International Journal of Bifurcation and Chaos

In this tutorial paper, we present some interesting phenomena from Chua's equation with a cubic nonlinearity as well as that with a piecewise-linear characteristic, where a cubic polynomial approximates the original three-segment piecewise-linear function. A gallery of attractors and bifurcation diagrams obtained by numerical simulations are presented. We hope this will motivate researchers to study the smooth version of this extremely simple yet versatile equation with more than 20 attractors.

The subject of moving curves (and surfaces) in three-dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a variety of physical problems in different disciplines. Making use of the underlying geometry, one can very often relate the associated evolution equations to many interesting nonlinear evolution equations, including soliton possessing nonlinear dynamical systems. Typical examples include dynamics of filament vortices in ordinary and superfluids, spin systems, phases in classical optics, various systems encountered in physics of soft matter, etc. Such interrelations between geometric evolution and physical systems have yielded considerable insight into the underlying dynamics. We present a succinct tutorial analysis of these developments in this article, and indicate further directions. We also point out how evolution equations for moving surfaces are often intimately related to soliton equations in higher dimensions.

In many industrial applications imbalance is a major cause for unwanted vibrations. One way to compensate for an unknown imbalance is the implementation of an automatic ball balancer. Oil-lubricated automatic ball balancers are applied in hand-held tools and washers. However, in applications such as optical drives fluid lubrication is highly undesirable since it may destroy the optical system upon leakage. Therefore, in this paper, the balancing performance of an automatic ball balancer without fluid lubrication is investigated. The absence of fluid lubrication gives rise to dry friction phenomena which cause the existence of equilibrium sets of the balls in the automatic ball balancer. A model of the system with dry friction, modeled by a set-valued force law, is built based on dedicated experiments. The resulting equilibrium sets and their dependency on system parameters are studied and the consequences for the balancing performance is assessed. Based on these results, it can be concluded that in parts of such equilibrium sets the balancing performance deteriorates when compared to the system without automatic ball balancer; in other words, the balancing performance is endangered by the presence of dry friction. This conclusion is supported by both numerical and experimental results.

In this paper, we discuss a new mechanism for chaos in light of some new developments in the theory of dynamical systems. It was shown in [Wang & Young, 2002b] that strange attractors occur when an autonomous system undergoing a generic Hopf bifurcation is subjected to a weak external forcing that is periodically turned on and off. For illustration purposes, we apply these results to the Chua's system. Derivation of conditions for chaos along with the results of numerical simulations are presented.

Nonlinear dynamical models are frequently used to approximate and predict observed physical, biological and economic systems. Such models will be subject to errors both in the model dynamics, and the observations of the underlying system. In order to improve models, it is necessary to understand the causes of error growth. A complication with chaotic models is that small errors may be amplified by the model dynamics. This paper proposes a technique for estimating levels of both dynamical and observational noise, based on the model drift. The method is demonstrated for a number of models, for cases with both stochastic and nonstochastic dynamical errors. The effect of smoothing or treating the observations is also considered. It is shown that use of variational smoothing techniques in the presence of dynamical model errors can lead to potentially deceptive patterns of error growth.

In this article, we propose and study an extension of the Chen–Lai algorithm for chaotification of discrete-time dynamical systems. The proposed method is a simple but mathematically rigorous feedback control design method that can gradually make all the Lyapunov exponents of the controlled system strictly positive for any given n-dimensional dynamical system that has a uniformly bounded Jacobian but otherwise could be originally nonchaotic or even asymptotically stable.

This paper first presents a theorem for constructing a unidirectionally coupled array of differential equations (ADE) such that a driving system and a driven system achieve generalized synchronization (GS) via a set of invertible C¹-diffeomorphism transformations. Secondly, a new secure communication scheme for digital image transformation is developed based on the theorem. Finally, a Chua's ADE with GS property is introduced, based on a prototype introduced by Chua as a two-dimensional dual-layer two-dimensional reaction–diffusion cellular neural network (CNN), and is applied to construct a secure image communication scheme.

In this paper, a Z₂-equivariant perturbed planar Hamiltonian polynomial vector field of degree 5 is used to show how to control parameter conditions such that this system has more limit cycles and various configurations of these limit cycles. By using the method of detection functions, we prove that under different determined parameter conditions, the given system has at least 20–23 limit cycles having different configurations of the compound eyes. We give a corollary to point out that the assertion is incorrect that this system exists as 29 limit cycles in a reference.

In this paper, we consider the Klein–Gordon–Schröodinger equation defined on Rⁿ (n ≤ 3) and Ω_m = {x ∈ Rⁿ : |x| ≤ m}. Let and be the global attractors of the equation corresponding to Rⁿ and Ω_m, respectively. Then we prove that for any neighborhood U of , the global attractor enters U when m is large enough.

A Cellular Neural Networks (CNN)-based nonlinear pattern classification algorithm utilizing the most likely path-finding feature of dynamic programming is proposed. Dynamic programming for the most likely path-finding algorithm can be implemented with CNN. If exemplars and test patterns are assigned as the goals and the start positions, respectively for our CNN-based dynamic programming, the paths from the test patterns to their closest exemplars are found with the optimality feature of CNN-based dynamic programming. Such paths are utilized as aggregating keys for classification. Our algorithm is suitable for patterns with nonlinear pattern boundaries. Simulation results are included.

The problem of stabilizing the Chua circuit about equilibrium points is addressed in this paper. Singular perturbation methods are used to prove that semiglobal stabilization of the circuit trajectories about equilibrium points can be achieved by means of current damping injection in a single circuit node. It is shown that such damping injection can be implemented with a standard high-pass filter. In this way, the resulting feedback control strategy has interesting robustness features since an a priori knowledge of the circuit equilibrium points is not required. Experimental work is used to illustrate the control performance.

In this study, we aimed to improve the VOA-based simple chaotic circuit proposed by Sprott in the literature by using FTFN-based circuit topology. And also, by choosing one of the FTFN-based circuits whose chaotic dynamics are very similar to that of Chua's circuit as a reference model, we tested the performance of this model at different frequencies and its extended frequency performance has been verified by laboratory experiments and PSpice simulations.

In this article, the secure transmission of data is being realized on the basis of the synchronization of two chaotic systems. The transmitted signal is only the sign of a projection of the state. This has the advantage of making the transmission not very sensitive to the noises of transmission and the attenuation of the transmitted signal without using a VCO (Voltage Control Oscillatory) into the transmitter and the receiver. It should be noted that for the considered type of chaotic system, the traditional conditions of observability and linearization by injection of outputs are not checked. Here, we can only carry out an estimator.

This paper investigates through experimental methods the dynamic characteristics of a farm tractor by changing the forward velocity from 0.63 m/s to 4.50 m/s. Nonlinear time series, frequency spectrum and continuous wavelet transform were used in the analysis.

From the nonlinear time series analysis, a nonlinear resonance of the tractor vibration occurred at forward velocity of 2.15 m/s. The existence of a subharmonic frequency at 4.32 m/s indicated chaotic dynamics that was confirmed by the trend of the Lyapunov exponent analysis with a positive exponent indicating chaos.

Continuous wavelet transform analysis results, presented graphically, called coefficient plots showed patterns composed of large and fine feature distribution in both time and scale. Quasi-periodic velocities of 1.95 m/s and 2.15 m/s were estimated by these coefficient plots. At forward velocity of 2.88 m/s, the coefficient plots showed dominant features that varied periodically and were estimated to be a period-doubling vibration. The coefficient plots during chaotic vibration at 3.52 m/s, 4.02 m/s and 4.32 m/s showed various feature distributions. At forward velocities of 0.95 m/s and 1.42 m/s, the existence of features with scale values of almost equal and half of the dominant feature was due to the influence of the gravel road surface and not to the artificial test track profile.

Experimentally obtained bifurcation was observed clearly from the coefficient plots that showed three different patterns from quasi-periodic vibration at 2.15 m/s, period-doubling at 2.88 m/s and chaos at 3.52 m/s.

The homoclinic bifurcation and chaos in a system of weakly coupled simple pendulum and harmonic oscillator subject to light dampings and weakly external and (or) parametric excitation of bounded noise is studied. The random Melnikov process is derived and mean-square criteria is used to determine the threshold amplitude of the bounded noise for the onset of chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated maximal Lyapunov exponent. The threshold amplitudes are further confirmed by using the Poincaré maps, which indicate the path from periodic motion to chaos or from random motion to random chaos in the system as the amplitude of bounded noise increases.

Based on the study of the dynamics of a dissipation-modified Toda anharmonic (one-dimensional, circular) lattice ring we predict here a new form of electric conduction mediated by dissipative solitons. The electron-ion-like interaction permits the trapping of the electron by soliton excitations in the lattice, thus leading to a soliton-driven current much higher than the Drude-like (linear, Ohmic) current. Besides, as we lower the values of the externally imposed field this new form of current survives, with a field-independent value.