A normal form method for the determination of oscillations characteristics near the primary Hopf bifurcation in bandpass optoelectronic oscillators: Theory and experiment

2019 ◽  
Vol 29 (3) ◽  
pp. 033104 ◽  
Author(s):  
Jimmi H. Talla Mbé ◽  
Paul Woafo ◽  
Yanne K. Chembo
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Optoelectronic Oscillators ◽  
Normal Form Method ◽  
Theory And Experiment

scholarly journals Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Stable Equilibrium ◽  
Feedback Loops ◽  
Asymptotically Stable ◽  
Center Manifold Theorem ◽  
Feedback Strength ◽  
Coupling Parameters ◽  
Normal Form Method

The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation properties.

scholarly journals Synchronized Hopf Bifurcation Analysis in a Delay-Coupled Semiconductor Lasers System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Semiconductor Lasers ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Stability Switches ◽  
Center Manifold Theorem ◽  
Coupling Parameters ◽  
Normal Form Method ◽  
The Stability

The dynamics of a system of two semiconductor lasers, which are delay coupled via a passive relay within the synchronization manifold, are investigated. Depending on the coupling parameters, the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies. Employing the center manifold theorem and normal form method, an algorithm is derived for determining the Hopf bifurcation properties. Some numerical simulations are carried out to illustrate the analysis results.

Mathematical analysis of a model for thymus infection with discrete and distributed delays

2014 ◽  
Vol 07 (06) ◽  
pp. 1450070 ◽  
Author(s):  
M. Prakash ◽  
P. Balasubramaniam
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Different Types ◽  
Normal Form Method ◽  
Bifurcation And Stability ◽  
Theoretical Results ◽  
Hiv 1

In this paper, the dynamics of mathematical model for infection of thymus gland by HIV-1 is analyzed by applying some perturbation through two different types of delays such as in terms of Hopf bifurcation analysis. Further, the conditions for the existence of Hopf bifurcation are derived by evaluating the characteristic equation. The direction of Hopf bifurcation and stability of bifurcating periodic solutions are determined by employing the center manifold theorem and normal form method. Finally, some of the numerical simulations are carried out to validate the derived theoretical results and main conclusions are included.

scholarly journals Turing–Hopf Bifurcation and Spatio-Temporal Patterns of a Ratio-Dependent Holling–Tanner Model with Diffusion

2018 ◽  
Vol 28 (09) ◽  
pp. 1850108 ◽  
Author(s):  
Qi An ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Neumann Boundary ◽  
Temporal Patterns ◽  
Positive Equilibrium ◽  
Spatially Inhomogeneous ◽  
Normal Form Method ◽  
Ratio Dependent ◽  
Spatio Temporal ◽  
Model Subject

In this paper, the dynamics of a diffusive ratio-dependent Holling–Tanner model subject to Neumann boundary conditions is considered. We derive the conditions for the existence of Hopf, Turing, Turing–Hopf, Turing–Turing, Hopf-double-Turing and triple-Turing bifurcations at the unique positive equilibrium. Furthermore, we study the detailed dynamics in the neighborhood of the Turing–Hopf bifurcation by using the normal form method. Our results show that the Turing–Hopf bifurcation can give rise to the formation of the temporal and spatio-temporal patterns. In particular, we theoretically prove the existence of the spatially inhomogeneous periodic and quasi-periodic solutions, which can be used to explain the phenomenon of spatio-temporal resonance of the populations. Finally, the numerical simulations are given to illustrate the analytical results.

Analysis of a 2DOF Nonlinear Mechanical System Using the Normal Form Method

2019 ◽  
Author(s):  
David Julian Gonzalez Maldonado ◽  
Peter Hagedorn ◽  
Thiago Ritto ◽  
Rubens Sampaio ◽  
Artem Karev
Keyword(s):  
Normal Form ◽  
Mechanical System ◽  
Nonlinear Mechanical ◽  
Nonlinear Mechanical System ◽  
Normal Form Method

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

Accurate determination of the noise figure of polarization-dependent optical amplifiers: theory and experiment

2006 ◽  
Vol 24 (3) ◽  
pp. 1499-1503 ◽  
Author(s):  
T. Briant ◽  
P. Grangier ◽  
R. Tualle-Brouri ◽  
A. Bellemain ◽  
R. Brenot ◽  
...  
Keyword(s):  
Noise Figure ◽  
Optical Amplifiers ◽  
Accurate Determination ◽  
Theory And Experiment

scholarly journals Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

scholarly journals Absolute Determination of the Single-Photon Optomechanical Coupling Rate via a Hopf Bifurcation

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Paolo Piergentili ◽  
Wenlin Li ◽  
Riccardo Natali ◽  
David Vitali ◽  
Giovanni Di Giuseppe
Keyword(s):  
Hopf Bifurcation ◽  
Single Photon ◽  
Absolute Determination ◽  
Coupling Rate ◽  
Optomechanical Coupling
Sign in / Sign up

Export Citation Format

Share Document