scholarly journals Hopf Bifurcation of Compound Stochastic van der Pol System

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Shaojuan Ma ◽  
Qianling Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Theory ◽  
Noise Intensity ◽  
Gaussian White Noise ◽  
Random Parameter ◽  
Critical Value ◽  
Deterministic System ◽  
Bifurcation Parameter ◽  
Van Der Pol ◽  
Stochastic Hopf Bifurcation

Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L) expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strengthδand noise intensityσon stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increasedδcan relocate the critical value of bifurcation parameter forward while increasedσmakes it backward and the influence ofδis more sensitive thanσ. The results are verified by numerical simulations.

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
Author(s):  
F. HU ◽  
W. Q. ZHU ◽  
L. C. CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bifurcation Parameter ◽  
Integrable Hamiltonian Systems ◽  
Fractional Derivative Damping ◽  
Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

Bifurcation and Control in a Singular Phytoplankton-Zooplankton-Fish Model with Nonlinear Fish Harvesting and Taxation

2018 ◽  
Vol 28 (03) ◽  
pp. 1850042 ◽  
Author(s):  
Xin-You Meng ◽  
Yu-Qian Wu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
State Feedback ◽  
Critical Value ◽  
Feedback Controller ◽  
Bifurcation Parameter ◽  
State Feedback Controller ◽  
Interior Equilibrium ◽  
Fish Model ◽  
Singularity Induced Bifurcation

In this paper, a delayed differential algebraic phytoplankton-zooplankton-fish model with taxation and nonlinear fish harvesting is proposed. In the absence of time delay, the existence of singularity induced bifurcation is discussed by regarding economic interest as bifurcation parameter. A state feedback controller is designed to eliminate singularity induced bifurcation. Based on Liu’s criterion, Hopf bifurcation occurs at the interior equilibrium when taxation is taken as bifurcation parameter and is more than its corresponding critical value. In the presence of time delay, by analyzing the associated characteristic transcendental equation, the interior equilibrium loses local stability when time delay crosses its critical value. What’s more, the direction of Hopf bifurcation and stability of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem, and nonlinear state feedback controller is designed to eliminate Hopf bifurcation. Furthermore, Pontryagin’s maximum principle has been used to obtain optimal tax policy to maximize the benefit as well as the conservation of the ecosystem. Finally, some numerical simulations are given to demonstrate our theoretical analysis.

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 Hopf Bifurcations of a Stochastic Fractional-Order Van der Pol System

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Stochastic System ◽  
Critical Parameter ◽  
Random Parameter ◽  
Order System ◽  
Hopf Bifurcations ◽  
Van Der Pol ◽  
Existence Conditions ◽  
Theoretical Results

The Hopf bifurcation of a fractional-order Van der Pol (VDP for short) system with a random parameter is investigated. Firstly, the Chebyshev polynomial approximation is applied to study the stochastic fractional-order system. Based on the method, the stochastic system is reduced to the equivalent deterministic one, and then the responses of the stochastic system can be obtained by numerical methods. Then, according to the existence conditions of Hopf bifurcation, the critical parameter value of the bifurcation is obtained by theoretical analysis. Then, numerical simulations are carried out to verify the theoretical results.

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.

scholarly journals Stability and Bifurcation of Two Kinds of Three-Dimensional Fractional Lotka-Volterra Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jinglei Tian ◽  
Yongguang Yu ◽  
Hu Wang
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Theoretical Analysis ◽  
Fractional Order ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Critical Value ◽  
The Other ◽  
Bifurcation Parameter ◽  
Volterra Systems

Two kinds of three-dimensional fractional Lotka-Volterra systems are discussed. For one system, the asymptotic stability of the equilibria is analyzed by providing some sufficient conditions. And bifurcation property is investigated by choosing the fractional order as the bifurcation parameter for the other system. In particular, the critical value of the fractional order is identified at which the Hopf bifurcation may occur. Furthermore, the numerical results are presented to verify the theoretical analysis.

Delay-Induced Bifurcation in High-Order Fractional Goodwin Models with Disparate Orders

2019 ◽  
Vol 29 (07) ◽  
pp. 1950090 ◽  
Author(s):  
Lingzhi Zhao ◽  
Chengdai Huang ◽  
Jinde Cao ◽  
Min Xiao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delays ◽  
Critical Value ◽  
High Order ◽  
High Dimensional ◽  
Bifurcation Parameter ◽  
Characteristic Roots ◽  
Goodwin Model ◽  
Linearized System

In the present paper, we first attempt to investigate the bifurcation for the delayed high-dimensional fractional Goodwin model with different orders. By taking the sum of time delays as a bifurcation parameter, the distribution of the characteristic roots of the linearized system is analyzed and the conditions of Hopf bifurcation are obtained. It is revealed that the model undergoes the Hopf bifurcation when the bifurcation parameter increases and exceeds the critical value. To verify the efficiency of our analytic findings, two simulation examples have been ultimately provided.

Sign in / Sign up

Export Citation Format

Share Document