International Journal of Bifurcation and Chaos

Chua Oscillator and its generalizations display a variety of chaotic behaviors, whose most startling manifestation is the presence of strange attractors. These come in many different shapes and sizes yet share a special kind of beauty. In the work reported in this paper, we explored the universe of these attractors — much of which is still virgin territory. We then recorded the most interesting and significant in a "Gallery of Chua attractors". In papers, published in previous issues of this journal, we showed how different versions of the Oscillator follow different "routes to chaos". Parts II–IV of the Gallery describe the attractors generated by the dimensional and dimensionless forms of the Oscillator as well as attractors generated by circuits with a cubic equation for the characteristic function of the Chua diode. Here, we provide a detailed discussion of single scroll Chua systems and present 248 attractors generated by such systems. For each attractor, we provide a three-dimensional image, time series and FFTs. We go on to describe the techniques used to create the Gallery and the main characteristics of the attractors it includes. We use techniques such as PCA to represent the gallery in parameter space. The same techniques allow us to manipulate the shape of the attractors by enlarging them along their main axes of development. We use Hausdorff distances to compare shapes, and exploit the results to create landscapes in parameter space. Finally, we present experimental data from a single scroll attractor, using inertial ellipsoids and Hausdorff distances to show how the shape of the attractor evolves.

In this paper, SEIS epidemic models with varying population size are considered. Firstly, we consider the case when births of population are throughout the year. A threshold σ is identified, which determines the outcome of disease, that is, when σ < 1, the disease dies out; whereas when σ > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable; when σ = 1, bifurcation occurs and leads to "the change of stability". Two other thresholds σ′ and are also identified, which determine the dynamics of epidemic model with varying population size, when the disease dies out or it is endemic. Secondly, we consider the other case, birth pulse. The population density is increased by an amount B(N)N at the discrete time nτ, where n is any non-negative integer and τ is a positive constant, B(N) is density-dependent birth rate. By applying the corresponding stroboscopic map, we obtain the existence of infection-free periodic solution with period τ. Lastly, through numerical simulations, we show the dynamic complexities of SEIS epidemic models with varying population size, there is a sequence of bifurcations, leading to chaotic strange attractors. Non-unique attractors also appear, which implies that the dynamics of SEIS epidemic models with varying population size can be very complex.

The primary purpose of this paper is to show that simple dissipation can bring about oscillations in certain kinds of asymptotically stable nonlinear dynamical systems; namely when the system is locally active where the dissipation is introduced. Furthermore, if these nonlinear dynamical systems are coupled with appropriate choice of diffusion coefficients, then the coupled system can exhibit spatio-temporal oscillations. The secondary purpose of this paper is to show that spatio-temporal oscillations can occur in spatially discrete reaction diffusion equations operating on the edge of chaos, provided the array size is sufficiently large.

We perform an analytical study of the external cavity modes of a semiconductor laser subject to conventional optical feedback as modeled by the well-known Lang–Kobayashi equations. Specifically, the bifurcation set is derived in the three-dimensional parameter space of feedback phase, feedback strength and pump current of the laser. Different open regions in this space correspond to different numbers of physically relevant external cavity modes of the laser. Some of their stability properties are determined from the characteristic equation.

The magnetic field line structure of tokamaks with reversed magnetic shear is analyzed by means of a nontwist map model that takes into account non-integrable perturbations that describe ergodic magnetic limiters. The map studied possess behavior expected of the standard nontwist map, a well-studied map, despite the different symmetries and the existence of coupled perturbations. A distinguising feature of nontwist maps is the presence of good surfaces in the reveresed shear region, and consequently the appearance of a transport barrier inside the plasma. Such barriers are observed in the present model and are seen to be very robust. Very strong perturbations are required to destroy them, and even after breaking, the transport turns out to be diffusive. Poloidal diffusion is found to be two orders of magnitude higher than radial diffusion.

A systematic experimental study of the Faraday instability in viscoelastic fluid is presented. We have used a shear thinning polymer solution in which the elastic effects are predominant within our work range. We have analyzed the dependence of the threshold instability as a function of the depth in layer. Depending on the fluid layer depth and the driving frequency, harmonic or subharmonic regimes are developed. We have focused our work on the subharmonic region and temporal and spatial behaviors were analyzed. In addition, we have used the onset acceleration to estimate the rheological properties of the fluid. These predictions are supported by experimental measurements.

This paper investigates the interaction of fast-scale and slow-scale bifurcations in the boost converter under current-mode control operating in continuous conduction mode. Effects of varying some chosen parameters on the qualitative behaviors of the system are studied in detail. Analysis is performed to identify the different types of bifurcation. Boundaries of stable region, slow-scale bifurcation region, fast-scale bifurcation region, interacting fast and slow-scale bifurcation regions are identified.

A frequency domain method is used to estimate the harmonic contents of a smooth oscillation arising from the Hopf bifurcation mechanism. The harmonic contents up to the eighth-order are well estimated, which agree with the results obtained from a completely different approach that measures the frequency content of a signal by using digital signal processing techniques such as the Fast Fourier Transform (FFT). The accuracy of the approximation is evaluated by computing the Floquet multipliers of the variational system based on the fact that for periodic solutions one multiplier must be +1. The separation from this theoretical value is proportional to the error of the approximation. A limitation of the frequency domain method is encountered when being used for continuing the secondary branch of limit cycle bifurcations, such as pitchfork and period-doubling bifurcations. Two examples are shown to illustrate the main results: Colpitts' oscillator with a pitchfork bifurcation of cycles, and Chua's circuit with a period-doubling bifurcation of cycles.

In this paper, higher-dimensional periodic and chaotic oscillations for a parametrically excited viscoelastic moving belt with multiple internal resonances are investigated for the first time. The external damping and internal damping of the material for the viscoelastic moving belt are considered simultaneously. First, the nonlinear governing equation of planar motion for the viscoelastic moving belt with the external damping is given. Then, the transverse nonlinear oscillations of the viscoelastic moving belt are considered. The method of multiple scales and the Galerkin approach are applied directly to the governing partial differential equation of motion for the viscoelastic moving belt to obtain an eight-dimensional averaged equation for the case of 1:2:3:4 internal resonances for the first-, the second-, the third- and the fourth-order modes and primary parametric resonance of the first-order mode. Finally, numerical method is used to investigate higher-dimensional periodic and chaotic motions of the viscoelastic moving belt. The results of numerical simulation demonstrate that there exist the period, period 2, period 4, multiple period and chaotic motions of the viscoelastic moving belt. The multipulse chaotic motions of the viscoelastic moving belt are observed from numerical simulations.

The least squares procedure is suggested to reveal weak modulation of the control parameters in a noisy chaotic map in the presence of both dynamical and observational noise. For a map of a known dynamical structure, which is supposed to depend linearly on modulated control parameters, the procedure allows to detect both harmonic and pulse modulation, reducing the problem of modulation extraction from noisy empirical time series to the classical scheme of signal detection in noisy data. As an example, detection of harmonic modulation in a logistic map is studied. It is shown that threshold amplitude is subjected to the influence of the noise enhancement phenomenon in chaotic systems. Despite noise enhancement, the method suggested is shown to be superior to direct spectral analysis of chaotic time series due to wideband noise-like spectrum of the chaotic map, which masks the weak control parameters modulation.

In this paper, we study the dynamical behavior of nonautonomous, almost-periodic discrete FitzHugh–Nagumo system defined on infinite lattices. We prove that the nonautonomous infinite-dimensional system has a uniform attractor which attracts all solutions uniformly with respect to the translations of external terms. We also establish the upper semicontinuity of uniform attractors when the infinite-dimensional system is approached by a family of finite-dimensional systems. This paper is based on a uniform tail method, which shows that, for large time, the tails of solutions are uniformly small with respect to bounded initial data as well as the translations of external terms. The uniform tail estimates play a crucial role for proving the uniform asymptotic compactness of the system and the upper semicontinuity of attractors.

New methods of identifying the transition rule of a Belousov–Zhabotinskii (BZ) reaction directly from experimental data using cellular automata (CA) models are investigated. The experimental set-up and new techniques for image pre-processing to ensure the identification of representative models are discussed including noise reduction, pixel and color calibration. Two kinds of models, the Greenberg–Hasting model (GHM) and the polynomial CA model are studied in detail. It is shown that the results of identifying a real BZ reacting system are very encouraging and the predicted patterns compare well with the imaged patterns both visually and quantitatively.

Video compression technologies have recently become an integral part of the way we create, communicate and consume visual information. The aim of this Letter is to show that the Cellular Neural Network (CNN) paradigm can be exploited for obtaining accurate video compression. In particular, the Letter presents an architecture that combines CNN algorithms and H.264 codec. The compression capabilities of the devised coding system are analyzed in detail using some benchmark video sequences, and comparisons are carried out between the CNN-based approach and the H.264 codec working alone. The outcome of the analysis is that the CNN-based coding approach outperforms the H.264 codec working alone, allowing to perceive the capabilities of the CNN paradigm.

We propose a CMOS device that is analogous to the reaction–diffusion system, a chemical complex system that produces various dynamic phenomena in the natural world. This electrical reaction–diffusion device consists of an array of pn junctions that are operated by CMOS reaction circuits and interact with each other through minority-carrier diffusion. Computer simulations reveal that the device can produce animated spatiotemporal carrier concentration patterns, e.g. expanding circular patterns and rotating spiral patterns that correspond to the dissipative structures produced by chemical reaction–diffusion systems.

We study the effects of surface stress on the dynamics of a solid interface which evolves by surface diffusion. In the absence of surface stress, it is known that a flat interface is unstable for long wavelength perturbations and that in the nonlinear regime this instability develops into the formation of a sharp cusp. We compute the stability spectrum for a circular pore and for a flat interface considering the surface stress contribution, and analyze its effect on the process of formation of the cusp. Analyzing the length scales involved in the problem, we show that, within the context of continuum elasticity theory, surface stress does not suppress the singularity formation.

The noise-like chaotic signal can be generated with very simple nonlinear circuits, and has broad bandwidth and aperiodic properties. These characteristics have drawn considerable attention in the radar community. In the past, the chaotic signal or its modulated version serves as a transmitting signal, and the traditional correlation-type receiver is used for processing. In this sense, the chaotic signal acts as a radar waveform in the noise signal radar, and hence the performance advantages are not distinct. Here, we present a scheme for processing chaotic radar signals. We find a simple relation between the target parameters (range and velocity) and the system parameters of chaos-generating system. With this relation, the measurement of the target parameters is transformed into estimation of the system parameters from the radar return signals. Equipped with high resolution parameter estimation techniques, the proposed principles provide a way to develop high resolution noise signal radars.

In this paper we consider the problem of estimating the parameters of a nonlinear dynamical system given a finite time series of observations that are contaminated by observational noise. The least squares method is a standard method for parameter estimation, but for nonlinear dynamical systems it is well known that the least squares method can result in biased estimates, especially when the noise is significant relative to the nonlinearity. In this paper, it is demonstrated that by combining nonlinear noise reduction and least squares parameter fitting it is possible to obtain more accurate parameter estimates.

Synchronization via multivariable PID control is studied. Based on the descriptor approach, the problem of PID controller design is transformed to that of static output feedback (SOF) controller design. The improvement of the solvability of the Linear Matrix Inequality (LMI) is achieved, in comparison with the existing literature on designing PID controller based on the LMI technique. With the aid of the free-weighting matrix approach and the S-procedure, the synchronization criterion for a general Lur'e system is established based on the LMI technique. The feasibility of the methodology is illustrated by the well-known Chua's circuit.

In this paper, we consider a new family of modified hyperchaotic Rössler systems, recently studied by Nikolov and Clodong using proper nonlinear feedback controllers. Particular attention is given to (i) globally exponential lag synchronization (GELS) for τ > 0; and (ii) globally exponential synchronization (GES) for τ = 0. As a representative example, one system of the family of modified hyperchaotic Rössler systems is particularly studied, and Lyapunov stability criteria for the GELS and GES are derived via eight families of proper nonlinear feedback controllers. Moreover, we also present some nonlinear feedback control laws for other modified hyperchaotic Rössler systems. Numerical simulations are used to illustrate the theoretical results.

We investigate in this Letter different dynamical states in the ring of four mutually coupled self-sustained electrical systems with time periodic coupling. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived using the Floquet theory and the Whittaker method. The effects of the amplitude of the periodic parametric perturbations of the coupling parameter on the stability boundaries are analyzed. Numerical simulations are then performed to complement the analytical results.

This paper proposes a module-based and unified approach to chaotic circuit design, where the description is based on the state equations without physical dimensions for simplicity of a general discussion. The main design process consists of transformation of state variables, transformation from differential to integral operations, and transformation of the time-scale. The designed circuit consists of anti-adder module integrator module, and inverter module. A novel 3-scroll Chua's circuit and a generalized Lorenz-like circuit are designed and implemented for verifying the effectiveness of this systematic circuit design methodology. Experimental observations are provided for confirmation. Comparing with the traditional circuit design methods, this new design approach has the following typical characteristics: (i) module-based and unified design; (ii) independent adjustment of system parameters; (iii) adjustment of distribution regions for the frequency spectra of chaotic signals; (iv) prominent observability.