Attractor and bifurcation of forced Lorenz-84 system

2019 ◽  
Vol 16 (01) ◽  
pp. 1950002 ◽  
Author(s):  
Yongjian Liu ◽  
Zhouchao Wei ◽  
Chunbiao Li ◽  
Aimin Liu ◽  
Lijie Li
Keyword(s):  
Random Attractor ◽  
Stochastic Bifurcation ◽  
Global Dynamics ◽  
Almost Periodic ◽  
Periodic Attractor ◽  
Stochastic Forcing ◽  
Bifurcation Phenomenon ◽  
Driven System ◽  
Theoretical Results ◽  
Bifurcation Behavior

In this paper, global dynamics of forced Lorenz-84 system are discussed, and some new results are presented. First of all, the periodic attractor is analyzed for the almost periodic Lorenz-84 system with almost periodically forcing, including the existence and the boundedness of those almost periodic solutions, and the bifurcation phenomenon in the driven system. Then, the random attractor set and the bifurcation phenomenon for the driven Lorenz-84 system with stochastic forcing are studied, including the globally exponentially attractive set, positive invariant set, random attractor and stochastic bifurcation behavior. Last but not least, some numerical simulations are also presented for verifying obtained theoretical results.

scholarly journals Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Yanju Xiao ◽  
Weipeng Zhang ◽  
Guifeng Deng ◽  
Zhehua Liu
Keyword(s):  
Sufficient Conditions ◽  
Global Dynamics ◽  
Sis Model ◽  
Delayed Treatment ◽  
Sis Epidemic Model ◽  
Treatment Function ◽  
Lyapunov Coefficient ◽  
Saturated Treatment ◽  
Theoretical Results ◽  
Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

Almost periodic attractor for Cohen–Grossberg neural networks with delay

2009 ◽  
Vol 373 (4) ◽  
pp. 434-440 ◽  
Author(s):  
Zhang Chen ◽  
Donghua Zhao ◽  
Jiong Ruan
Keyword(s):  
Neural Networks ◽  
Almost Periodic ◽  
Periodic Attractor

Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

2021 ◽  
Vol 31 (03) ◽  
pp. 2150050
Author(s):  
Demou Luo ◽  
Qiru Wang
Keyword(s):  
Growth Rate ◽  
Allee Effect ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Growth ◽  
Trivial Equilibrium ◽  
Existence And Stability ◽  
Stable Equilibria ◽  
Theoretical Results

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

scholarly journals Analysis of the Mathematical Model for the Spread of Pine Wilt Disease

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xiangyun Shi ◽  
Guohua Song
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Pine Wilt Disease ◽  
Wilt Disease ◽  
Endemic Equilibrium ◽  
Feasible Region ◽  
Global Dynamics ◽  
Pine Wilt ◽  
Theoretical Results ◽  
The Basic Reproduction Number

This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction numberℛ0and the other valueℛcwhich is larger thanℛ0. Ifℛ0andℛcare both less than one, the disease-free equilibrium is asymptotically stable and the pine wilt disease always dies out. If one is between the two values, though the pine wilt disease could occur, the outbreak will stop. If the basic reproduction number is greater than one, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists. Numerical simulations are carried out to illustrate the theoretical results, and some disease control measures are especially presented by these theoretical results.

scholarly journals Global Dynamics of Some Exponential Type Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-24
Author(s):  
A. Q. Khan ◽  
H. M. Arshad ◽  
B. A. Younis ◽  
KH. I. Osman ◽  
Tarek F. Ibrahim ◽  
...  
Keyword(s):  
Fixed Point ◽  
Lyapunov Function ◽  
Exponential Type ◽  
Type Systems ◽  
Global Dynamics ◽  
Dynamical Properties ◽  
Boundedness And Persistence ◽  
Invariant Rectangle ◽  
Theoretical Results ◽  
Positive Fixed Point

We explore the boundedness and persistence, existence of an invariant rectangle, local dynamical properties about the unique positive fixed point, global dynamics by the discrete-time Lyapunov function, and the rate of convergence of some 2,3-type exponential systems of difference equations. Finally, theoretical results are numerically verified.

scholarly journals Bifurcating energy-angular spectrum of electrons accelerated by intense laser pulse

2007 ◽  
Vol 25 (3) ◽  
pp. 365-370 ◽  
Author(s):  
D. Lin ◽  
Y.K. HO ◽  
Q. Kong ◽  
Z. Chen ◽  
P.X. Wang ◽  
...  
Keyword(s):  
Laser Pulse ◽  
Angular Correlation ◽  
Potential Model ◽  
Angular Spectrum ◽  
Intense Laser ◽  
Bifurcation Phenomenon ◽  
Correlation Spectrum ◽  
Laser Acceleration ◽  
Ponderomotive Potential ◽  
Theoretical Results

Bifurcation phenomenon in the energy-angular correlation spectrum of the vacuum laser acceleration has been observed with computer simulation. Concerning a focused laser pulse, the classical single-valued energy-angular correlation spectrum for a plane wave is, besides broadened to a band, bifurcated with the classical value in between the two branches. Analytic expression to describe the correlation has been derived and physical explanations based on the ponderomotive potential model and Lorentz-Newton force analyses are presented. The theoretical results are supported by numerical simulations which have been compared with the experimental results. This study is helpful in designing vacuum laser acceleration experiments.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

Bifurcation and attractor of the stochastic Rabinovich system with jump

2015 ◽  
Vol 12 (09) ◽  
pp. 1550092 ◽  
Author(s):  
Yongjian Liu ◽  
Lijie Li ◽  
Xiong Wang
Keyword(s):  
Stochastic Stability ◽  
Geometrical Structure ◽  
Necessary Condition ◽  
Stochastic Bifurcation ◽  
Sufficient Condition ◽  
Random Attractors ◽  
The Rich ◽  
Global Attractive Set ◽  
Bifurcation Behavior

In this paper, the bifurcation and attractor of the stochastic Rabinovich system with jump are discussed, and some new results for the system are presented. First, the sufficient condition and necessary condition for stochastic stability of the system are given. Second, the estimation of the global attractive set of system is obtained. The existence of random attractors of the stochastic Rabinovich system with jump is also discussed. Finally, stochastic bifurcation behavior for the system is analyzed. It is hoped that the investigation of this paper can help understanding the rich dynamic of the stochastic Rabinovich system and the true geometrical structure of the original amazing Rabinovich attractor.

TOWARD AN UNDERSTANDING OF STOCHASTIC HOPF BIFURCATION

1996 ◽  
Vol 06 (11) ◽  
pp. 1947-1975 ◽  
Author(s):  
LUDWIG ARNOLD ◽  
N. SRI NAMACHCHIVAYA ◽  
KLAUS R. SCHENK-HOPPÉ
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Lyapunov Exponents ◽  
Variational Equation ◽  
Van Der Pol Oscillator ◽  
Stochastic Bifurcation ◽  
Van Der Pol ◽  
Truncated Normal ◽  
Stochastic Hopf Bifurcation ◽  
Bifurcation Behavior

In this paper, asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through the study of a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied. In the first part of this paper, we present an introduction to the theory of random dynamical systems (in particular, their generation, their invariant measures, the multiplicative ergodic theorem, and Lyapunov exponents). We then present the two concepts of stochastic bifurcation theory: Phenomenological (based on the Fokker-Planck equation), and dynamical (based on Lyapunov exponents). The method of stochastic averaging of the nonlinear system yields a set of equations which, together with its variational equation, can be explicitly solved and hence its bifurcation behavior completely analyzed. We augment this analysis by asymptotic expansions of the Lyapunov exponents of the variational equation at zero. Finally, the stochastic normal form of the noisy Duffing-van der Pol oscillator is derived, and its bifurcation behavior is analyzed numerically. The result is that the (truncated) normal form retains the essential bifurcation characteristics of the full equation.

scholarly journals Global Dynamics of Infectious Disease with Arbitrary Distributed Infectious Period on Complex Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Xiaoguang Zhang ◽  
Rui Song ◽  
Gui-Quan Sun ◽  
Zhen Jin
Keyword(s):  
Complex Networks ◽  
Reproduction Number ◽  
Epidemic Models ◽  
Infectious Period ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
Different Types ◽  
Theoretical Results ◽  
Disease Free

Most of the current epidemic models assume that the infectious period follows an exponential distribution. However, due to individual heterogeneity and epidemic diversity, these models fail to describe the distribution of infectious periods precisely. We establish a SIS epidemic model with multistaged progression of infectious periods on complex networks, which can be used to characterize arbitrary distributions of infectious periods of the individuals. By using mathematical analysis, the basic reproduction numberR0for the model is derived. We verify that theR0depends on the average distributions of infection periods for different types of infective individuals, which extend the general theory obtained from the single infectious period epidemic models. It is proved that ifR0<1, then the disease-free equilibrium is globally asymptotically stable; otherwise the unique endemic equilibrium exists such that it is globally asymptotically attractive. Finally numerical simulations hold for the validity of our theoretical results is given.

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