International Journal of Bifurcation and Chaos

The stability of the state of motion in which a collection of coupled oscillators are in identical synchrony is often a primary and crucial issue. When synchronization stability is needed for many different configurations of the oscillators the problem can become computationally intense. In addition, there is often no general guidance on how to change a configuration to enhance or diminsh stability, depending on the requirements. We have recently introduced a concept called the Master Stability Function that is designed to accomplish two goals: (1) decrease the numerical load in calculating synchronization stability and (2) provide guidance in designing coupling configurations that conform to the stability required. In doing this, we develop a very general formulation of the identical synchronization problem, show that asymptotic results can be derived for very general cases, and demonstrate that simple oscillator configurations can probe the Master Stability Function.

We study some periodic and quasiperiodic behaviors exhibited by the Chua's equation with a cubic nonlinearity, near a Hopf–pitchfork bifurcation. We classify the types of this bifurcation in the nondegenerate cases, and point out the presence of a degenerate Hopf–pitchfork bifurcation. In this degenerate situation, analytical and numerical study shows a diversity of bifurcations of periodic orbits. We find a secondary Hopf bifurcation of periodic orbits, where invariant torus appears. This secondary Hopf bifurcation is bounded by a Takens–Bogdanov bifurcation of periodic orbits. Here, a sequence of period-doubling bifurcations of invariant tori is detected. Resonance phenomena are also analyzed. In the case of strong resonance 1:4, we show a new sequence of period-doubling bifurcations of 4T invariant tori.

Bandpass sigma-delta modulation is a technique used to convert high-frequency narrowband electronic signals from analog form to digital, or vice versa. It is a fundamentally nonlinear technique, and therefore can be analyzed only approximately by application of linear analysis. The aim of this work is to present certain results concerning the dynamical behavior of bandpass sigma-delta modulators with zero input based on application of methods of nonlinear dynamics and noninvertible map theory. We will consider the effect of certain circuit imperfections on the behavior of the system, and will also consider the effect of operating in a chaotic regime. While many of our results arise from application of standard methods of nonlinear dynamics, we will emphasize in particular those results that can be obtained by application of the critical lines tool of Mira. The paper, therefore, has two aims: To investigate the nonlinear behavior of bandpass sigma-delta modulators and to provide a demonstration of the applicability of the critical lines tool to a system of engineering importance.

I focus on the temporal dynamics generated by a life cycle consisting of two contiguous stages developing under the influence of a stimulus which pulsates between on and off. I ask: under what general conditions does a population held in exposure to this kind of periodic stimulus achieve life cycle synchrony? The situation is represented by a dynamical system consisting of a nondecreasing circle map whose plot is made up of 45° and horizontal piecewise-linear sections. These features permit the iterative dynamics (itineraries) followed by successive generations to be derived and algebraic conditions for high-ordered synchronization to be derived. Using development data obtained for the phytoplankton Thalassiorira pseudonana and mean daily irradiation intensities recorded over different months at the latitude of Oban (west coast of Scotland), I apply the model to investigate how seasonal change in daily irradiance may directly influence the synchronous dynamics of such populations.

A chaotic impulse radio system is an ultrawide-band communication system that uses a train of very narrow baseband impulses as a carrier. In the transmitter of a chaotic impulse radio system, a message signal is modulated by two kinds of pulse carriers. Firstly, a frequency modulation is used to modulate the message signal into a subcarrier that functions as the clock pulses of a chaotic circuit. Driven by the modulated clock pulses, the chaotic circuit outputs a chaotic impulse positioning sequence which generates the positions of the carrier impulses. The specially designed chaotic circuit in the transmitter guarantees that the time intervals between the carrier impulses are chaotic. Thus the energy of the impulse carrier is distributed evenly over the entire bandwidth. In the receiver of a chaotic impulse radio system the message signal is demodulated in two stages. At the first stage, the time interval between two consecutive impulses is recovered. At the second stage, a simple algorithm based on the knowledge of the chaotic circuit in the transmitter is used to calculate partially the locations of the inner clock pulses which in turn are used to demodulate the message signal. No synchronization at any level is needed in this chaotic impulse radio system. The security of this chaotic impulse radio system depends on the hardware parameters of the chaotic circuit and the inner clock pulse train. Simulation results are presented to illustrate the design procedure of an example of this chaotic impulse radio system.

This paper is concerned with the study of nonlinear phenomena in a closed loop voltage-controlled DC–DC Buck–Boost converter when suitable parameters are varied. The dynamics is analyzed using both the continuous-time model and the numerically computed stroboscopic map. The analysis of the one-dimensional bifurcation diagram shows that Neimarck–Sacker bifurcation occurs at certain values of the parameters. Phase-locking periodic windows, the period-adding sequence, and transition from quasiperiodicity to period-doubling via torus breakdown are also obtained. The two-dimensional bifurcation diagram is carefully computed. This shows that phase-locking orbits produce so-called Arnold tongues in the parameter space. It is shown that the winding number plotted as a function of the bifurcation parameter is a devil's staircase. As typically occurs in general circle maps, the fine structures of the Arnold tongues and the devil's staircase show self-similarity.

The nonlinear dynamics of PWM DC–DC switching regulators operating in the continuous conduction mode are investigated. A quick review of the existing analysis techniques and their limitations are first presented. A discrete nonlinear time-domain model is then derived for open-loop DC–DC converters. This model is then extended for closed-loop regulator systems implementing any type of compensation scheme. The equilibrium solutions of the closed-loop system are then identified. The eigenvalues of the Jacobian matrix evaluated at the equilibrium solution are used to assess its stability. The methods developed are used to study the dynamic behavior of a DC–DC buck regulator implementing different types of compensation design: proportional, integral, proportional–integral, and proportional–integral–derivative type feedback control. A detailed bifurcation analysis of the dynamic solutions as a design or a control parameter is changed is presented. A period-doubling route to chaos is shown to exist in voltage-mode regulators, depending on the values of the parameters of the compensator and the input voltage. A further investigation of the behavior of the converter in the instability regions was carried out to improve the understanding of this interesting behavior.

This paper is primarily a numerical study of the fixed-point bifurcation loci — saddle-node, period-doubling and Hopf bifurcations — present in the family:

More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time.

We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well.

Two new computing models based on information coding and chaotic dynamical systems are presented. The novelty of these models lies on the blending of chaos theory and information coding to solve complex combinatorial problems. A unique feature of our computing models is that despite the nonpredictability property of chaos, it is possible to solve any combinatorial problem in a systematic way, and with only one dynamical system. This is in sharp contrast to methods based on heuristics employing an array of chaotic cells. To prove the computing power and versatility of our models, we address the systematic solution of classical NP-complete problems such as the three colorability and the directed Hamiltonian path in addition to a new chaotic simulated annealing scheme.

This paper investigates the effects of the insertion of a flip–flop type "latch" on the dynamics exhibited by a class of power electronic circuits. It is shown that, despite preventing multiple-switching phenomena, this device does not cause the expected suppression of chaotic behavior but actually introduces novel destabilizing effects which make the occurrence of chaos in this system to take place on a larger scale. A detailed comparison between the dynamics of the nonlatched voltage controlled converter and the latched one is presented. Also, the implications on the existence of so-called sliding orbits and the associated bifurcation diagrams are discussed.

We investigate the local bifurcations experienced by a time-discrete dynamical system from population biology when there is an attractor in an invariant subspace that loses stability. The system describes competition between two species in a constant environment; invariant subspaces contain single-species attractors; the loss of stability of the attractor in one invariant subspace means that the corresponding species (i.e. the "resident" species) becomes invadable by its competitor. The global dynamics may be understood by examining the sign structure of Lyapunov exponents transverse to the invariant subspace. When the transverse Lyapunov exponent (computed for the natural measure) changes from negative to positive on varying a parameter, the system experiences a so-called blowout bifurcation. We unfold two generic scenarios associated with blowout bifurcations: (1) a codimension-2 bifurcation involving heteroclinic chaos and on–off intermittency and (2) a sequence of riddling bifurcations that cause asymptotic indeterminacy. An ingredient that both scenarios have in common is the fact that the "resident" species subspace contains multiple invariant sets with transverse Lyapunov exponents that do not change sign simultaneously. This simple model adds on a short list of archetypical systems that are needed to investigate the structure of blowout bifurcations. From a biological viewpoint, the results imply that mutual invasibility in a constant environment is neither a necessary nor a sufficient condition for coexistence.

In this work we study synchronization in a network of nonlinear elements with global couplings. We consider a family of maps and for each model we find sufficient conditions for synchronization. We also study cluster solutions and the corresponding low-dimensional maps.

Nonlinear system identification and analysis methods are employed to study the low-frequency oscillations present in time-series data obtained from reflectance imagery of microvasculature. Using the method of surrogate data testing the analysis reveals the deterministic nature of these oscillations believed by many to be chaotic. Further investigations by means of nonlinear system identification techniques indicate however that the underlying dynamics can described by a periodically driven nonlinear dynamical model exhibiting quasiperiodic behavior.

Recently, along with diminishing numbers of students, it has been observed that some graduate programs still have lots of students, whereas other fields of study suffer from a great loss of students. Apparently, students do not distribute evenly over the fields that a university offers. The relations among the number of students in different fields of concentration are not constant. In particular, the harder subjects appear to suffer more from a diminishing number of students than the fields with a reputation of being somewhat "softer."

The reasons for such trends must be expected to be related to the size of the market, and to psychological effects. Our study has been motivated by attempts to model economic mechanisms that are related to advertising [Feichtinger, 1992]. We have constructed a mathematical model that is able to explain trends such as reported above.

The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest-size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.

A growing body of work suggests an important unifying perspective in the study of stochastic resonance: Examining the "prehistory probability density" of the ensemble of pathways by which a system approaches the trigger of the nonlinear oscillator. This view, in the context of the Shannon–McMillan Theorem of information theory, leads to a draconian simplification of the complex of "signal," "noise" and oscillator in terms of a single object, an ergodic information source dual to a certain class of resonances.

For "spatial" stochastic resonator arrays, in the most general sense, the Shannon–McMillan Theorem further provides an analogy between the Shannon uncertainty of the dual information source and the free energy density of a physical system which allows imposition of real-space renormalization to give essential results in a straightforward manner. We find in particular threshold behavior in the onset of an epileptiform spatiotemporal coherence, and a likely usefulness of tuned spatial arrays for the detection of very subtle pattern.

These simplifications may provide, in addition, a natural context for discussing hierarchical structure in neural systems.

In sum, we attempt to bring stochastic resonance into the contemporary framework of applied mathematics which has begun to unify the study of fluctuations, statistical mechanics and information theory. Our critical generalization lies in the transfer of renormalization methods from statistical mechanics to information theory, via a parametization of source uncertainty.

The railway bogie model introduced by Cooperrider [1972] exhibits a Hopf bifurcation as the speed of the train v_tr is increased. This Hopf bifurcation is unusual because the amplitude of the family of periodic solutions that arises, behaves locally like rather than like . The explanation seems to lie with the form of the creep forces. Because of these forces, the model is of class C¹ but not C² at the origin.