scholarly journals The Yamada Model for a Self-Pulsing Laser: Bifurcation Structure for Nonidentical Decay Times of Gain and Absorber

2020 ◽  
Vol 30 (14) ◽  
pp. 2030039
Author(s):  
Robert Otupiri ◽  
Bernd Krauskopf ◽  
Neil G. R. Broderick
Keyword(s):  
Three Dimensional ◽  
Equation Model ◽  
Decay Rates ◽  
Phase Portraits ◽  
Stable Phase ◽  
Bifurcation Structure ◽  
Decay Times ◽  
Comprehensive Picture ◽  
Two Parameter ◽  
Parameter Bifurcation

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as [Formula: see text]-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits. The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on [Formula: see text]-switching lasers with different kinds of saturable absorbers.

CHAOS IN A THREE-DIMENSIONAL GENERAL MODEL OF NEURAL NETWORK

2002 ◽  
Vol 12 (10) ◽  
pp. 2271-2281 ◽  
Author(s):  
A. DAS ◽  
PRITHA DAS ◽  
A. B. ROY
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
General Model ◽  
Three Dimensional ◽  
Stationary Regime ◽  
Decay Rates ◽  
Phase Portraits ◽  
Study Phase ◽  
Bifurcation Diagrams ◽  
Parameter Values

The dynamics of a network of three neurons with all possible connections is studied here. The equations of control are given by three differential equations with nonlinear, positive and bounded sigmoidal response function of the neurons. The system passes from stable to periodic and then to chaotic regimes and returns to stationary regime with change in parameter values of synaptic weights and decay rates. We have developed programs and used Locbif package to study phase portraits, bifurcation diagrams which confirm the result. Lyapunov Exponents have been calculated to confirm chaos.

Two-parameter bifurcation analysis of longitudinal flight dynamics

2003 ◽  
Vol 39 (3) ◽  
pp. 1103-1112 ◽  
Author(s):  
Der-Cherng Liaw ◽  
Chau-Chung Song ◽  
Yew-Wen Liang ◽  
Wen-Ching Chung
Keyword(s):  
Bifurcation Analysis ◽  
Flight Dynamics ◽  
Two Parameter ◽  
Parameter Bifurcation

scholarly journals Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 876
Author(s):  
Wieslaw Marszalek ◽  
Jan Sadecki ◽  
Maciej Walczak
Keyword(s):  
Equilibrium Points ◽  
Autonomous Systems ◽  
Cytosolic Calcium ◽  
Calcium Oscillations ◽  
Dynamical Model ◽  
Bifurcation Diagrams ◽  
Step Size ◽  
Oscillatory Systems ◽  
Two Parameter ◽  
Parameter Bifurcation

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation.

scholarly journals Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

Meccanica ◽  
2021 ◽  
Author(s):  
Gábor Csernák ◽  
Gábor Licskó
Keyword(s):  
Dry Friction ◽  
Continuation Method ◽  
Friction Model ◽  
Lugre Friction Model ◽  
Kinetic Coefficients ◽  
Friction Oscillator ◽  
Lugre Friction ◽  
Friction Parameters ◽  
Two Parameter ◽  
Parameter Bifurcation

AbstractThe responses of a simple harmonically excited dry friction oscillator are analysed in the case when the coefficients of static and kinetic coefficients of friction are different. One- and two-parameter bifurcation curves are determined at suitable parameters by continuation method and the largest Lyapunov exponents of the obtained solutions are estimated. It is shown that chaotic solutions can occur in broad parameter domains—even at realistic friction parameters—that are tightly enclosed by well-defined two-parameter bifurcation curves. The performed analysis also reveals that chaotic trajectories are bifurcating from special asymmetric solutions. To check the robustness of the qualitative results, characteristic bifurcation branches of two slightly modified oscillators are also determined: one with a higher harmonic in the excitation, and another one where Coulomb friction is exchanged by a corresponding LuGre friction model. The qualitative agreement of the diagrams supports the validity of the results.

scholarly journals The holographic conformal manifold of 3d $$ \mathcal{N} $$ = 2 S-fold SCFTs

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Nikolay Bobev ◽  
Friðrik Freyr Gautason ◽  
Jesse van Muiden
Keyword(s):  
String Theory ◽  
Three Dimensional ◽  
Parameter Family ◽  
Global Symmetry ◽  
Maximal Supergravity ◽  
All Solutions ◽  
Operator Spectrum ◽  
Type Iib String Theory ◽  
Two Parameter

Abstract We employ a non-compact gauging of four-dimensional maximal supergravity to construct a two-parameter family of AdS4 J-fold solutions preserving $$ \mathcal{N} $$ N = 2 supersymmetry. All solutions preserve $$ \mathfrak{u} $$ u (1) × $$ \mathfrak{u} $$ u (1) global symmetry and in special limits we recover the previously known $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{u} $$ u (1) invariant $$ \mathcal{N} $$ N = 2 and $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{su} $$ su (2) invariant $$ \mathcal{N} $$ N = 4 J-fold solutions. This family of AdS4 backgrounds can be uplifted to type IIB string theory and is holographically dual to the conformal manifold of a class of three-dimensional S-fold SCFTs obtained from the $$ \mathcal{N} $$ N = 4 T [U(N)] theory of Gaiotto-Witten. We find the spectrum of supergravity excitations of the AdS4 solutions and use it to study how the operator spectrum of the three-dimensional SCFT depends on the exactly marginal couplings.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

scholarly journals Luminescence of SiO2-BaF2:Tb3+, Eu3+ Nano-Glass-Ceramics Made from Sol–Gel Method at Low Temperature

Nanomaterials ◽  
2022 ◽  
Vol 12 (2) ◽  
pp. 259
Author(s):  
Natalia Pawlik ◽  
Barbara Szpikowska-Sroka ◽  
Tomasz Goryczka ◽  
Ewa Pietrasik ◽  
Wojciech A. Pisarski
Keyword(s):  
Sol Gel ◽  
Three Dimensional ◽  
Glass Ceramics ◽  
Emission Spectra ◽  
Scanning Calorimetry ◽  
Light Emitting ◽  
Co Doping ◽  
Dsc Measurements ◽  
Decay Times

The synthesis and characterization of multicolor light-emitting nanomaterials based on rare earths (RE3+) are of great importance due to their possible use in optoelectronic devices, such as LEDs or displays. In the present work, oxyfluoride glass-ceramics containing BaF2 nanocrystals co-doped with Tb3+, Eu3+ ions were fabricated from amorphous xerogels at 350 °C. The analysis of the thermal behavior of fabricated xerogels was performed using TG/DSC measurements (thermogravimetry (TG), differential scanning calorimetry (DSC)). The crystallization of BaF2 phase at the nanoscale was confirmed by X-ray diffraction (XRD) measurements and transmission electron microscopy (TEM), and the changes in silicate sol–gel host were determined by attenuated total reflectance infrared (ATR-IR) spectroscopy. The luminescent characterization of prepared sol–gel materials was carried out by excitation and emission spectra along with decay analysis from the 5D4 level of Tb3+. As a result, the visible light according to the electronic transitions of Tb3+ (5D4 → 7FJ (J = 6–3)) and Eu3+ (5D0 → 7FJ (J = 0–4)) was recorded. It was also observed that co-doping with Eu3+ caused the shortening in decay times of the 5D4 state from 1.11 ms to 0.88 ms (for xerogels) and from 6.56 ms to 4.06 ms (for glass-ceramics). Thus, based on lifetime values, the Tb3+/Eu3+ energy transfer (ET) efficiencies were estimated to be almost 21% for xerogels and 38% for nano-glass-ceramics. Therefore, such materials could be successfully predisposed for laser technologies, spectral converters, and three-dimensional displays.

scholarly journals Neimark-Sacker Bifurcation in Demand-Inventory Model with Stock-Level-Dependent Demand

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Piotr Hachuła ◽  
Magdalena Nockowska-Rosiak ◽  
Ewa Schmeidel
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Inventory Model ◽  
Phase Portraits ◽  
Dimensional System ◽  
Planar System ◽  
Stock Level ◽  
Lyapunov Coefficient ◽  
Analysis Of Dynamics ◽  
Dependent Demand

An analysis of dynamics of demand-inventory model with stock-level-dependent demand formulated as a three-dimensional system of difference equations with four parameters is considered. By reducing the model to the planar system with five parameters, an analysis of one-parameter bifurcation of equilibrium points is presented. By the analytical method, we prove that nondegeneracy conditions for the existence of Neimark-Sacker bifurcation for the planar system are fulfilled. To check the sign of the first Lyapunov coefficient of Neimark-Sacker bifurcation, we use numerical simulations. We give phase portraits of the planar system to confirm the previous analytical results and show new interesting complex dynamical behaviours emerging in it. Finally, the economical interpretation of the system is given.

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