International Journal of Bifurcation and Chaos

Mixing of granular materials provides fascinating examples of pattern formation and self-organization. More mixing action — for example, increasing the forcing with more vigorous shaking or faster tumbling — does not guarantee a better-mixed final system. This is because granular mixtures of just barely different materials segregate according to density and size; in fact, the very same forcing used to mix may unmix. Self-organization results from two competing effects: chaotic advection or chaotic mixing, as in the case of fluids, and flow-induced segregation, a phenomenon without parallel in fluids. The rich array of behaviors is ideally suited for nonlinear-dynamics-based inspection. Moreover, the interplay with experiments is immediate. In fact, these systems may constitute the simplest example of coexistence between chaos and self-organization that can be studied in the laboratory. We present a concise summary of the necessary theoretical background and central physical ideas accompanied by illustrative experimental results to aid the reader in exploring this fascinating new area.

In this paper we describe the winning entry of the time-series prediction competition which was part of the International Workshop on Advanced Black-Box Techniques for Nonlinear Modeling, held at K. U. Leuven, Belgium on July 8–10, 1998. We also describe the source of the data set, a nonlinear transform of a 5-scroll generalized Chua's circuit. Participants were given 2000 data points and were asked to predict the next 200 points in the series.

The winning entry exploited symmetry that was discovered during exploratory data analysis and a method of local modeling designed specifically for the prediction of chaotic time-series. This method includes an exponentially weighted metric, a nearest trajectory algorithm, integrated local averaging, and a novel multistep ahead cross-validation estimation of model error for the purpose of parameter optimization.

We describe the simplest bifurcations of spiral waves in reaction–diffusion systems in the plane and present the list of model systems. One-parameter bifurcations of one-armed spiral waves are fold and Hopf bifurcations. Multiarmed spiral waves may additionally undergo a period-doubling pitchfork bifurcation, when two congruent spiral wave solutions, having the "double" period, branch from the original spiral wave at the bifurcation point.

In this paper we consider both the dynamical and parameter planes for the complex exponential family E_λ(z)=λe^z where the parameter λ is complex. We show that there are infinitely many curves or "hairs" in the dynamical plane that contain points whose orbits under E_λ tend to infinity and hence are in the Julia set. We also show that there are similar hairs in the λ-plane. In this case, the hairs contain λ-values for which the orbit of 0 tends to infinity under the corresponding exponential. In this case it is known that the Julia set of E_λ is the entire complex plane.

Consider the Henon's map T_b: (x→ 1-ax²+y, y→bx), the parameters (a, b) being such that |b|<1, with the existence of an attracting set A. This paper deals with an approximate implicit analytical representation of the stable manifold W^S(q₁) of the saddle fixed point q₁ belonging to the basin boundary of the attracting set A. A method of successive approximations of iterative type is used from the definition of a "generating approximation" g₀(x, y)=0 (approximation of order zero). In the case of absence of homoclinic points to q₁, a generating approximation is defined from the two parabolas constituting the degenerate stable manifold in the (x, y) plane when b=0. Formally the result is extended when homoclinic points to q₁ are created for b=0, i.e. the degenerate stable manifold W^S(q₁) is made up of infinitely many parabolas.

A model of a simple electric power system consisting of two generators operating onto an infinite bus is studied by numerical simulations. In many practical situations the proper concern of the engineer is the extent of the basin of attraction of the safe operating condition in phase space. However traditional analysis using energy function method can capture only the regions of sufficient conditions, so that geometrical concepts and theories of dynamical systems are recognized to be necessary tools to understand power system stability. In this paper, all equilibrium points and limit cycles are listed, and their locations are investigated with relation to basins. It will be helpful to globally grasp the total structure of the basins of attractions in phase space. Periodic orbits correspond to one or both generators operating in a desynchronized steady state, and if realized, rotors are destroyed. However their basins concern the study of the basin of the stable equilibrium point. Focusing on one of the generators, that is, identifying states of another generator, the basin portrait is similar to that of the single degree of freedom swing equations system. And in this view, basin erosions of synchronized state are understood in connection with an escape phenomena. Global basin structures are studied by taking systematic basin portrait sections. Note that we do not treat situations where inertia ratio between generators is very large as is seen in some reports using Melnikov's method.

A new chaotic system introduced in a recent paper [Banks & McCaffrey, 1998] is analyzed in detail. It is based on the theory of Lie algebras and is inspired by the well-known systems of Lorenz and Chua. The dynamics of an invariant semisimple part is first studied and then the full system is considered to be a perturbation of this invariant part.

We consider a simplified neural network model for a ring of four neurons where each neuron receives two time delayed inputs: One from itself and another from the previous neuron. Local stability analysis of the positive equilibrium leads to a characteristic equation containing products of four transcendental functions. By analyzing the equivalent system of four scalar transcendental equations, we obtain sufficient conditions for the linear stability of the positive equilibrium. Furthermore, we show that a Hopf bifurcation can occur when the positive equilibrium loses stability.

A chaotic neuron model with the linear saturating activation function is analyzed. The model accounts for the property of relative refractoriness, that is, gradual recovery of responsiveness of a biological neuron after a stimulus is applied to the neuron. A neural network model composed of chaotic neurons with the linear saturating activation functions, which includes the generalized Brain-State-in-a-Box (gBSB) model as a special case, is proposed and analyzed. The proposed model is then used to implement associative memory. The existence and stability of equilibrium points of the model are analyzed. Fuzzy logic is used to tune associative memory parameters for the purpose of directing the network trajectory to visit memory patterns with sought features. Simulation results are presented to illustrate the effectiveness of the memory retrieval capability.

This work is based on the concept of order parameters of synergetics. The order parameter equations describe the behavior of a system in the vicinity of an instability and are used here not only for the analysis but also for the control of nonlinear time discrete dynamical systems. Usually, the dimensionality of the evolution equations of the order parameters is less than the dimensionality of the original evolution equations. It is, therefore, convenient to introduce control mechanisms, first in the order parameter equations, and then to use the obtained results for the control of the original system. The aim of the control in this case is to avoid chaotic behavior of the system. This is achieved by shifting appropriate bifurcation points of a period-doubling cascade. In this work we concentrate on the shifting of only the first bifurcation point. The used control mechanisms are delayed feedback schemes. As an example the well-known Hénon map is investigated. The order parameter equation is calculated using both the adiabatic elimination procedure and the center manifold theory. Using the order parameter concept two types of control mechanisms are constructed, analyzed and compared.

We consider a lattice of coupled logistic maps with periodic boundary condition. We prove that synchronization and almost synchronization occur for the case of 1D lattice with lattice size n=2, 3, 4 provided the coupling strength c is chosen in a suitable open interval contained in . For the case of lattice size n≥4, we also show the numerical results of (almost) synchronized chaotic behavior of the coupled map lattice. For each fixed parameter γ∈ [3.57, 4] of the logistic maps, the lattice sizes and the ranges of the coupling strengths c so that the coupled map lattice is synchronized, are given.

Bifurcation problems with the symmetry group Z₂⊕Z₂ of the rectangle are common in applied science, for example, whenever a Euclidean invariant PDE is posed on a rectangular domain. In this work we derive normal forms for one-parameter bifurcations of steady states with symmetry of the group Z₂⊕Z₂. We study degeneracies of Z₂⊕Z₂-codimension 3 and modality 1. We also deduce persistent bifurcation diagrams when the system is subject to symmetry-preserving perturbations.

The effect of n-bit precision on the orbit-periods of second-order lossless overflowing digital filters is discussed. It is observed that there are of the order of 2ⁿ orbits with periods O(2ⁿ), whereas in a random finite-state machine with the same number (2ⁿ)² of states and with an invertible next-state function there are a few large orbits with periods O((2ⁿ)²). This behavior persists when the coefficient multiplication is replaced by a random mapping. This "randomized" model is readily understood and so provides a rough theory of the orbit-periods.

This paper presents the discussion of experimental results and numerical simulation of nonlinear phenomena in a flexible rotor system under mechanical resonance. The whirling motion of the first flexural mode of the rotor causes the hunting of its rotating speed. Hysteretic jump phenomena exist between the charge and the discharge of the rotation. It is shown that the quasi-periodicity of the whirling motion is closely related to the hunting of the rotating speed. When the rotor is coupled to a generator, a continuous change of output power cannot be achieved under mechanical resonance. These results show the necessity to stabilize the whirling motion and the hunting, for example in a flywheel energy storage system.