International Journal of Bifurcation and Chaos

We consider stochastic delay differential equations of the form

Rather diverse components of mathematical background are necessary to understand the questions of interest. In this paper we first review some deterministic results and some basic elements of the stochastic analysis that (i) suggests lines of investigation for the stochastic case and (ii) are expected to facilitate the theoretical investigation of the stochastic problem. We then present the results of numerical experiments that illustrate some of the complexities that arise when considering bifurcations in stochastic delay differential equations. They give prima facie evidence for certain convergence properties of the bifurcation points estimated using the Euler–Maruyama method for the equations considered. We conclude by drawing attention to a number of open questions in the field.

In this paper, an approach based on ergodic properties for classifying chaotic sequences is given. It is particularly robust due to the open-loop structure of the detector. Unlike previous works in this direction, such as those concerning the inverse system approach, the detector is not uniquely determined by the transmitter (identical or subsystem thereof), but instead depends on the measurement noise model. The classification is optimized under such imperfect observation conditions. The method is introduced in general for the case of chaotic sequences generated by ergodic maps, and a special case is analyzed in detail to illustrate the method. This special example resorts to Tchebychev maps and some additional symmetries to make up a simple signaling scheme which is low in complexity on both transmitter and receiver sides, while at the same time relatively robust, due to the open-loop structure of the detector.

This paper illustrates the recent phenomenon of chaotic beats in a modified version of Chua's circuit, driven by two sinusoidal inputs with slightly different frequencies. In order to satisfy the constraints imposed by the beats dynamics, a novel implementation of the voltage-controlled characteristic of the Chua diode is proposed. By using Pspice simulator, the behavior of the designed circuit is analyzed both in time-domain and state-space, confirming the chaotic nature of the phenomenon and the effectiveness of the approach.

In this paper, a class of MIMO nonlinear systems are studied. Some frequency domain conditions are established for the property of dichotomy. These kinds of systems can also be viewed as a class of interconnected systems composed of SISO systems through some linear and nonlinear interconnections. A class of nonlinear input and output interconnections are presented. The corresponding condition for testing dichotomy is given. Furthermore, Chua's circuit and interconnected Chua's circuit are studied to illustrate the theoretical results.

This paper investigates limit cycling behavior of observer-based controlled mechanical systems with friction compensation. The limit cycling is induced by the interaction between friction and friction compensation, which is based on the estimated velocity. The limit cycling phenomenon, which is experimentally observed in a rotating arm manipulator, is analyzed through computational bifurcation analysis. The computed bifurcation diagram confirms that the limit cycles can be eliminated by enlarging observer gains and controller gains at the cost of a steady state error. The numerical results match well with laboratory experiments.

A model which allows a double impacting regime for a particle undergoing simple harmonic motion is considered in some detail. The behavior of the particle in the weak spring limit is considered. Symmetries of the motion are found and the extent of the resonant dynamical behavior is considered. Control equations are developed and strategies are described for both the preservation and the annihilation of experimental and analytical resonant periodic orbits.

A system that consists of a damped oscillator impacting two immovable fenders has been considered in this paper. In the first part a method of analytical determination of the existence of periodic solutions to the equations of motion and a method of analysis of the stability of these solutions have been presented. The results of the computations carried out by means of these methods have been illustrated by a few examples. In the second part of the paper, the results of some numerical investigations have been presented. The goal of these studies was to determine, in which regions of parameters characterizing the system, the motion of the oscillator is periodic and stable.

In a normal power system, many generators are operating in synchrony. That is, they all have the same speed or frequency, the system frequency. In case some accident occurs, a situation might arise in which one or more generators are running at a different speed, much faster than the system frequency. They are said to be running away or stepping out, or in a state of accelerated stepping out. We have been engaged in a series of studies of this situation, and have found global attractor-basin portraits. In the course of this program, we have observed the phenomenon of decelerated stepping out, in which one or more generators deviate from the system frequency toward lower speeds. These kinds of behavior cannot be explained with the well-known model involving one generator operating on to an infinite bus. Rather, we require a model in which robust subsystems — for example, generator/motor combination, which we call swing pairs — are connected by interconnecting transmission lines. In this more general context, the deviant behaviors we are considering may be regarded as forms of desynchronization of subsystems. We therefore begin this paper with the derivation of a new mathematical model, in which there is no infinite bus nor fixed system frequency. In the simple case of two subsystems (each a swing pair) weakly coupled by an interconnecting transmission line, we develop a system of seven differential equations which include the variation of frequency in a fundamental way. We then go on to study the behavior of this model, using our usual methods of computer simulation to draw the attractor-basin portraits. We have succeeded in finding both accelerated and decelerated stepping out in this new model. In addition, we discovered an unexpected subharmonic swinging of the whole system.

The orbital evolution and stability of planetary systems with interaction from the belts is studied using the standard phase-plane analysis. In addition to the fixed point which corresponds to the Keplerian orbit, there are other fixed points around the inner and outer edges of the belt. Our results show that for the planets, the probability to move stably around the inner edge is larger than the one to move around the outer edge. It is also interesting that there is a limit cycle of semi-attractor for a particular case. Applying our results to the Solar System, we find that our results could provide a natural mechanism to do the orbit rearrangement for the larger Kuiper Belt Objects and thus successfully explain the absence of these objects beyond 50 AU.

By applying the undetermined coefficient method, this paper finds homoclinic and heteroclinic orbits in the Chen system. It analytically demonstrates that the Chen system has one heteroclinic orbit of Ši'lnikov type that connects two nontrivial singular points. The Ši'lnikov criterion guarantees that the Chen system has Smale horseshoes and the horseshoe chaos. In addition, there also exists one homoclinic orbit joined to the origin. The uniform convergence of the series expansions of these two types of orbits are proved in this paper. It is shown that the heteroclinic and homoclinic orbits together determine the geometric structure of Chen's attractor.

This study investigates bifurcations and chaos in two-cell Cellular Neural Networks (CNN) with periodic inputs. Without the inputs, the time periodic solutions are obtained for template A=[r,p,s] with p>1, r>p-1 and -s>p-1. The number of periodic solutions can be proven to be no more than two in exterior regions. The input is bsin2πt/T with period T>0 and amplitude b>0. The typical trajectories Γ(b,T,A) and their ω-limit set ω(b,T,A) vary with b, T and A are also considered. The asymptotic limit cycles Λ_∞(T,A) with period T of Γ(b,T,A) are obtained as b→∞. When (given in (67)), Λ_∞ and -Λ_∞ can be separated. The onset of chaos can be induced by crises of ω(b,T,A) and -ω(b,T,A) for suitable T and b. The ratio , of largest amplitude a₁(b) except for T-mode and amplitude of the T-mode of the Fast Fourier Transform (FFT) of Γ(b,T,A), can be used to compare the strength of sustained periodic cycle Λ₀(A) and the inputs. When , Λ₀(A) dominates and the attractor ω(b,T,A) is either a quasi-periodic or a periodic. Moreover, the range b of the window of periodic cycles constitutes a devil's staircase. When , finitely many chaotic regions and window regions exist and interweave with each other. In each window, the basic periodic cycle can be identified. A sequence of period-doubling is observed to the left of the basic periodic cycle and a quasi-periodic region is observed to the right of it. For large b, the input dominates, ω(b,T,A) becomes simpler, from quasi-periodic to periodic as b increases.

Phase synchronization and entrainment of a linear array of loss modulated Nd:YAG lasers are experimentally investigated. The temporally shifting nature of phase synchronization with increasing modulation depth is studied. Chaotic intensity dynamics changes dramatically with increasing modulation, and exhibit entrainment to the modulation signal. Frequency filtered phase variables, defined from the laser intensity time series, uncover phenomena of phase synchronization and competition for phase synchronization that are otherwise hidden.

Standard Cellular Automata (CA) are ahistoric (memoryless): i.e. the new state of a cell depends on the neighborhood configuration only at the preceding time step. This article introduces an extension to the standard framework of CA by considering automata implementing memory capabilities. While the update rules of the CA remains the same, each site remembers a weighted mean of all its past states, with a decreasing weight of states farther in the past. The historic weighting is defined by a geometric series of coefficients based on a memory factor (α). This paper considers the time evolution of one-dimensional range-two CA with memory.

A new and fast algorithm is presented in this paper for the automatic generation of aesthetic patterns on aperiodic tilings by means of dynamical systems. The Penrose tilings are used as illustration of which invariant mappings are constructed for the creation of striking patterns. A modified convergence time scheme is described to enhance the artistic appeal of generated images. This algorithm can be used to create a great variety of exotic patterns with various aperiodicities.

Intrinsic fluctuation of Chua system is studied with master equation method. Our results have shown that the intrinsic noise indeed exerted considerable influence on Chua system. In contrast to that of deterministic equation the patterns of time evolution and attractor have been greatly altered under the influence of intrinsic noise.

The impulsive synchronization method has been applied to several well-known chaotic circuits and systems such as Chua's circuit, Lorenz system and hyperchaotic circuit in the literature. In this paper, we also present two impulsive synchronization studies using SC-CNN-based circuit and Chua's circuit. In the first study, we have investigated the impulsive synchronization between two SC-CNN-based circuits. Pspice simulation results show that two SC-CNN-based circuits can be synchronized impulsively via x₁ and x₂ cell dynamics for different impulse width and impulse period values. And in the second study, we have investigated the impulsive synchronization between SC-CNN-based circuit and Chua's circuit. Pspice simulation results verify that two chaotic circuits, which have identical dynamical systems via appropriate parameter transformations but having quite different hardware implementations, can be synchronized impulsively for different impulse width and impulse period values.

This paper derives some sufficient conditions for the stabilization and synchronization of n-scroll attractors via impulsive control with varying impulsive intervals. Some sufficient conditions for the stabilization and synchronization of n-scroll attractors are obtained. The results are illustrated to be efficient through a numerical example and simulation results.

The quenching of alternans is considered using a nonlinear cardiac conduction model. The model consists of a nonlinear discrete-time piecewise smooth system. Several authors have hypothesized that alternans arise in the model through a period-doubling bifurcation. In this work, it is first shown that the alternans exhibited by the model actually arise through a period-doubling border collision bifurcation. No smooth period-doubling bifurcation occurs in the parameter region of interest. Next, recent results of the authors on feedback control of border collision bifurcation are applied to the model, resulting in control laws that quench the bifurcation, and hence result in alternan suppression.

This Letter improves the Chen–Lai chaotification algorithm for discrete-time systems from multiple-input to single-input state feedback, while preserving its mathematical rigor.

In this paper, we study the number of limit cycles in a family of polynomial systems. Using bifurcation methods, we obtain the maximal number of limit cycles in global bifurcation.

A sufficient condition for the uniqueness of the Nth order normal form is provided. A new grading function is proposed and used to prove the uniqueness of the first-order normal forms of generalized Hopf singularities. Recursive formulas for computation of coefficients of unique normal forms of generalized Hopf singularities are also presented.

We discuss the experimental implementation of a chemical controller for a robotic hand. In the present case study, we have designed a closed system where a Belousov–Zhabotinsky (BZ) thin layer chemical reactor was linked to the robotic hand via an array of photo-sensors and the fingers of the hand were able to stimulate the excitation dynamics in the BZ medium via the local addition of an activator species. A principal working loop of the chemo-robotic system is that oxidation wave fronts traveling in the medium are detected by photo-sensors and cause (via a microcontroller) the fingers of the hand to bend. When a finger bends, it is set up to apply a small quantity of colloid silver to the reaction and thus causes an additional excitation wave. The traveling and interacting waves stimulate further movements of the fingers and patterns of behavior are observed. These patterns of behavior have been modeled using a cellular automaton. In the paper, we offer an experimental set-up, including algorithms and interfacing, for an experimental chemical robotic hand controller, which contributes to the fields of nonclassical computation, nonlinear physics, and unconventional robotics.

We present an experimental study of the generation of strongly localized structures that propagate on the surface of a dissipative fluid. We excited a layer of fluid with a vertical periodic acceleration field, and a parametric instability occurs when a certain threshold value is achieved. This process is known as Faraday Instability and the temporal evolution of the system obeys a period-doubling route. For a highly dissipative fluid we observed two new interesting phenomena: the generation of high spatially localized structures which propagate on top of the stripes of stationary pattern, and a periodic window which occurs after the system reached spatiotemporal chaos.

There is considerable spatial heterogeneity in the electrical properties of the heart muscle and there are indications that anisotropic conduction may play an important role in the pathogenesis of clinical cardiac arrhythmias. Spiral waves of electrical activity are related to reentrant cardiac arrhythmias as ventricular tachycardia and ventricular fibrillation, and the generation of a wave breakup is hypothesized to underlie the transition from ventricular tachycardia to ventricular fibrillation — the leading cause of sudden cardiac death. Here we investigate the effect of heterogeneity on spiral wave reentry in a two-dimensional modified FitzHugh–Nagumo membrane model. Spiral wave breakup induced by the heterogeneity is found. The spiral wave dynamics is invariant under translational and rotational transformations in homogeneous tissue, but for heterogeneous tissue, this symmetry is broken due to the heterogeneity. The reentry dynamics depends on the degree of heterogeneity and the point where the reentry is initiated within the simulated tissue. This study may open potentially exciting new diagnostic and therapeutic possibilities in a clinical context.

Some sufficient conditions for the asymptotic stability of cellular neural networks with time delay are derived using the Lyapunov–Krasovskii stability theory for functional differential equations as well as the linear matrix inequality (LMI) approach. The analysis shows how some well-known results can be refined and generalized in a straightforward manner. Moreover, the stability criteria obtained are delay-independent. They are less conservative and restrictive than those reported so far in the literature, and provide a more general set of criteria for determining the stability of delayed cellular neural networks.