International Journal of Bifurcation and Chaos

The 11th part of our tour through one-dimensional binary Cellular Automata concerns period-2 rules, which form the second group in our classification of the 88 globally-independent CA rules according to the properties of their periodic orbits. In this article, we display the basin tree diagrams of all period-2 rules along with their time-2 characteristic functions, and then we prove that all rules belonging to group 2 have robust period-2 ω-limit orbits for any finite, and infinite, bit string length. This rigorous result, which pairs with the one about period-1 rules given in the tenth installment of our chronicle, confirms what we stated about period-2 rules on the basis of empirical evidence. In the second part of this tutorial, we introduce the notion of quasi global-equivalence and prove that there are only 82 quasi globally-independent CA rules. For the first time, we show that the space-time patterns of globally-independent local rules can depend on each other, and we present an example of quasi-global transformation. We also define the super string 𝄞, and its unique decimal representation x_𝄞, dubbed the super decimal, which provides a completely transparent yet rigorous proof that rule is chaotic when L → ∞. Moreover, we present the basin tree generation formulas, which uncover the analytical relationships between basin trees of globally-equivalent rules. Last but not least, for pedagogical and epistemological reasons, we conclude this paper with the selection of rule, instead of rule , as the prototypic universal Turing machine for our future discourse.

In this paper, a modified generalized Lorenz-type system is introduced, which is state-equivalent to a simple and special form, and is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, two classes of new chaotic attractors are found for the first time in the literature, which are similar but nonequivalent in topological structure. To further understand the complex dynamics of the new system, some basic properties such as Lyapunov exponents, Hopf bifurcations and compound structure of the attractors are analyzed and demonstrated with careful numerical simulations.

We study the effect of different periodic excitations like sine, square, triangle and sawtooth waves on Chua's circuit and show that the circuit can undergo distinctly modified bifurcation structure, generation of new periodic regimes, induction of crises and so on. In particular, we point out that under the influence of different periodic excitations, a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent route to chaos and period-adding sequences, reverse bifurcations, remerging chaotic band attractors, a large number of coexisting periodic attractors exist in the system. The analysis is carried out numerically using phase portraits, two-parameter phase diagrams in the forcing amplitude-frequency plane and one-parameter bifurcation diagrams. The chaotic dynamics of this circuit is also realized experimentally.

This paper focuses on periodic motions and chaos relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in [Luo & O'Connor, 2007] are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.

By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.

In this paper, a scalar sign function-based digital design methodology is presented to develop a digital tracking controller for the continuous-time chaotic systems with absolute value state constraints. A scalar sign function, which is the counterpart of the well-known matrix sign function, is utilized to approximately represent the absolute value state term by a rational function. As a result, the original state constrained nonsmooth nonlinear system becomes a smooth nonlinear system having rational nonlinear terms. Then, an optimal linearization technique is applied to the afore-mentioned nonlinear system for finding an optimal linearization model, which has the exact dynamics of the original nonlinear system at any operating point of interest with minimal modeling error in the vicinity of the operating point on the trajectory. To overcome the effect of modeling errors and to quickly track the desired reference signals, a high-gain optimal analog tracker is designed for the obtained linear model. For practical implementation of the high-gain analog tracker, the prediction-based digital redesign technique is utilized to obtain a low-gain digital tracker for digital control of the sampled-data nonlinear system with constrained states. Chua's chaotic circuits are used to demonstrate the effectiveness of the proposed approach.

In this paper, a new method for controlling spatiotemporal chaos in discrete-time spatially extended systems modeled by coupled map lattices is proposed. This method is based on quasi-sliding mode using a Lyapunov function. This method enables the system to drive the chaotic motion toward any desired trajectory via a sliding surface in the error space. The controller also guarantees finite time convergence of the state trajectory. The main advantage of this method is its robustness with respect to additive uncertainties and its applicability for all types of coupled map lattices. A diffusively coupled map lattice is used as an example to demonstrate the method. Simulation results reveal the robustness and the effectiveness of the method in controlling spatiotemporal chaos in coupled map lattices.

In this paper, we study nonlinear and bifurcation behavior of a two-cell DC-DC buck power electronic converter. The system shows nonsmooth period doubling bifurcation and chaotic phenomena in a certain zone of parameter space. This zone is located both analytically and from numerical simulations. One-dimensional, two-dimensional bifurcation diagrams and Lyapunov exponent spectrum are used to detect the different dynamic behaviors of the system. The Fixed Point Induced Control (FPIC) technique is applied to the system in order to widen the stability zone. The performance of the FPIC technique applied to the stabilization of a two-cell DC-DC buck converter is analyzed. With this technique, stabilization is achieved without changing the fixed point. The robustness in the presence of a noisy environment is checked by numerical simulations by considering different noise levels.

In this paper we present a contribution toward a dynamic theory of the consumer. Canonical economic theory assumes a perfectly rational agent that acts according to optimization principles of a utility function subject to budget constraints. Here we propose a two-dimensional discrete dynamical system that describes the choice problem of a consumer under the assumption of bounded rationality. The map representing this adaptive process is characterized by the presence of a denominator that can vanish. We use recent results on the global bifurcations of this kind of maps in order to explain the coexistence of different attractors and the structure of the corresponding basins of attraction. The stationary equilibria of the map represent the rational choices of the consumer in the static setting, i.e. solutions of the utility maximization problem under budget constraints. We use geometric and numerical methods to study the problem of coexistence of different attractors and the related problem of the basins of attraction and their global bifurcations.

This paper proposes a new three-dimensional chaotic generator, for which the nonlinearity is based on sinusoidal function. Based on a practical circuit design for realizing a wide angular range of sinusoidal functions, chaotic attractors with large number of scrolls have been obtained. Different types of attractors are available, governed by the choices of variables used in the sinusoidal function. The design is illustrated with both simulations and experiments.

This article gives a topological and measure-dynamics based analysis on the deciphering of cipher texts encoded by the basic chaos shift keying scheme. Our analysis gives a proof of the ambiguity involved in deciphering and also shows that the deciphering error is a source of practical concern.

In this paper, an investigation on nonlinear dynamical behaviors of a transmission system with a gear pair is conducted. The transmission system is described through an impact model with a possible stick between the two gears. From the theory of discontinuous dynamical systems, the motion mechanism of impacting chatter with stick is investigated. The onset and vanishing conditions of stick motions are developed, and the condition for maintaining the stick motion is achieved as well. The corresponding physics interpretation is given for a better understanding of nonlinear behaviors of gear transmission systems. Furthermore, such an understanding may be very helpful to improve the efficiency of gear transmission systems.

In this paper we deal with a centrally symmetric quintic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have ten limit cycles by using a homoclinic bifurcation method based on stability-change.

This paper proves that a Z₅-equivariant planar polynomial vector field of degree 5 has at least five symmetric centers, if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z₅-equivariant perturbations, the conclusion that the perturbed system has at least 25 limit cycles with the scheme 〈5 ∐ 5 ∐ 5 ∐ 5 ∐ 5〉 is rigorously proved.

In this work we present the alternated Julia sets, obtained by alternated iteration of two maps of the quadratic family and prove analytically and computationally that these sets can be connected, disconnected or totally disconnected verifying the known Fatou–Julia theorem in the case of polynomials of degree greater than two. Some examples are presented.