scholarly journals 1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Bo Li ◽  
Zhimin He
Keyword(s):  
Normal Form ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Two Dimensional ◽  
Invariant Circle ◽  
Dynamical Behaviors ◽  
Normal Form Method

1 : 3 resonance of a two-dimensional discrete Hindmarsh-Rose model is discussed by normal form method and bifurcation theory. Numerical simulations are presented to illustrate the theoretical analysis, which predict the occurrence of a closed invariant circle, period-three saddle cycle, and homoclinic structure. Furthermore, it also displays the complex dynamical behaviors, especially the transitions between three main dynamical behaviors, namely, quiescence, spiking, and bursting.

scholarly journals Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay

2021 ◽  
Vol 26 (3) ◽  
pp. 375-395
Author(s):  
Rina Su ◽  
Chunrui Zhang
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Distribution Of Eigenvalues ◽  
Normal Form Method ◽  
The Stability

In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method.

CHAOS BEHAVIOR IN THE DISCRETE BVP OSCILLATOR

2002 ◽  
Vol 12 (03) ◽  
pp. 619-627 ◽  
Author(s):  
ZHUJUN JING ◽  
ZHIYUAN JIA ◽  
RUIQI WANG
Keyword(s):  
Theoretical Analysis ◽  
Periodic Orbit ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Period Doubling ◽  
Euler Method ◽  
Dynamical Behaviors ◽  
Chaotic Orbits ◽  
Marotto’S Chaos ◽  
Definition Of

The discrete BVP oscillator obtained through the Euler method is investigated, and also first proved that there exist chaotic phenomena in the sense of Marotto's definition of chaos and two-period cycles. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including the ten-periodic orbit, a cascade of period-doubling bifurcation, quasiperiodic orbits and the chaotic orbits in Marotto's chaos and intermitten's chaos. The computations of Lyapunov exponents confirm the existence of dynamical behaviors.

scholarly journals Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Lingling Li ◽  
Jianwei Shen
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Dynamical Behaviors ◽  
Theoretical Results

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.

Bifurcation Analysis of a Bounded Rational Duopoly Game with Consumer Surplus

2021 ◽  
Vol 31 (07) ◽  
pp. 2150097
Author(s):  
Wei Zhou ◽  
Yinxia Cao ◽  
Amr Elsonbaty ◽  
A. A. Elsadany ◽  
Tong Chu
Keyword(s):  
Normal Form ◽  
Theoretical Analysis ◽  
Center Manifold ◽  
Profit Maximization ◽  
Consumer Surplus ◽  
Normal Form Theory ◽  
Nonlinear Dynamical ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Form Theory

The nonlinear dynamical behaviors of economic models have been extensively examined and still represented a great challenge for economists in recent and future years. A proposed boundedly rational game incorporating consumer surplus is introduced. This paper aims at studying stability and bifurcation types of the presented model. The flip and Neimark–Sacker bifurcations are analyzed via applying the normal form theory and the center manifold theorem. This study helps determine an appropriate choice of decision parameters which have significant influences on the behavior of the game. The duopoly game that is formed by considering bounded rationality and consumer surplus is more realistic than the ordinary duopoly game which only has profit maximization. And then, some numerical simulations are provided to verify the theoretical analysis. Finally, we compare the dynamical behaviors of the built model with that of Bischi–Naimzada model so as to better understand the performance of the duopoly game with consumer surplus.

Bifurcation Analysis of a Network of Three Neurons with Time Delays

2011 ◽  
Vol 105-107 ◽  
pp. 147-150
Author(s):  
Xiao Chen Mao
Keyword(s):  
Hopf Bifurcation ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Trivial Equilibrium ◽  
Nice Agreement ◽  
Theoretical Results

The paper studies the dynamical behaviors of a recurrent neural network model consisting of three neurons with time delays through theoretical analysis and numerical simulations. The local stability of the trivial equilibrium of the network is analyzed and the sufficient conditions of the existence of Hopf bifurcation are given by discussing the associated characteristic equation. The direction and stability of the bifurcated periodic oscillations arising from Hopf bifurcation, which depend on the nonlinear terms of the network, are determined by means of the normal form and the center manifold theorem. Afterwards, numerical examples are performed to validate the theoretical analysis. The case studies of numerical simulations reach nice agreement with the theoretical results.

An Improved Way of Detecting the Threshold Value of Chaotic Motion on a Parametrical Excited Rectangular Thin Plate

2011 ◽  
Vol 66-68 ◽  
pp. 833-837
Author(s):  
Gen Ge ◽  
Zhi Wen Zhu
Keyword(s):  
Normal Form ◽  
Theoretical Analysis ◽  
Fundamental Frequency ◽  
Time Scale ◽  
Chaotic Motion ◽  
Threshold Value ◽  
Scale Transformation ◽  
Rectangular Thin Plate ◽  
Frequency Method ◽  
Normal Form Method

One dynamical model of a thin rectangular plate subject to in-plate parametrical excitation is proposed based on elastic theory and Galerkin’s approach. At first, the undermined fundamental frequency and normal form method was utilized to study the influence of the disturbing parameters to the fundamental frequency. Secondly, the improved Melnikov expression for the oscillator was built based on the results of the undermined fundamental frequency method and time scale transformation to improve the approximate threshold value of chaotic motion in the Homoclinicity. Finally, the numerical results show the efficiency of the theoretical analysis.

A Hyperchaotic System and its Synchronization via Scalar Controller

2011 ◽  
Vol 50-51 ◽  
pp. 254-257
Author(s):  
Wu Chun Dai ◽  
Zheng Fu Cheng
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Mathematical Proof ◽  
Hyperchaotic System ◽  
Scalar Control ◽  
Response System ◽  
Dynamical Behaviors ◽  
Synchronization Scheme ◽  
New Hyperchaotic System

In this paper, a 4D hyperchaotic system is proposed. Some basic dynamical behaviors are explored by calculating its Lyapunov exponents, Poincar´e mapping, etc.. Finally, synchronization for this new hyperchaotic system is achieved via scalar control. The nonlinear terms in the response system are not dropped. The proposed synchronization scheme is simple and theoretically rigorous. The mathematical proof of this method is provided. Some numerical simulations are obtained. The numerical simulations coincide with the theoretical analysis.

Contour Controller Design for Two-Dimensional Stage System with Friction

2006 ◽  
Vol 505-507 ◽  
pp. 1267-1272 ◽  
Author(s):  
Chen Hsieh ◽  
K.C. Lin ◽  
Chieh Li Chen
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Controller Design ◽  
Coordinate Frame ◽  
Two Dimensional ◽  
Contouring Control ◽  
Stage System

For multiple-axis stages, it is required to operate the axes simultaneously, such that the resulting trajectory of platform follows a given contour. For most stage systems, friction acts as the major disturbance which degrades the precision of system motion and its effect should be compensated. In this paper, contouring control of a two-dimensional stage system subjected to friction is investigated. A systematic contour controller design based on task coordinate frame is proposed and its effectiveness is studied through theoretical analysis and numerical simulations.

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.

Sign in / Sign up

Export Citation Format

Share Document