International Journal of Bifurcation and Chaos

In this paper we study briefly several chaos indicators and we focus our attention on the recently developed , presenting a numerical scheme for its calculation based on the Taylor method. Using this new indicator we present several sensitivity plots for several classical problems: the Hydrogen Atom in a magnetic field, an Extensible Pendulum, the Heavy-Top, the (N + 1)-body Ring problem and the Duffing-Oscillator. Most of these plots are new and they provide a global picture of the evolution of such a dynamical system. Besides, we show plots of variations in the parameters of the problem, giving a new perspective of the evolution of the problem.

To give rigorous mathematical proofs of chaotic behaviors in a given system, it is necessary to identify the homoclinic structures in the system. In this tutorial review, methods for constructing explicit solutions for nonlinear partial differential equations are presented, with more emphasis placed on those utilizing complete integrability associated with soliton equations. As an extended application, homoclinic orbits to spatial uniform plane waves of coupled modified nonlinear Schrödinger equations are obtained via the dressing method. During the procedure, it is necessary to introduce the Lax pair for these coupled equations, as well as its Floquet spectral analysis and corresponding Bloch functions.

In this paper, we first give a constructive proof for the existence of globally exponential attractive set of Chua's system with a smooth nonlinear function. Then, we derive a series of simple algebraic sufficient conditions under which two same type of smooth Chua's systems are globally exponentially synchronized using simple linear feedback controls. Also, as the special cases of chaos synchronization, we consider global tracking and global exponentially tracking of periodic motions, as well as global stabilization and globally exponential stabilization of equilibrium points in smooth Chua's systems. We construct a series of simple, easily applicable feedback control laws. Computer simulation results are presented to verify the theoretical predictions.

The problem of phase synchronization of Chua's chaotic oscillators is investigated. We consider Chua's circuit when it exhibits a chaotic attractor and apply a single pulse stimulus. It is shown that under certain conditions the system displays self-referential phase reset (SPR) phenomenon. This is a case when the reset phase of the chaotic oscillation is independent on the initial phase, hence on the time moment when the stimulus has been applied. In an ensemble of chaotic oscillators simultaneously stimulated, the SPR yields mutual phase coherence or synchronization between the units. We describe basic dynamical mechanisms of the effect and show how it can be used for controllable cluster formation and for the control of chaotic dynamics.

Based on the generalized Lorenz system, a conjugate Lorenz-type system is introduced, and a new unified Lorenz-type system containing these two classes of systems is naturally constructed in the paper. Such a unified system is state-equivalent to a simple special form, which is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, three new chaotic attractors, called conjugate attractors, are found for the first time, which are conjugate to the Lorenz attractor, the Chen attractor, and the Lü attractor, respectively.

In this paper we describe a rigorous method to find for a continuous system all low-period cycles enclosed in a certain region of the state space. As an example of the application of this technique we present the complete analysis of the Rössler attractor in terms of short periodic orbits. We construct the trapping region for the associated Poincaré map. The trapping region encloses the numerically observed attractor. We find all periodic orbits of the Poincaré map up to period 20. We prove that these periodic orbits correspond to all cycles of the flow with period shorter than 120. Using the number of short periodic orbits we compute estimates for topological entropy of the Poincaré map and the flow.

Pendulum equation with parametric and external excitations is investigated in (I) and (II). In (I), by applying Melnikov's method, we prove the criterion of existence of chaos under periodic perturbation. The numerical simulations, including bifurcation diagram of fixed points, bifurcation diagram of system in three- and two-dimensional space, homoclinic and heteroclinic bifurcation surface, Maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors including the period-n (n = 2 to 6, 10, 15 and 20) orbits in different chaotic regions, interlocking periodic orbits, symmetry-breaking of periodic orbit, cascade of period-doubling bifurcations from period-5 and -10 orbits, reverse period-doubling bifurcation, onset of chaos which occurs more than once for a given external frequency or parametric frequency and chaos suddenly converting to periodic orbits, sudden jump in the size of attractors which is associated with the transverse intersection of stable and unstable manifolds of perturbed saddle, hopping behavior of chaos, transient chaos with complex periodic windows and interior crisis, varied chaotic attractors including the more than three-band and eight-band chaotic attractors, chaotic attractor after strange nonchaotic attractor. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting damping δ, spring constant α and frequency Ω of parametric excitation which can be considered as a control strategy. In (II), we will investigate the complex dynamics under quasi-periodic perturbation.

A synchronized neural network model with delays is considered. The bifurcations arising from the zero root of the corresponding characteristic equation have been studied by employing the center manifold theorem, normal form method and bifurcation theory. It is shown that the system may exhibit transcritical/pitchfork bifurcation, or Bogdanov–Takens bifurcation. Some numerical simulation examples are given to justify the theoretical results.

In this paper, several sufficient conditions are established for the persistence and extinction in a Lotka–Volterra system with time delay. Based on the use of Lyapunov functionals techniques, necessary and sufficient conditions are also given for global asymptotic stability of the positive equilibrium for autonomous systems.

This paper investigates synchronization dynamics of a large class of chaotic delayed systems with all the parameters unknown. By a simple combination of adaptive control and linear feedback with the updated laws, some simple yet generic criteria for determining global synchronization based on parameter identification of uncertain chaotic delayed systems are derived by using the invariance principle of functional differential equations. It is shown that the approaches developed here further extend the ideas and techniques presented in recent literature. Furthermore, the theoretical results are applied to the well-known Chua's circuit and a typical class of chaotic delayed Hopfied neural networks, and numerical simulations also demonstrate the effectiveness and feasibility of the proposed techniques.

This paper considers stabilization of unstable periodic solutions of nonlinear systems. Based on differential geometry method, a nonlinear recursive delayed feedback controller is designed. The concept of γ dynamics is introduced and the stability of the periodic solution of the closed-loop system is proved rigorously. The proposed control method does not have the odd number limitation. Simulation results are also presented for validating the effectiveness of the proposed method.

A novel multiple-output pseudo-random-bit generator (PRBG) based on a coupled map lattice (CML) consisting of skew tent maps, which generates spatiotemporal chaos, is presented. In order to guarantee PRBG highly effective, avoiding synchronization among the sites in the CML is discussed. The cryptographic properties, such as probability distribution, auto-correlation and cross-correlation, of the PRBG with various parameters, are investigated numerically. The randomness of the PRBG is verified via FIPS 140-2. In addition, as compared with the PRBG based on the CML consisting of the logistic maps, which are often used in chaos-based PRBGs by many other researchers, the ranges of the parameters within which this multiple-output PRBG have good cryptographic properties are much bigger in terms of their cryptographic properties. It lays a foundation for designing a faster and more secure encryption.

We study the nonlinear dynamics of a reaction–diffusion equation where the nonlinearity presents a discontinuity. We prove the upper semicontinuity of solutions and the global attractor with respect to smooth approximations of the nonlinear term. We also give a complete description of the set of fixed points and study their stability. Finally, we analyze the existence of heteroclinic connections between the fixed points, obtaining information on the fine structure of the global attractor.

We study a binary-cell-state eight-cell neighborhood two-dimensional cellular automaton model of a quasi-chemical system with a substrate and a reagent. Reactions are represented by semi-totalistic transitions rules: every cell switches from state 0 to state 1 depending on if the sum of neighbors in state 1 belongs to some specified interval, cell remains in state 1 if the sum of neighbors in state 1 belong to another specified interval. We investigate space-time dynamics of 1296 automata, establish morphology-bases classification of the rules, explore precipitating and excitatory cases and scrutinize collisions between mobile and stationary localizations (gliders, cycle life and still-life compact patterns). We explore reaction–diffusion like patterns produced as a result of collisions between localizations. Also, we propose a set of rules with complex behavior called Life 2c22.

Edge smoothing and noise removing for images are a common method of image processing. By designing CNN genes, edges can be smoothed and particles can be removed from a binary image. However, a satisfying result cannot be obtained by choosing only one CNN gene. In this paper, a group of edge smoothing and noise removing CNN genes is proposed as a synthetic disposal for a binary image. Disposed by the group of CNN genes, the characteristics of the original image can be preserved as much as possible. Two examples of edge smoothing and noise removing for a binary image are well illustrated by this method in this paper.

Reinforcement learning is usually required in the process of trial and error called exploration, and the uniform pseudorandom number generator is considered effective in that process. As a generator for the exploration, chaotic sources are also useful in creating a random-like sequence such as in the case of stochastic sources. In this research, we investigate the efficiency of the deterministic chaotic generator for the exploration in learning a nonstationary shortcut maze problem. As a result, it is found that the deterministic chaotic generator based on the logistic map is better in the performance of the exploration than in the stochastic random generator. This has been made clear by analyzing the difference of the performances between the two generators in terms of the patterns of exploration occurrence. We also examine the tent map, which is homeomorphic to the logistic map, compared with other generators.

One purpose of this paper is to document the fact that, in dynamical systems described by ordinary differential equations, the trajectories can be organized not only around fixed points (steady states), but also around lines. In 2D, these lines are the nullclines themselves, in 3D, the intersections of the nullclines two by two, etc.

We precise the concepts of "partial steady states" (i.e. steady states in a subsystem that consists of sections of phase space by planes normal to one of the axes) and of "partial multistationarity" (multistationarity in such a subsystem).

Steady states, nullclines or their intersections are revisited in terms of circuits, defined from nonzero elements of the Jacobian matrix. It is shown how the mere examination of the Jacobian matrix and the sign patterns of its circuits can help interpreting (and often predicting) aspects of the dynamics of systems.

The results reinforce the idea that chaotic dynamics requires both a positive circuit, to provide (if only partial) multistationarity, and a negative circuit, to provide sustained oscillations. As shown elsewhere, a single circuit may suffice if it is ambiguous (i.e. positive or negative depending on the location in phase space).

The description in terms of circuits is by no means exclusive of the classical description. In many cases, a fruitful approach involves repeated feedback between the two viewpoints.

We study the problem of existence of homoclinic and heteroclinic orbits of Chen's system. For the case of 2c > a > c > 0 and b ≥ 2a, we prove that the system has no homoclinic orbit but has two and only two heteroclinic orbits.

Bifurcations of a forced Duffing oscillator in the presence of fuzzy noise are studied by means of the fuzzy generalized cell mapping (FGCM) method. The FGCM method is first introduced. Two categories of bifurcations are investigated, namely catastrophic and explosive bifurcations. Fuzzy bifurcations are characterized by topological changes of the attractors of the system, represented by the persistent groups of cells in the context of the FGCM method, and by changes of the steady state membership distribution. The fuzzy noise-induced bifurcations studied herein are not commonly seen in the deterministic systems, and can be well described by the FGCM method. Furthermore, two conjectures are proposed regarding the condition under which the steady state membership distribution of a fuzzy attractor is invariant or not.

This paper (II) is a continuation of "Complex dynamics in pendulum equation with parametric and external excitations (I)." By applying second-order averaging method and Melnikov's method, we obtain the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 1, 2, 4 and cannot prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5–15 by Melnikov's method, where ν is not rational to ω. However, we show the occurrence of chaos in the averaged and original systems under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5 by numerical simulation. The numerical simulations, include the bifurcation diagram of fixed points, bifurcation diagrams in three- and two-dimensional spaces, homoclinic bifurcation surface, maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors, including period-3 orbits in different chaotic regions, interleaving occurrence of chaotic behaviors and quasi-periodic behaviors, a different kind of interior crisis, jumping behavior of quasi-periodic sets, different nice quasi-periodic attractors, nonchaotic attractors and chaotic attractors, coexistence of three quasi-periodic sets, onset of chaos which occurs more than once for a given external frequency or amplitudes, and quasi-periodic route to chaos. We do not find the period-doubling cascade. The dynamical behaviors under quasi-periodic perturbation are different from that of periodic perturbation.

Most of the systems in an organism (human included) function in a regular daily rhythm. Hypothalamo-pituitary-adrenocortical (HPA) axis, although mostly known for its role in stress response, probably has a role in conveying rhythmic signals from the major pacemaker, suprachiazmatic nucleus (SCN), to the periphery. A general qualitative nonphenomenological mathematical model of the HPA axis is constructed and its dynamics is examined using linear stability analysis and Roushe's theorem. The results show that this system is asymptotically stable, i.e. it does not generate circadian oscillations, but only responds to the external pacemaker.

A master-slave synchronization scheme for Lur'e systems is studied for a known delay existing between master and slave systems. Based on the latest development of stability studies for time-delay systems, a new delay-dependent synchronization criterion is derived by the free-weighting matrix approach. The criterion shown by example is less conservative than the existing synchronization criteria.

This paper proposes a novel complex network with disassortative property based on multicenter networks. The average path length and clustering coefficient of the network are calculated, and the impact on the network topology is investigated. A simple dynamic system established on the proposed network is used to analyze how the disassortative property of the network affects synchronization.

Techniques such as detrended fluctuation analysis (DFA) and its extensions have been widely used to determine the nature of scaling in nucleotide sequences. In this brief communication we show that tandem repeats which are ubiquitous in nucleotide sequences can prevent reliable estimation of possible long-range correlations. Therefore, it is important to investigate the presence of tandem repeats prior to scaling exponent estimation.

This letter introduces two new concepts of S-stability and stationary oscillation for impulsive differential system. Based on the concept of S-stable, we present the relationship of S-stability and stationary oscillation for an impulsive system. Some criterions of stationary oscillation for the impulsive system are established. A numerical example is presented to illustrate the efficiency of the proposed result.

A sliding mode control technique is introduced for generalized fractional chaotic systems. These systems are governed by a set of fractional differential equations of incommensurate orders. The proposed design method relies on the fact that the stability region of a fractional system contains the stability region of its underlying integer-order model. A sliding mode controller designed for an equivalent integer-order chaotic system is used to stabilize all its corresponding fractional chaotic systems. The design technique is demonstrated using two generalized fractional chaotic models; a chaotic oscillator and the Chen system. The effect of the total fractional order is investigated with respect to the controller effort and the convergence rate of the system response to the origin. Numerical simulations validate the main results of this work.