scholarly journals Dynamics in the Sakaguchi-Kuramoto model with bimodal frequency distribution

PLoS ONE ◽  
2020 ◽  
Vol 15 (12) ◽  
pp. e0243196
Author(s):  
Shuangjian Guo ◽  
Yuan Xie ◽  
Qionglin Dai ◽  
Haihong Li ◽  
Junzhong Yang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Natural Frequency ◽  
Frequency Distribution ◽  
Bimodal Distribution ◽  
Kuramoto Model ◽  
Phase Lag ◽  
Phase Oscillators ◽  
Different Types ◽  
Low Dimensional

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.

CHAOS AND HYPERCHAOS IN A CLASS OF SIMPLE CELLULAR NEURAL NETWORKS MODELED BY O.D.E.

2006 ◽  
Vol 16 (09) ◽  
pp. 2729-2736 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Exponents ◽  
Cellular Neural Networks ◽  
New Class ◽  
Connection Matrices ◽  
Low Dimensional

This paper presents a new class of chaotic and hyperchaotic low dimensional cellular neural networks modeled by ordinary differential equations with some simple connection matrices. The chaoticity of these neural networks is indicated by positive Lyapunov exponents calculated by a computer.

scholarly journals Solving Oscillating Problems Using Modifying Runge-Kutta Methods

2021 ◽  
Vol 34 (4) ◽  
pp. 58-67
Author(s):  
Zainab Khaled Ghazal ◽  
Kasim Abbas Hussain
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lag Phase ◽  
Phase Lag ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Runge Kutta ◽  
Order Four

     This paper develop conventional Runge-Kutta methods of order four and order five to solve ordinary differential equations with oscillating solutions. The new modified Runge-Kutta methods (MRK) contain the invalidation of phase lag, phase lag’s derivatives, and amplification error. Numerical tests from their outcomes show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientific literature.

CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL

2005 ◽  
Vol 15 (11) ◽  
pp. 3457-3466 ◽  
Author(s):  
YURI L. MAISTRENKO ◽  
OLEKSANDR V. POPOVYCH ◽  
PETER A. TASS
Keyword(s):  
Chaotic Attractor ◽  
Nonlinear Dynamical System ◽  
Kuramoto Model ◽  
Phase Oscillators ◽  
Bifurcation And Chaos ◽  
Nonlinear Dynamical ◽  
Transition To Chaos ◽  
Quasiperiodic Dynamics ◽  
The Kuramoto Model ◽  
Low Dimensional

The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos. We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.

scholarly journals A Survey of Recent Results for the Generalizations of Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Daniel C. Biles ◽  
Márcia Federson ◽  
Rodrigo López Pouso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Time Scales ◽  
Review Paper ◽  
Different Types ◽  
Discontinuous Equations ◽  
Generalized Ordinary Differential Equations

This is a review paper on recent results for different types of generalized ordinary differential equations. Its scope ranges from discontinuous equations to equations on time scales. We also discuss their relation with inclusion and highlight the use of generalized integration to unify many of them under one single formulation.

A Posterior Model Reduction Method Based on Weighted Residue for Linear Partial Differential Equations

2014 ◽  
Vol 684 ◽  
pp. 34-40
Author(s):  
Jie Sha ◽  
Li Xiang Zhang ◽  
Chui Jie Wu
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Partial Differential Equation ◽  
Galerkin Projection ◽  
Lagrangian Multiplier ◽  
Partial Differential ◽  
Projection Equation ◽  
Dimensional Dynamical System ◽  
Low Dimensional

This paper is concerned with a new model reduced method based on optimal large truncated low-dimensional dynamical system, by which the solution of linear partial differential equation (PDE) is able to be approximate with highly accuracy. The method proposed is based on the weighted residue of PDE under consideration, and the weighted residue is used as an alternative optimal control condition (POT-WR) while solving the PDE. A set of bases is constructed to describe a dynamical system required in case. The Lagrangian multiplier is introduced to eliminate the constraints of the Galerkin projection equation, and the penalty function is used to remove the orthogonal constraint. According to the extreme principle, a set of the ordinary differential equations is obtained by taking the variational operation on generalized optimal function. A conjugate gradients algorithm on FORTRAN code is developed to solve these ordinary differential equations with Fourier polynomials as the initial bases for iterations. The heat transfer equation under a potential initial condition is used to verify the method proposed. Good agreement between the simulations and the analytical solutions of example was obtained, indicating that the POT-WR method presented in this paper provides the most effective posterior way of capturing the dominant characteristics of an infinite-dimensional dynamical system with only finitely few bases.

Low-Dimensional Discrete Kuramoto Model: Hierarchy of Multifrequency Quasiperiodicity Regimes

2014 ◽  
Vol 24 (07) ◽  
pp. 1430022 ◽  
Author(s):  
Alexander P. Kuznetsov ◽  
Yuliya V. Sedova
Keyword(s):  
Coupling Parameter ◽  
Kuramoto Model ◽  
Parameter Plane ◽  
Phase Oscillators ◽  
Repulsive Interactions ◽  
Discrete Phase ◽  
Model Hierarchy ◽  
Frequency Coupling ◽  
Low Dimensional ◽  
Two Parameter

The dynamics of a low-dimensional ensemble consisting of a network of five discrete phase oscillators is considered. A two-parameter synchronization picture, which appears instead of the Arnol'd tongues with an increase of the system dimension, is discussed. An appearance of the Arnol'd resonance web is detected on the "frequency–coupling" parameter plane. The cases of attractive and repulsive interactions are discussed.

scholarly journals THE FOURIER-ASYMPTOTIC APPROXIMATION AND ITS APPLICATIONS

1998 ◽  
Vol 3 (1) ◽  
pp. 14-24
Author(s):  
Mihails Belovs ◽  
Jānis Smotrovs
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Galerkin Method ◽  
Ordinary Differential Equations ◽  
Asymptotic Approximation ◽  
Fourier Coefficients ◽  
High Accuracy ◽  
Numerical Examples ◽  
Different Types ◽  
Accuracy Of Results

The Fourier ‐ asymptotic approximation can be obtained for different types of Fourier series by replacing the Fourier coefficients with their asymptotic (n → + 8) approximations beginning with some index n. We obtain some generalization of the classical Galerkin method for the solution of boundary and spectral problems of ordinary differential equations. The numerical examples show that the addition of asymptotic correction allows us to obtain a high accuracy of results.

A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations

2015 ◽  
Vol 7 (6) ◽  
pp. 754-779 ◽  
Author(s):  
Jie Sha ◽  
Lixiang Zhang ◽  
Chuijie Wu
Keyword(s):  
Dynamical System ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galerkin Projection ◽  
Reference Database ◽  
Lagrangian Multiplier ◽  
Partial Differential ◽  
Dimensional Dynamical System ◽  
Low Dimensional

AbstractThis paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only.

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