scholarly journals Hopf Bifurcation Analysis in a Delayed Kaldor-Kalecki Model of Business Cycle

2008 ◽  
Vol 13 (4) ◽  
pp. 439-449 ◽  
Author(s):  
A. Kaddar ◽  
H. Talibi Alaoui
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Business Cycle ◽  
Bifurcation Analysis ◽  
Capital Stock ◽  
Capital Accumulation ◽  
Local Stability ◽  
Critical Value

In this paper, we analyze the model of business cycle with time delay set forth by A. Krawiec and M. Szydłowski [1]. Our goal in this model is to introduce the time delay into capital stock and gross product in capital accumulation equation. The dynamics are studied in terms of local stability and of the description of the Hopf bifurcation, that is proven to exist as the delay (taken as a parameter of bifurcation) cross some critical value. Additionally we conclude with an application.

BIFURCATION ANALYSIS OF A DETRITUS-BASED ECOSYSTEM WITH TIME DELAY

2000 ◽  
Vol 08 (03) ◽  
pp. 255-261 ◽  
Author(s):  
DEBASIS MUKHERJEE ◽  
SANTANU RAY ◽  
DILIP KUMAR SINHA
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Mangrove Ecosystem ◽  
Stability Criteria ◽  
Delay Effect

This article concentrates on the study of delay effect of a mangrove ecosystem of detritus, detritivores and predator of detritivores. Local stability criteria are derived in the absence of delays. Conditions are found out for which the system undergoes a Hopf bifurcation. Further conditions are derived for which there can be no change in stability.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

scholarly journals Hopf Bifurcation in an Augmented IS-LM Linear Business Cycle Model with Two Time Delays

2020 ◽  
Vol 5 (3) ◽  
pp. 518-528
Author(s):  
Firdos Karim ◽  
Sudipa Chauhan ◽  
Sumit Kaur Bhatia ◽  
Joydip Dhar
Keyword(s):  
Hopf Bifurcation ◽  
Business Cycle ◽  
Capital Accumulation ◽  
Local Stability ◽  
Time Delays ◽  
Equilibrium Points ◽  
Scale Production ◽  
Cycle Model ◽  
Investment Function ◽  
Business Cycle Model

This paper deals with the amalgamated basic IS-LM business cycle model with Kaldor’s growth model to form an augmented model. Pertaining to substantial evidence, IS-LM model in paradigm with a specific economic extension (Kaldor-Kalecki Business cycle model in our case) provides an adept explanation of a developing but strong economy like that of our country. Occurring in the equation of capital accumulation, the two time delays are a result of the assumption in the investment function being both income and capital stock dependent in past period and maturity period. Investigating a model combined with capital accumulation is both interesting and important. From economist point of view, production without capital is impossible to even imagine. Moreover capital accumulation is impeccable to large-scale production, specialisation and creation of employment opportunities. In our model ‘I’ the investment function, ‘S’ the savings function and ‘L’ the demand for money are depending linearly on their arguments. We adhere to a linear model, contrary to the popular belief of non- linear models being the undisputed style for modern economics. The model is first shown to be mathematically and economically poised. The local stability of boundary and interior equilibrium points has been investigated. Three cases arise, pertaining to two time delays. System dynamics exhibits mutation under the influence of time delays and may clinch or discharge its local stability when subjected to the latter. Hopf bifurcation occurs when the delay parameter crosses a critical value.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

Time-delay-induced instabilities and Hopf bifurcation analysis in 2-neuron network model with reaction–diffusion term

2018 ◽  
Vol 313 ◽  
pp. 306-315 ◽  
Author(s):  
Swati Tyagi ◽  
Subit K Jain ◽  
Syed Abbas ◽  
Shahlar Meherrem ◽  
Rajendra K Ray
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Network Model ◽  
Bifurcation Analysis ◽  
Reaction Diffusion ◽  
Neuron Network ◽  
Diffusion Term

Hopf Bifurcation Analysis of a Three-Stage-Structured Prey-Predator System with Multi-Delays

2013 ◽  
Vol 756-759 ◽  
pp. 2857-2862
Author(s):  
Shun Yi Li ◽  
Wen Wu Liu
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Characteristic Equations ◽  
Numerical Examples ◽  
Predator Model ◽  
Predator System

A three-stage-structured prey-predator model with multi-delays is considered. The characteristic equations and local stability of the equilibrium are analyzed, and the conditions for the positive equilibrium occurring Hopf bifurcation are obtained by applying the theorem of Hopf bifurcation. Finally, numerical examples and brief conclusion are given.

scholarly journals Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV, Delay, and Harvesting

2018 ◽  
Vol 2018 ◽  
pp. 1-19
Author(s):  
Xin-You Meng ◽  
Jiao-Guo Wang ◽  
Hai-Feng Huo
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Dynamical Behaviour ◽  
Optimal Harvesting ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Type Iv ◽  
Local Hopf Bifurcation ◽  
Plankton Model

In this paper, a Holling type IV nutrient-plankton model with time delay and linear plankton harvesting is investigated. The existence and local stability of all equilibria of model without time delay are given. Regarding time delay as bifurcation parameter, such system around the interior equilibrium loses its local stability, and Hopf bifurcation occurs when time delay crosses its critical value. In addition, the properties of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem. What is more, the global continuation of the local Hopf bifurcation is discussed by using a global Hopf bifurcation result. Furthermore, the optimal harvesting is obtained by the Pontryagin’s Maximum Principle. Finally, some numerical simulations are given to confirm our theoretical analysis.

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.

Sign in / Sign up

Export Citation Format

Share Document