International Journal of Bifurcation and Chaos

Mathematical problems arising from the study of complex dynamics in Chua’s circuit are discussed. An explanation of the extreme complexity of the structure of attractors of Chua’s circuit is given. This explanation is based upon recent results on systems with homoclinic tangencies. A number of new dynamical phenomena is predicted for those generalizations of Chua’s circuits which are described by multidimensional systems of ordinary differential equations.

This article provides an introduction to a technique for formulating nonlinear models of mechanical systems composed of interconnected and constrained rigid body systems such as those encountered in vehicle technology, biomechanics, spacecraft design and robotics. The approach is based on an algorithm developed by Kane to treat nonholonomic systems, for example systems with rolling constraints. The algorithm is interpreted geometrically in terms of tangent vectors to the instantaneous configuration manifold embedded in the space of nonconstrained motions for the system. The level and style of the presentation is intended to be understood by scientifically literate readers with minimal knowledge in mechanics beyond the introductory level. Examples also show how computer algebra can be used to reduce the effort required for treating complex systems. An annotated reference list, which includes a discussion of computer software, is also provided.

In this paper we study the dynamics of a biological oscillator forced at two incommensurate frequencies which, as our results show, can exhibit a variety of novel dynamical phenomena, including two-frequency quasi-periodicity, three-frequency quasi-periodicity, chaos and, in particular, strange nonchaotic attractors. We discuss the significance and observable signatures of these phenomena, and suggest biological systems in which these phenomena may be experimentally investigated.

The nonlinear small amplitude excitations of a domain wall (DW) in the ferromagnet with high uniaxial anisotropy are investigated using reduced magnetodynamic equations. The “life-time” of soliton-like oscillations of DW in the nondissipative media is defined. The conditions of direct and parametric stabilization of self-localized DW oscillations by alternating magnetic fields are analyzed in the presence of dissipation.

The paper presents results of numerical simulations on a model free boundary problem which is qualitatively equivalent to the free interface problems describing solid combustion and exothermic phase transitions. The model problem has been recently shown to exhibit transition to chaotic oscillations via a sequence of period doubling, assuming an Arrhenius type boundary kinetics. In the present paper we demonstrate that for a slightly different class of kinetics the behavior pattern, while retaining the above scenario, may undergo a drastic change. This behavior is characterized by slowly expanding oscillations followed by a powerful burst, after which the system returns to near equilibrium and the scenario is repeated periodically. As the bifurcation parameter approaches the stability threshold, the total period tends to infinity due to an increasingly prolonged “accumulation phase.” Additional scenarios corresponding to increasing supercriticality of the bifurcation parameter include finite period doubling sequences that return to simple periodic regime via a reversed cascade, infinite sequences followed by an interval of chaos and reversed sequences, or a combination of period doubling, chaos and “Shilnikov type” orbits. Our observations suggest that major dynamical patterns are qualitatively kinetics independent and, moreover, may be essentially finite-dimensional. We argue that for a deeper understanding of the mechanisms responsible for the formation of various patterns a further analytical study (including perhaps a reduction to a finite-dimensional model) of the free boundary problem is desirable.

The breakdown of three-dimensional tori T³ and the transition to chaos are studied numerically in a periodically forced system of two coupled logistic maps. We have shown that for small perturbations the breakdown of T³ possesses the same features as in the T² case described by the theorem of Afraimovich and Shil’nikov. Remarkably, some indications for a chaotic motion on T³ are found.

Induced templates for two members of Chua’s attractors: spiral-type and double-scroll chaotic attractors are computed using the orbits of lower period. The template describes the topological properties of periodic orbits embedded in the attractor. It is identified by a set of integers which characterize the attractor. The templates are confirmed by investigating orbits of higher period.

Bifurcational precedences are obtained for systems which permit escape from a potential well by considering the three striped horseshoe formed by the invariant manifolds of the hilltop saddles. Considerable information regarding the bifurcational structure is obtained by considering the crossings of periodic orbits in an idealised suspension of the flow in terms of this three striped horseshoe.

An infinite set of stationary traveling waves are found in a discrete analog of the nonlinear diffusion Huxley equation. Their profiles are determined by the trajectories of Bernoulli scheme containing two symbols. It is proved that these waves are stable with respect to perturbations moving with the same speed. Thus it is shown that chaos of traveling waves is realized in this system.

Spatiotemporal dynamics of a one-dimensional cellular neural network (CNN) made of Chua’s circuits which mimics a reaction-diffusion medium is considered. An approach is presented to analyse the properties of this reaction-diffusion CNN through the characteristics of the attractors of an associated infinite-dimensional dynamical system with a matrix phase space. Using this approach, the spatiotemporal correlation dimension of the CNN’s spatiotemporal patterns is computed over various ranges of the diffusion coupling parameter, length of the chain, and initial conditions. It is shown that in a finite-dimensional projection of the matrix phase space of the CNN, both low- and high-dimensional attractors corresponding to different initial conditions coexist.

The Maxwell-Bloch equations describing the transverse modes of a large aspect ratio homogeneously broadened single-longitudinal-mode laser are solved numerically. The solutions are visualized by color-coding the intensity and phase of the transverse electric field at each time step and a video showing the detailed time evolution has been made. As the gain is increased optical vortices, or defects, are observed. These vortices interact and for very high gain optical turbulence exists. The orbits of the single- and two-defect solutions are analyzed and the effective force law coupling the defects is numerically calculated. The best fit force is a modified Bessel function which is proportional to r^−1/2exp(−r/α) at large distances and ln(α/r) at close range where α is approximately 2. This interaction force law for optical defects is similar to the force law between Abrikosov vortices in a superconductor.

Knowledge of precursors and predictors of human parturition would be important both for our understanding of the controlling mechanisms and for practical use in the detection and diagnosis of various abnormalities of the birth process. They involve a multitude of genetic, metabolic, nutritional, hormonal and environmental factors. Present research is however hindered by the lack of a clear recognized correlation between the time evolution of these various variables and the initiation of parturition. Here, we propose a coherent logical framework which allows us to rationalize the various laboratory and clinical observations on maturation, the triggering mechanisms of parturition, the existence of various abnormal patterns as well as the effect of external stimuli of various kinds. Within the proposed mathematical model, parturition is seen as a ”critical” instability or phase transition from a state of quietness, characterized by a weak incoherent activity of the uterus in its various parts as a function of time (state of activity of many small incoherent intermittent oscillators), to a state of globally coherent contractions where the uterus functions as a single macroscopic oscillator. Our approach gives a number of new predictions and suggests a strategy for future research and clinical studies, which present interesting potentials for improvements in predicting methods and in describing various abnormal prenatal situations.

We investigate the dynamics of an oscillator with gain with switching between two cosinusoidal drives. The drive is assumed to be dependent on the state of the system and its prehistory. We show that frequency switching alone can lead to a chaotic response. Moreover, we demonstrate that resonance phenomena inhibit chaos and lead to unbounded motion. We also point out the crucial role played by the initial phases of the drives.

This paper deals with the spectral properties of pseudo-random signals generated in maximum-length shift registers. It is shown that infinitely long registers are comparable in behavior with one-dimensional discrete-time chaotic maps. Therefore the theory, results and tools developed for chaotic systems can be applied to the design of the spectral properties of maximum-length shift register sequences.

We investigate numerically and analytically the nonlinear dynamics of a system consisting of two self-synchronizing pulse-coupled nonlinear oscillators with delay. The particular system considered consists of connected digital phase-locked loops. We find mapping equations that govern the system and determine the synchronization properties. We study the bifurcation diagrams, which show regions of periodic, quasiperiodic and chaotic behavior, with unusual bifurcation diagrams, depending on the delay. We show that depending on the parameter that is varied, the delay will have a synchronizing or desynchronizing effect on the locked state. The stability of the system is studied by determining the Liapunov exponents, indicating marked differences compared to coupled systems without delay.

A systematic approach is developed for synthesizing dissipative chaotic arrays that possess the self-synchronization property. The ability to synthesize high-dimensional chaotic arrays further enhances the usefulness of synchronized chaotic systems for communications, signal processing, and modeling of physical processes.

A three-parameter numerical bifurcation analysis of the forced Brusselator model is performed. Keeping one parameter fixed the bifurcation structure in the frequency-amplitude parameter plane may, or may not, contain Arnol’d tongues. The bifurcation scenario that destroys the dominant Arnol’d tongues is described.

In this letter, we propose a new experimental method for converting a chaotic attractor in Chua’s circuit to a periodic orbit. A tunnel mechanism is used to achieve this conversion. Using this method, we were able to demonstrate experimentally that periodic orbits of very high periods (e.g., greater than 30) can be robustly stabilized.