Periodic Oscillations in a Chemostat Model with Two Discrete Delays

Periodic oscillations of solutions of a chemostat-type model in which a species feeds on a limiting nutrient are considered. The model incorporates two discrete delays representing the lag in nutrient recycling and nutrient conversion. Through the study of characteristic equation associated with the linearized system, a unique positive equilibrium is found and proved to be locally asymptotically stable under some conditions. Meanwhile, a Hopf bifurcation causing periodic solutions is also obtained. Numerical simulations illustrate the theoretical results

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions

Bifurcation of an Orbit Homoclinic to a Hyperbolic Saddle of a Vector Field in R4

We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field in R4. We give an expression of the gap between returning points in a transverse section by renormalizing system, through which we find the existence of homoclinic-doubling bifurcation in the case 1+α>β>ν. Meanwhile, after reparametrizing the parameter, a periodic-doubling bifurcation appears and may be close to a saddle-node bifurcation, if the parameter is varied. These scenarios correspond to the occurrence of chaos. Based on our analysis, bifurcation diagrams of these bifurcations are depicted

Problems of homoclinic flips bifurcation in four-dimensional systems

LYON-ENS Sciences (693872304) / SudocSudocFranceF

Two-Dimensional Analytic Modeling Of Acoustic Diffraction For Ultrasonic Beam Steering By Phased Array Transducers

Phased array ultrasonic transducers enable modulating the focal position of the acoustic waves, and this capability is utilized in many applications, such as medical imaging and non-destructive testing. This type of transducers also provides a mechanism to generate tilted wavefronts in acousto-optic deflectors to deflect laser beams for high precision advanced laser material processing. In this paper, a theoretical model is presented for the diffraction of ultrasonic waves emitted by several phased array transducers into an acousto-optic medium such as TeO2 crystal. A simple analytic expression is obtained for the distribution of the ultrasonic displacement field in the crystal. The model prediction is found to be in good agreement with the results of a numerical model that is based on a non-paraxial multi-Gaussian beam (NMGB) model

Dynamic Two-Dimensional Refractive Index Modulation For High Performance Acousto-Optic Deflector

The performance of an acousto-optic deflector is studied for two-dimensional refractive index that varies as periodic and sinc functions in the transverse and longitudinal directions, respectively, with respect to the direction of light propagation. Phased array piezoelectric transducers can be operated at different phase shifts to produce a two-dimensionally inhomogeneous domain of phase grating in the acousto-optic media. Also this domain can be steered at different angles by selecting the phase shift appropriately. This mechanism of dynamically tilting the refractive indexmodulated domain enables adjusting the incident angle of light on the phase grating plane without moving the light source. So the Bragg angle of incidence can be always achieved at any acoustic frequency, and consequently, the deflector can operate under the Bragg diffraction condition at the optimum diffraction efficiency. Analytic solutions are obtained for the Bragg diffraction of plane waves based on the second order coupled mode theory, and the diffraction efficiency is found to be unity for optimal index modulations at certain acoustic parameters

Effect Of Large Deflection Angle On The Laser Intensity Profile Produced By Acousto-Optic Deflector Scanners In High Precision Manufacturing

Laser beam scanners have found wide applications in a variety of laser-assisted advanced microprocessing technologies such as printing, patterning, and doping. Traditional galvo-scanners affect the accuracy of beam positioning and repeatability in high precision manufacturing due to mechanical motion of the mirrors and backlash errors. The purpose of this study is to analyze an acousto-optic deflector (AOD) to achieve high diffraction efficiency and high deflection angle. Conventional AODs are operated with one-dimensional refractive index variation induced by modulating only the acoustic wave frequency. The effect of two-dimensional refractive index variations, which can be achieved by modulating both the phase and frequency of the acoustic waves, is analyzed. This study, therefore, advances the current AOD technology from just acoustic wave frequency modulation to a new class of AODs involving both phase and frequency modulations. This new type of AOD is designed by developing an analytic model based on the propagation of a laser beam through the AOD in which the refractive index varies in two dimensions. The model predicts a specific index modulation strength to achieve 100% diffraction efficiency. Also a fairly large diffraction efficiency can be obtained over a large angle of incidence, which is not possible in conventional AODs where the maximum diffraction efficiency occurs only at Bragg angle of incidence

Two-Dimensional Refractive Index Modulation By Phased Array Transducers In Acousto-Optic Deflectors

Acousto-optic deflectors are photonic devices that are used for scanning high-power laser beams in advanced microprocessing applications such as marking and direct writing. The operation of conventional deflectors mostly relies on one-dimensional sinusoidal variation of the refractive index in an acousto-optic medium. Sometimes static phased array transducers, such as step configuration or planar configuration transducer architecture, are used to tilt the index modulation planes for achieving higher performance and higher resolution than a single transducer AO device. However, the index can be modulated in two dimensions, and the modulation plane can be tilted arbitrarily by creating dynamic phase gratings in the medium using phased array transducers. This type of dynamic two-dimensional acousto-optic deflector can provide better performance using, for example, a large deflection angle and high diffraction efficiency. This paper utilizes an ultrasonic beam steering approach to study the two-dimensional strain-induced index modulation due to the photoelastic effect. The modulation is numerically simulated, and the effects of various parameters, such as the operating radiofrequency of the transducers, the ultrasonic beam steering angle, and different combinations of pressure on each element of the transducer array, are demonstrated

Gaussian Beam Diffraction By Two-Dimensional Refractive Index Modulation For High Diffraction Efficiency And Large Deflection Angle

The propagation of Gaussian beams is analyzed for an acousto-optic deflector by varying the refractive index in two-dimensions with a row of phased array piezoelectric transducers. Inhomogeneous domains of phase grating are produced by operating the transducers at different phase shifts, resulting in two-dimensional index modulation of periodic and sinc function profiles. Also different phase shifts provide a mechanism to steer the grating lobe in various directions and, therefore, the incident angle of the laser beam on the grating plane is automatically modified without moving the beam. Additionally, the acoustic frequency can be varied to achieve the Bragg condition for the new incident angle of the laser beam so that the diffraction efficiency of the deflector is maximized. The Gaussian beam is expressed in terms of planes and the second order coupled mode theory is implemented to analyze the diffraction of the beam. The diffraction efficiency is found to be nearly unity for optimal operating parameters of the acousto-optic device