Multiple period-doubling bifurcation route to chaos in periodically pulsed Murali-Lakshmanan-Chua circuit-controlling and synchronization of chaos

We consider a simple nonautonomous dissipative nonlinear electronic circuit consisting of Chua's diode as the only nonlinear element, which exhibit a typical period doubling bifurcation route to chaotic oscillations. In this paper, we show that the effect of additional periodic pulses in this Murali-Lakshmanan-Chua (MLC) circuit results in novel multiple-period-doubling bifurcation behavior, prior to the onset of chaos, by using both numerical and some experimental simulations. In the chaotic regime, this circuit exhibits a rich variety of dynamical behavior including enlarged periodic windows, attractor crises, distinctly modified bifurcation structures, and so on. For certain types of periodic pulses, this circuit also admits transcritical bifurcations preceding the onset of multiple-period-doubling bifurcations. We have characterized our numerical simulation results by using Lyapunov exponents, correlation dimension, and power spectrum, which are found to be in good agreement with the experimental observations. Further controlling and synchronization of chaos in this periodically pulsed MLC circuit have been achieved by using suitable methods. We have also shown that the chaotic attractor becomes more complicated and their corresponding return maps are no longer simple for large n-periodic pulses. The above study also indicates that one can generate any desired n-period-doubling bifurcation behavior by applying n-periodic pulses to a chaotic system.     

Holes and chaotic pulses of traveling waves coupled to a long-wave mode

It is shown that localized traveling-wave pulses and holes can be stabilized by a coupling to a long-wave mode. Simulations of suitable real Ginzburg-Landau equations reveal a small parameter regime in which the pulses exhibit a breathing motion (presumably related to a front bifurcation), which subsequently becomes chaotic via period-doubling bifurcations.     

Running pulses of complex shape in a reaction-diffusion model

In a one-dimensional reaction-diffusion model of an active medium, stable steady-state wave pulses of a new type are described. They are called multihumped because their waveforms contain several maxima of similar size. Presumably, the multihumped pulses arise via a bifurcation at which an unstable trigger wave disappears. The parameter governing this bifurcation is the diffusion coefficient for the model inhibitor. The model is analyzed by varying this parameter to determine the conditions for the emergence of multihumped pulses. The results of this analysis show how their waveform and dynamics of excitation depend on the inhibitor diffusion coefficient.     

Bifurcations and chaos in a parametrically damped two-well Duffing oscillator subjected to symmetric periodic pulses

We study a parametrically damped two-well Duffing oscillator, subjected to a periodic string of symmetric pulses. The order-chaos threshold when altering solely the width of the pulses is investigated theoretically through Melnikov analysis. We show analytically and numerically that most of the results appear independent of the particular wave form of the pulses provided that the transmitted impulse is the same. By using this property, the stability boundaries of the stationary solutions are determined to first approximation by means of an elliptic harmonic balance method. Finally, the bifurcation behavior at the stability boundaries is determined numerically.     

Bistability in pulse propagation in networks of excitatory and inhibitory populations

We study the propagation of traveling solitary pulses in one-dimensional networks of excitatory and inhibitory integrate-and-fire neurons. Slow pulses, during which inhibitory cells fire well before neighboring excitatory cells, can propagate along the network at intermediate inhibition levels. At higher levels, they destabilize via a Hopf bifurcation. There is a bistable parameter regime in which both fast and slow pulses can propagate. Lurching pulses with spatiotemporal periodicity can propagate in regimes for which continuous pulses do not exist.     

A normal form for excitable media

We present a normal form for traveling waves in one-dimensional excitable media in the form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behavior of single pulses in a periodic domain and also the richer behavior of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.     

Numerical study on the discharge characteristics and nonlinear behaviors of atmospheric pressure coaxial electrode dielectric barrier discharges

The discharge characteristics and temporal nonlinear behaviors of the atmospheric pressure coaxial electrode dielectric barrier discharges are studied by using a one-dimensional fluid model. It is shown that the discharge is always asymmetrical between the positive pulses and negative pulses. The gas gap severely affects this asymmetry. But it is hard to acquire a symmetrical discharge by changing the gas gap. This asymmetry is proportional to the asymmetric extent of electrode structure, namely the ratio of the outer electrode radius to the inner electrode radius. When this ratio is close to unity, a symmetrical discharge can be obtained. With the increase of frequency, the discharge can exhibit a series of nonlinear behaviors such as period-doubling bifurcation, secondary bifurcation and chaotic phenomena. In the period-doubling bifurcation sequence the period-n discharge becomes more and more unstable with the increase of n. The period-doubling bifurcation can also be obtained by altering the discharge gas gap. The mechanisms of two bifurcations are further studied.It is found that the residual quasineutral plasma from the previous discharges and corresponding electric field distribution can weaken the subsequent discharge, and leads to the occurrence of bifurcation.     

Bifurcation to chaos in clocked digital systems containing autonomous timing circuits

Observations of chaos and period doubling in a repetitively triggered monostable multivibrator circuit are reported, with the time between trigger pulses as the bifurcation parameter. A theory is presented which predicts the first bifurcation point. Measurements of an experimental circuit confirm the predictions of this theory. The consequences for concurrent digital systems are briefly considered.     

Collisions of counter-propagating pulses in coupled complex cubic-quintic Ginzburg–Landau equations

We discuss the results of the interaction of counter-propagating pulses for two coupled complex cubic-quintic Ginzburg–Landau equations as they arise near the onset of a weakly inverted Hopf bifurcation. As a result of the interaction of the pulses we find in 1D for periodic boundary conditions (corresponding to an annular geometry) many different possible outcomes. These are summarized in two phase diagrams using the approach velocity, v, and the real part of the cubic cross-coupling, c_r, of the counter-propagating waves as variables while keeping all other parameters fixed. The novel phase diagram in the limit v ↦0, c_r ↦0 turns out to be particularly rich and includes bound pairs of 2 π holes as well as zigzag bound pairs of pulses.     

Delayed pulse dynamics in single mode class B lasers

Single slow–fast intensity pulses are generated by quickly increasing the pump from a below to an above threshold value during a finite time interval. Under particular conditions, a pulse appears after the pump perturbation has ended and with a significant delay. We demonstrate that this delayed pulse is not a classical turn-on pulse and that it verifies unusual properties. Experimental and numerical observations of the amplitude and the delay of these pulses are compared quantitatively and indicate that they emerge from zero at a bifurcation point. PACS 42.60.Mi; 42.65.Sf     

Observation of Periodic Multiplication and Chaotic Phenomena in Atmospheric Cold Plasma Jets

We investigate the temporal evolution of the current pulses from an ac Fie cold plasma jet at atmospheric pressure and with driving frequency in the range 14.76-15.30 kHz. The driving frequency is used as the plasma system＇s bifurcation parameter in analogy with the evolution in which the current pulses undergoes multiplication and chaos. Such time-domain nonlineaxity is important for controlling instabilities in atmospheric glow discharges. In addition, the observation can provide some data to support the simulation results reported previously [Appl. Phys. Lett. 90 （2007） 071501].     

An investigation of dark soliton behaviors within the nonlinear micro and nano ring resonators

We firstly propose the interesting results of a dark soliton pulse propagating within the nonlinear micro and nano waveguides. The system consists of nonlinear micro and nano ring resonators, whereas the dark soliton is input into the system and traveling within the waveguide. A continuous dark soliton pulse is chopped to be the smaller pulses by the nonlinear effects known as chaos. The nonlinear behaviors such as chaos, bistability and bifurcation are analyzed and discussed. The power amplification is the property that can be used to perform the long distance link, where the security is the dominant reason.     

On propagation failure in one- and two-dimensional excitable media

We present a nonperturbative technique to study pulse dynamics in excitable media. The method is used to study propagation failure in one-dimensional and two-dimensional excitable media. In one-dimensional media we describe the behavior of pulses and wave trains near the saddle node bifurcation, where propagation fails. The generalization of our method to two dimensions captures the point where a broken front (or finger) starts to retract. We obtain approximate expressions for the pulse shape, pulse velocity, and scaling behavior. The results are compared with numerical simulations and show good agreement.     

Solitary pulses in linearly coupled Ginzburg-Landau equations

This article presents a brief review of dynamical models based on systems of linearly coupled complex Ginzburg-Landau (CGL) equations. In the simplest case, the system features linear gain, cubic nonlinearity (possibly combined with cubic loss), and group-velocity dispersion (GVD) in one equation, while the other equation is linear, featuring only intrinsic linear loss. The system models a dual-core fiber laser, with a parallel-coupled active core and an additional stabilizing passive (lossy) one. The model gives rise to exact analytical solutions for stationary solitary pulses (SPs). The article presents basic results concerning stability of the SPs; interactions between pulses are also considered, as are dark solitons (holes). In the case of the anomalous GVD, an unstable stationary SP may transform itself, via the Hopf bifurcation, into a stable localized breather. Various generalizations of the basic system are briefly reviewed too, including a model with quadratic (second-harmonic-generating) nonlinearity and a recently introduced model of a different but related type, based on linearly coupled CGL equations with cubic-quintic nonlinearity. The latter system features spontaneous symmetry breaking of stationary SPs, and also the formation of stable breathers.     

Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study

The bifurcation structure of localized stationary radial patterns of the Swift-Hohenberg equation is explored when a continuous parameter n is varied that corresponds to the underlying space dimension whenever n is an integer. In particular, we investigate how 1D pulses and 2-pulses are connected to planar spots and rings when n is increased from 1 to 2. We also elucidate changes in the snaking diagrams of spots when the dimension is switched from 2 to 3.     

Structurally stable phase portraits for the five-dimensional Lorenz equations

We discuss the universal unfolding of a planar codimension four singularity which occurs in the five dimensional Lorenz equations. All structurally stable phase portraits are given and some representative bifurcation diagrams are displayed. These phase portraits have a rich structure including up to four limit cycles. The bifurcation sets in unfolding space — where the phase portraits undergo a qualitative change — are determined and new types of saddle loops are found. We show that the codimension four singularity occurs stably in a model for the laser with saturable absorber. Solution branches indicating birhythmicity and saddle loops for the pulsed mode of laser operation are found in bifurcation diagrams corresponding to the universal unfolding of the codimension four singularity. This explains numerical solutions of other authors which so far have not been related to a bifurcation analysis. Hints to other Lorenz models are given.     

Multipulse excitability in a semiconductor laser with optical injection

An optically injected semiconductor laser can produce excitable multipulses. Homoclinic bifurcation curves confine experimentally accessible regions in parameter space where the laser emits a certain number of pulses after being triggered from its steady state by a single perturbation. This phenomenon is organized by a generic codimension-two homoclinic bifurcation and should also be observable in other systems.     

Experimental bifurcations and homoclinic chaos in a laser with a saturable absorber

The shape and the peak values of the pulses from a passive Q-switching CO2 laser with SF6 as saturable absorber were detected while the laser was tuned in frequency across a longitudinal mode. A succession of stability windows, typical for bifurcation diagrams in the homoclinic scenario, was observed and the widths of those windows were measured. The expansion rate of the undulations in individual pulses was also obtained and compared to Floquet's multipliers given by the ratio of widths in consecutive windows. The dynamics is consistent with a homoclinic tangency to a periodic orbit.     

Experimental researches of propagation of acoustic compression pulses in copper wires

The results of experimental research related to the propagation of elastic-plastic compression pulses in polycrystalline copper wires are presented. We have developed a technique of generating compression pulses with amplitudes of pressure exceeding 100 MPa; this essentially exceeds the elasticity limit of copper. This allows us to carry out the research for the propagation of compression pulses in copper wire (as simples). Significant influence of the pulse amplitude on its spectrum, velocity and absorption will be discussed. The experiments showed that as the amplitude of the pulse increase the pulses velocity is decreased and absorption will be increased. The pulse spectrum also changes correspondingly. The analysis and discussion of experimental results will be presented through the context.