with transitional drops at certain parameter values. It is analyzed that such behaviors are well explained in the context of synchronization between two chaotic patterns of stimulus and ISI sequence. The result implies that a new scheme of temporal coding at low firing rate regime can be achieved which exploits the preservation of nonlinear deterministic structures in stimulus.

Dynamical behaviors of a new chaotic attractor is investigated in this paper. Some basic properties, bifurcations, routes to chaos, and periodic windows of the new system are studied either analytically or numerically. Meanwhile, the transition between the Lorenz attractor and Chen's attractor through the new system is explored.

Based on the delayed feedback chaotic anticontrol systems, a new chaotic phase shift keying (CPSK) scheme is proposed for secure communications in this paper. The chaotic transmitter is a linear system with nonlinear delayed feedback in which a trigonometric function cos(·) is used. Such system can exhibit rich chaotic behavior with the choice of appropriate parameters. For an M-ary communication system where M=2ⁿ, each of these M possible symbols (n-bits) is firstly mapped to 2(m-1)π/M (with m=1, 2, …, M) which is used as the phase argument for the cos(·) function in the nonlinear feedback. Two different kinds of signals can be transmitted. In the first one, an appropriate linear combination of state variables is chosen as the transmitting signal based on the observer theory. In another one, a nonlinear component in the transmitter state equation is chosen. In both schemes, only a scalar chaotic signal is transmitted through the channel. Demodulation is based on the synchronization of the transmitter and the receiver, and different decoded phases correspond to different information signals.

In a chaos-shift-keying (CSK) digital communication system, correlators followed by a comparator are typically used for coherent detection of the signal. Such a detection method, however, does not take the temporal variation of the bit energy into consideration. In this paper, an optimum detection for a binary CSK system is derived, taking into account the temporal variation of the bit energy for minimizing the error rates. Simulations are carried out to compare the performance between the optimum receiver and a typical receiver. The results provide theoretical performance benchmarks of coherent CSK systems for future references.

In this paper, the reduction approach based on the center manifold theory for parameter-dependent nonlinear dynamical systems is applied to simplify the analysis of the fold bifurcation relevant to voltage collapse in a simple electric power system. This technique enables us to obtain a lower-dimensional and topologically equivalent system in the neighborhood of the bifurcation value, which is a subset in the state-parameter space. Explicit formulas are presented for the computation of quadratic coefficients of the Taylor approximations to the center manifold for the fold bifurcation in parameter-dependent dynamical systems. It is demonstrated that it is the fold bifurcation but not other static bifurcations that occurs in the reduced power system, via validating the nondegenerate conditions for the fold bifurcation. The dynamics of the reduced system are visualized by two- and three-dimensional plots, which show that the reduction method is applicable and accurate to analyze local bifurcations in the power system. Furthermore, time domain simulation and modal analysis technique for linear systems are applied to distinguish the voltage stability from the rotor (angle) stability in the electric power system.

Delayed feedback control (DFC) is a powerful method for stabilizing unstable periodic orbits embedded in chaotic attractors, which uses a small control input fed by the difference between the current state and the delayed state. One drawback of the DFC is known as the odd number limitation; that is, DFC can never stabilize a target unstable fixed point of a chaotic discrete-time system, if the Jacobian of its linearized system around the unstable fixed point has an odd number of real eigenvalues greater than unity. To overcome it, in this paper we propose a dynamic DFC method using output measurements of the chaotic systems. The proposed dynamic DFC is realized by using an output feedback controller with a minimal-order observer that has the least order for estimating the state of the chaotic system from the control input and the output measurements. In addition to the design procedure of the controller, we derive a necessary and sufficient condition for the existence of such controllers.

In this paper, a time-delayed chaos control method based on repetitive learning is proposed. A general repetitive learning control structure based on the invariant manifold of the chaotic system is given. The integration of the repetitive learning control principle and the time-delayed chaos control technique enables adaptive learning of appropriate control actions from learning cycles. In contrast to the conventional repetitive learning control, no exact knowledge (analytic representation) of the target unstable periodic orbits is needed, except for the time delay constant, which can be identified via either experiments or adaptive learning. The controller effectively stabilizes the states of the continuous-time chaos on desired unstable periodic orbits. Simulations on the Duffing and Lorenz chaotic systems are provided to verify the design and analysis.

We propose an application of state predictor to delayed feedback control (DFC) for continuous-time dynamical systems. First, we define the DFC for continuous-time systems clearly. Next, on the basis of this definition, we propose a DFC method with state-predictor to stabilize unstable equilibrium points in autonomous systems and derive a sufficient condition for stabilization. Then we extend this method to stabilizing unstable periodic orbits and also derive a sufficient condition for stabilization. Finally, we briefly discuss the relation between the conditions for stabilization of our methods and those of the dynamic DFC and the prediction-based feedback control for discrete-time systems.

Prediction, smoothing, filtering and synchronization or observer design given finitely many measurements and a given (possibly nonlinear) dynamical map are discussed from a computational complexity point of view. All these problems are particular instances of finding a zero of an appropriately defined function. The recognition of this fact enables one to approach these questions from a computational complexity point of view. For polynomial maps the computational complexity of a global Newton algorithm adapted to identify the finite trajectory of the dynamical system's state over the desired window scales in a polynomial manner with the condition number (an invariant for the problem at hand) and the degree of the polynomials required to describe the models. The computational complexity analysis allows one to identify the most efficient manner to approach synchronization (prediction, smoothing, filtering) problems. Moreover differences between adaptive and nonadaptive formulations are revealed based on the condition number of the associated zero finding problem. The advocated formulation, with the associated global Newton algorithm has good robustness properties with respect to measurement errors and model errors for both adaptive and nonadaptive problems. These aspects are illustrated through a simulation study based around the Hénon map.

Though chaotic behaviors are exhibited in many simple nonlinear models, physical chaotic systems are much more complex and contain many types of uncertainties. This paper presents a robust adaptive neural control scheme for a class of uncertain chaotic systems in the disturbed strict-feedback form, with both unknown nonlinearities and uncertain disturbances. To cope with the two types of uncertainties, we combine backstepping methodology with adaptive neural design and nonlinear damping techniques. A smooth singularity-free adaptive neural controller is presented, where nonlinear damping terms are used to counteract the disturbances. The differentiability problem in controlling the disturbed strict-feedback system is solved without employing norm operation, which is usually used in robust control design. The proposed controllers can be applied to a large class of uncertain chaotic systems in practical situations. Simulation studies are conducted to verify the effectiveness of the scheme.

In this paper, we propose a general method for controlling chaos in a nonlinear dynamical system containing a state-dependent switch. The pole assignment for the corresponding discrete system derived from such a nonsmooth system via Poincaré mapping works effectively. As an illustrative example, we consider controlling the chaos in the Rayleigh-type oscillator with a state-dependent switch, which is changed by the hysteresis comparator. The unstable one- and two-periodic orbits in the chaotic attractor are stabilized in both numerical and experimental simulations.

This paper studies the chaotification problem of driving a continuous-time system to a chaotic state by using an impulsive control input. The controller is designed to ensure the controlled orbit be bounded and, meanwhile, have positive Lyapunov exponents. This is proved to be not only possible but also implementable near a stable limit cycle of the given system. Two numerical examples are given to illustrate the effectiveness of the proposed chaotification method.

We investigate the differences among several definitions of the snap-back-repeller, which is always regarded as an inducement to produce chaos in nonlinear dynamical system. By analyzing the norms in different senses and the illustrative examples, we clarify why a snap-back-repeller in the neighborhood of the fixed point, where all eigenvalues of the corresponding variable Jacobian Matrix are absolutely larger than 1 in norm, might not imply chaos. Furthermore, we theoretically prove the existence of chaos in a discrete neural networks model in the sense of Marotto with some parameters of the systems entering some regions. And the following numerical simulations and corresponding calculation, as concrete examples, reinforce our theoretical proof.

Partial synchronization in Hamiltonian systems is investigated based on the concept of measure-synchronization. The classical φ⁴ model is used for the investigation. A macroscopic observable of long-term average of particle energy is computed to describe transitions between desynchronization, different partial synchronization, and complete synchronization structures. It is found that, prior to the entire synchronization of all oscillators, partial measure-synchronization for some clusters of oscillators is stable within certain regions. Moreover, transition from quasiperiodicity to chaos is observed to be associated with the measure-synchronization as the coupling strength is increased.

This paper studies the single-input controllable nonlinear system and explores its invariant structure in terms of internal feedback loops. The invariant structure provides a useful framework for the analysis and design of nonlinear systems. An application of the invariant structure to controlling Lorenz chaos is presented.

In this paper, by means of case studies we discuss an observable feature of 3D continuous-time autonomous chaotic systems through scalar output and its time derivatives. We observe that the Lorenz system, the Rössler system, the Chua circuit and the Chen system are all observable based on scalar output and its derivatives. This leads to our conjecture that chaotic motion described by 3D continuous-time autonomous dynamical system is observable based on a scalar output and its first- and second-order derivatives. Finally, we present some mathematical analysis and put up some theoretical questions for future studies.

In this paper, a novel method for locating and stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems is proposed. The main idea of the method is to formulate the UPO locating problem as an optimization issue by using some inherent properties of UPOs of chaotic systems. The global optimal solution of this problem yields the desired UPO. To avoid a local optimal solution, the state of the controlled chaotic system is absorbed into the initial condition of the optimization problem. The ergodicity of chaotic dynamics guarantees that the optimization process does not stay forever at any local optimal solution. When the chaotic orbit approaches the global optimal solution, which is the desired UPO, the controller will stabilize it at the UPO, and the optimization process will cease simultaneously. The method has been developed for both discrete-time and continuous-time systems, and validated for some typical chaotic systems such as the Hénon map and the Duffing oscillator, among others.

This paper proposes a new iteration method for chaotifying and controlling dynamical systems. By applying this iteration method, the dimension of the given dynamical system can be reduced from to n to n-1. Moreover, the chaotified system is not necessarily Hurwitz stable originally. The iteration method is applied to three-dimensional systems for demonstration, for which a sufficient condition is obtained for chaotification. In addition, the iteration method can be used to control a class of chaotic systems. These results are illustrated via simulations on the Duffing oscillator and the Chen system.

This paper studies an impulsive control problem. By utilizing the method of Lyapunov functions, a set of impulsive stabilization criteria are established. These results are then applied to the Lorenz system. It is shown that by using impulsive feedback control, all the solutions of the Lorenz system will converge to an equilibrium point.

In this paper, impulsive control of nonlinear systems and its application to Chen's chaotic system are considered. A new impulsive control method for chaos suppression, using Chen's system as an example, is developed. Some new general criteria for exponential stability and asymptotical stability of nonlinear impulsive systems are established and, particularly, some simple conditions sufficient for driving the chaotic state of Chen's system to its zero equilibrium are presented.

We investigate the phase transition between solutions with distinct symmetrical property observed in a system of coupled three-oscillators with hard characteristics and state coupling. By a symmetry-breaking bifurcation, a symmetrical in-phase solution bifurcates into synchronized modes with a partially in-phase solution and an almost in-phase solution. Moreover, by using the definition of symmetrical and asymmetrical three-phase solutions, we confirmed the existence of a stable symmetrical three-phase quasi-periodic solution and an asymmetrical three-phase chaotic solution in the coupled system.

We consider quantized chaotic dynamics for a spiking oscillator with two periodic inputs. As the first input is applied, the oscillator generates various periodic and chaotic pulse-trains governed by a pulse position map. As the second input is added, the oscillator produces pulse positions restricted on a lattice, and the pulse position map is quantized. Then the oscillator generates a set of super-stable periodic pulse-trains (SSPTs). The oscillator has various coexisting SSPTs and generates one of them depending on the initial state condition. In order to characterize the set of SSPTs, we elucidate the number and the minimum pulse interval of the SSPTs theoretically. By presenting a simple test circuit, we then verify some typical phenomena in the laboratory environment.

In this paper, the open-plus-closed-loop control strategy is adapted to synchronize discrete chaos. Two synchronization problems of chaos are studied: one is to drive a chaotic map with the aim of obtaining desirable chaotic dynamics; the other is to identify chaotic behaviors of a nonlinear map for different initial conditions. It is shown that in the latter case the needed control signal can be arbitrarily small. Two numerical examples, namely, the Gaussian map and a hyperchaotic map, are investigated in detail for demonstration of the effectiveness of the approach. The results show that synchronization of discrete chaos can be realized if the control is activated in the basin of entrainment.