International Journal of Bifurcation and Chaos

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G⁽¹⁾-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

In this communication the relationship between emerging patterns generated by using CNN Universal Machine and patterns appearing in visual art masterpieces is discussed. The template based programming tools are outlined and a gallery of impressive examples is presented. The paper remarks the impressive role of Cellular Nonlinear Networks in covering the gap among art, science and information technology.

We study perturbations of cubic planforms, proving there exists perturbations with homoclinic cycles between persistent steady states. Our results do not depend on the representation of the symmetry group of the lattice, and are thus quite general.

The problem is studied using group theory rather than direct methods. We use the abstract action of the symmetry group of the perturbation on the group orbit to determine the existence of zero- and one-dimensional flow-invariant subspaces. The residual symmetry of the perturbation constrains the flows on these subspaces and, in certain cases, homoclinic cycles are guaranteed to exist.

Cubic planforms are physically interesting due to their relevance to certain physical systems. Applications to reaction–diffusion systems, nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction are discussed.

In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behavior. This is achieved by studying the appearance of inversely unstable solutions within each invariant set.

Intermittent instability is commonly observed in switching power supplies during the design and development phase. It manifests as symmetrical period-doubling bifurcation in the time domain with long intermittent periods. Such intermittent operation is considered undesirable in practice and is usually avoided by appropriate adjustments of circuit parameters. This paper explores the mechanism and conditions for the emergence of intermittency in a common voltage-mode controlled buck converter. It is found that interference at frequencies near the switching frequency or its rational multiples will induce intermittent operation. The strengths and frequencies of the interfering signals determine the type and period of intermittency. The problem is analyzed by transforming the conventional parameter-bifurcation analysis to a time-bifurcation analysis. Analytical results are verified by simulations and experimental measurements.

Whether or not the human cardiac system is chaotic has long been a subject of interest in the application of nonlinear time series analysis. The surrogate data method, which identifies an observed time series against three common kinds of hypotheses, does not provide sufficient evidence to confirm the existence of deterministic chaotic dynamics in cardiac time series, such as electrocardiogram data and pulse pressure propagation data. Moreover, these methods fail to exclude all but the most trivial hypothesis of linear noise. We present a recently suggested fourth algorithm for testing the hypotheses of a noise driven periodic orbit to decide whether these signals are consistent with deterministic chaos. Of course, we cannot exclude all other alternatives but our test is certainly stronger than the those applied previously. The algorithmic complexity is used as the discriminating statistic of the surrogate data method. We then perform nonlinear modeling for the short-term prediction between ECG and pulse data to provide further evidence that they conform to deterministic processes. We demonstrate the application of these methods to human electrocardiogram recordings and blood pressure propagation in the fingertip of seven healthy subjects. Our results indicate that bounded aperiodic determinism exists in both ECG and pulse time series. The addition of (the inevitable) dynamic noise means that it is not possible to conclude the underlying system is chaotic.

The density of the measure-theoretic directional entropy for the lattice dynamical system is introduced and it is shown that for a lattice dynamical transformation commuting with the shift transformation the density coincides with the entropy of the ℤ²-action generated by these two transformations, i.e. it is a constant function with respect to the direction. It is also proved that in the noncommutative case this result fails to be true.

In this paper, a new comparison method is developed by using increasing and decreasing mechanisms, which are inherent in time-delay systems, to decompose systems. Based on the new method, whose expected performance is compared with the state of the original system, some new conditions are obtained to guarantee that the original system tracks the expected values. The locally exponential convergence rate and the convergence region of the polynomial differential equations with time-varying delays are also investigated. In particular, the comparison method is used to improve the 3/2 stability theorems of differential systems with pure delays. Moreover, the comparison method is applied to identify a threshold, and to consider the disease-free equilibrium points of an HIV endemic model with stages of progress to AIDs and time-varying delay. It is shown that if the threshold is smaller than 1, the equilibrium point of the model is globally, exponentially stable. Another application of the comparison method is to investigate the global, exponential stability of neural networks, and some new theoretical results are obtained. Numerical simulations are presented to verify the theoretical results.

In this paper, a new approach to analyze the asymptotic, exponential and robust stability of the master-slave synchronization for Lur'e systems using time-varying delay feedback control is proposed. The discussion is motivated by the problem of transmitting information in optical communication systems using chaotic lasers. The approach is based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique with the use of a recent Leibniz–Newton model based transformation, without including any additional dynamics. Using the problem of synchronizing coupled Chua's circuits, three examples are given to illustrate the effectiveness of the proposed methodology.

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays x_n = f(n - 1, x_n-k). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p, k) nonautonomous p/gcd(p, k)-periodic difference equations. We give formulas for calculating the number of different periodic orbits under certain conditions. In addition, when p and k are relatively prime integers, we introduce what we call the pk-Sharkovsky's ordering of the positive integers, and extend Sharkovsky's theorem to periodic difference equations with delays. Finally, we characterize global stability and show that the period of a globally asymptotically stable orbit must be divisible by p.

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

In this paper, a chaotic attractor gallery obtained from a new higher dimensional chaotic circuit combining LMT (Lindberg–Murali and Tamasevicious) circuit and Chua's circuit is presented. This circuit is capable of exhibiting a rich variety of chaotic attractors including temporary multiscroll attractors.

Information signal embedded in a chaotic transmitter can be recovered by a receiver if it is a replica of the transmitter. In this paper, a new aspect of chaotic communication is introduced. A sliding-mode observer replaces the conventional chaotic system at the receiver side, which does not need information from the transmitter. So the uncertainties in the transmitter and the transmission line do not affect the synchronization, the proposed communication scheme is robust with respect to some disturbances and uncertainties. Three chaotic systems, Duffing equation, Van der Pol oscillator and Chua's circuit, are provided to illustrate the effectiveness of the chaotic communication.

This paper considers the problem of stability analysis for neural networks with time-varying delays. The time-varying delays under consideration are assumed to be bounded but not necessarily differentiable. In terms of a linear matrix inequality, a delay-dependent asymptotic stability condition is developed, which ensures the existence of a unique equilibrium point and its global asymptotic stability. The proposed stability condition is easy to check and less conservative. An example is provided to show the effectiveness of the proposed condition.

In recent years, a growing number of cryptosystems based on chaos have been proposed. But most of them have encountered many problems such as small key space and weak security. In the present paper, a new kind of chaotic cryptosystem based on Composition of Trigonometric Chaotic Maps is proposed. These maps which are defined as ratios of polynomials of degree N, have interesting properties such as invariant measure, ergodicity, variable chaotic region with respect to the control parameters and ability to construct composition form of maps. We have used a composition of chaotic map to shuffle the position of image pixels. Another composition of chaotic map is used in diffusion process. According to the performed analysis, the introduced algorithm can satisfy the required performances such as high level security, large key space and the acceptable encryption speed.

In this paper, we propose a fuzzy model-based methodology to deal with various control objectives for discrete-time chaotic systems from a unified viewpoint. With intent to unify the design process, we introduce a new design concept called virtual-desired-variable synthesis. Then, both the chaotic control and chaotification are eventually treated as a stabilization problem. Consequently, the conditions concerning the stability of the closed-loop system are formulated into LMIs. A feasible solution of the LMI problem guarantees the quadratical stability and gives the state feedback gains as well. The well-known Hénon map is used to demonstrate the unified approach.

In this paper we analyze a simple mathematical model which describes the interaction between proteins P53 and Mdm2. For the stationary state we discuss the local stability and the existence of a Hopf bifurcation. We study the direction and stability of the bifurcating periodic solutions by choosing the delay as a bifurcation parameter. Finally, we will offer some numerical simulations and present our conclusions.