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.

scholarly journals Analysis of two- and three-dimensional fractional-order Hindmarsh-Rose type neuronal models

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Eva Kaslik
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Three Dimensional ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Fast System ◽  
Stability Properties ◽  
Theoretical Results

AbstractA theoretical analysis of two- and three-dimensional fractional-order Hindmarsh-Rose neuronal models is presented, focusing on stability properties and occurrence of Hopf bifurcations, with respect to the fractional order of the system chosen as bifurcation parameter. With the aim of exemplifying and validating the theoretical results, numerical simulations are also undertaken, which reveal rich bursting behavior in the three-dimensional fractional-order slow-fast system.

scholarly journals Bifurcation of a Cohen-Grossberg Neural Network with Discrete Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Qiming Liu ◽  
Wang Zheng
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Explicit Formulas ◽  
Global Hopf Bifurcation ◽  
Discrete Delays ◽  
Bifurcation And Stability

A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

Stability Analysis of a Biological Model of a Marine Resources Allowing Density Dependent Migration

2017 ◽  
Vol 12 ◽  
pp. 22-34 ◽  
Author(s):  
Meriem Bentounsi ◽  
Imane Agmour ◽  
Naceur Achtaich ◽  
Youssef El Foutayeni
Keyword(s):  
Numerical Simulation ◽  
Stability Analysis ◽  
Asymptotic Stability ◽  
Food Chain ◽  
Sufficient Conditions ◽  
The Other ◽  
Marine Resources ◽  
Temporal Models ◽  
Spatio Temporal ◽  
Rigorous Mathematical Analysis

Biology of a marine resources is a descriptive science. The description is the first step towards understanding a system. However, the main objective is to present a rigorous mathematical analysis and numerical simulation of these spatio temporal models. In the present paper, we consider a two species food chain, i.e. a prey and predator populations modeled in a two-patch environment, one of which is a free fishing zone and the other one is protected zone. We study the qualitative analysis of solutions and we establish sufficient conditions under which the endemic and trivial equilibria are asymptotically stable.The asymptotic stability corresponding to the equilibria is graphically shown.

scholarly journals Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. C. Wei ◽  
J. F. Wang ◽  
A. Akgul
Keyword(s):  
Hopf Bifurcation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Financial System ◽  
Invariant Set ◽  
Phase Portraits ◽  
Lyapunov Coefficient ◽  
Stable Periodic ◽  
Analytical Proof ◽  
Positively Invariant Set

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

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.

scholarly journals Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances

2016 ◽  
Vol 14 (1) ◽  
pp. 1157-1173 ◽  
Author(s):  
Fengde Chen ◽  
Xiaoxing Chen ◽  
Shouying Huang
Keyword(s):  
Functional Response ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
The Other ◽  
Toxic Substances ◽  
Two Dimensional ◽  
Competitive System ◽  
Volterra Systems ◽  
Appl Math

AbstractA two species non-autonomous competitive phytoplankton system with Beddington-DeAngelis functional response and the effect of toxic substances is proposed and studied in this paper. Sufficient conditions which guarantee the extinction of a species and global attractivity of the other one are obtained. The results obtained here generalize the main results of Li and Chen [Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances, Appl. Math. Comput. 182(2006)684-690]. Numeric simulations are carried out to show the feasibility of our results.

scholarly journals Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Normal Form Method ◽  
Predator Prey Models ◽  
The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

The dynamic lateral instability of beams

1954 ◽  
Vol 226 (1164) ◽  
pp. 111-128
Keyword(s):  
Theoretical Analysis ◽  
Reasonable Agreement ◽  
Vertical Plane ◽  
Bending Moment ◽  
Critical Value ◽  
The Other ◽  
Lateral Bending ◽  
Lateral Instability ◽  
Maximum Stiffness ◽  
The Moment

The vibrations of a deep slender beam, bent to uniform curvature by in variant moments acting in a vertical plane, which is also the plane of maximum stiffness, have been studied. It is shown that the moments couple up the lateral bending and torsional modes of the beam, those modes being replaced by two independent modes, each involving torsion and flexure. One of these m odes is associated with a frequency which decreases with increasing bending moment, the frequency becoming zero when the moment reaches the critical value for lateral instability. The other mode is associated with a frequency which increases with bending moment. Experiments were carried out on an I-section cantilever carrying an end mass. Owing to the varying bending moment, the theoretical analysis of this case is more complicated, and an iterative method, originated by Schwarz (1890), has been employed. Results are in reasonable agreement with experiment.

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