scholarly journals Global Existence of Periodic Solutions in a Delayed Kaldor-Kalecki Model

2009 ◽  
Vol 14 (4) ◽  
pp. 463-472 ◽  
Author(s):  
A. Kaddar ◽  
H. Talibi Alaoui
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Business Cycle ◽  
Periodic Solutions ◽  
Cycle Model ◽  
Global Hopf Bifurcation ◽  
Numerical Examples ◽  
Existence Of Periodic Solutions ◽  
Business Cycle Model ◽  
Theoretical Results

This paper is concerned with a delayed Kaldor-Kalecki non-linear business cycle model in income. By applying a global Hopf bifurcation result due to Wu, the global existence of periodic solutions is investigated. Numerical examples will be given in the end, to illustrate our theoretical results.

scholarly journals Local Hopf Birurcation and Stability of Limit Cycle in a Delayed Kaldor-Kalecki Model

2009 ◽  
Vol 14 (3) ◽  
pp. 333-343 ◽  
Author(s):  
A. Kaddar ◽  
H. Talibi Alaoui
Keyword(s):  
Hopf Bifurcation ◽  
Business Cycle ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Cycle Model ◽  
Explicit Algorithm ◽  
Local Hopf Bifurcation ◽  
Business Cycle Model ◽  
The Stability

We consider a delayed Kaldor-Kalecki business cycle model. We first consider the existence of local Hopf bifurcation, and we establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions using the methods presented by O. Diekmann et al. in [1]. In the end, we conclude with an application.

scholarly journals Global existence of periodic solutions in a simplified four-neuron BAM neural network model with multiple delays

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Xiang-Ping Yan ◽  
Wan-Tong Li
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Global Existence ◽  
Periodic Solutions ◽  
Network Model ◽  
Neural Network Model ◽  
Multiple Delays ◽  
Global Hopf Bifurcation ◽  
Existence Of Periodic Solutions ◽  
Bam Neural Network

We consider a simplified bidirectional associated memory (BAM) neural network model with four neurons and multiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to Wu and Bendixson's criterion for high-dimensional ordinary differential equations due to Li and Muldowney. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the sum of two delays. Numerical simulations supporting the theoretical analysis are also included.

scholarly journals Analysis of the Stability and Hopf Bifurcation of Currency Supply Delay in an Opened Kaldorian Business Cycle Model

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Liming Zhao ◽  
Zhipei Zhao
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Business Cycle ◽  
Three Dimension ◽  
Cycle Model ◽  
Macroeconomic Policies ◽  
Existence Of Equilibrium ◽  
Business Cycle Model ◽  
The Stability ◽  
Theoretical Results

First of all, we establish a three-dimension open Kaldorian business cycle model under the condition of the fixed exchange rate. Secondly, with regard to the model, we discuss the existence of equilibrium point and the stability of the system near it with a time delay in currency supply as the bifurcating parameters of the system. Thirdly, we discuss the existence of Hopf bifurcation and investigate the stability of periodic solution generated by the Hopf bifurcation; then the direction of the Hopf bifurcation is given. Finally, a numerical simulation is given to confirm the theoretical results. This paper plays an important role in theoretical researching on the model of business cycle, and it is crucial for decision-maker to formulate the macroeconomic policies with the conclusions of this paper.

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 The Kaldor–Kalecki stochastic model of business cycle

2011 ◽  
Vol 16 (2) ◽  
pp. 191-205 ◽  
Author(s):  
Gabriela Mircea ◽  
Mihaela Neamt¸u ◽  
Dumitru Opris
Keyword(s):  
Hopf Bifurcation ◽  
Business Cycle ◽  
Stochastic Model ◽  
Local Stability ◽  
Mean Values ◽  
Numerical Examples ◽  
Investment Demand ◽  
The Mean ◽  
Theoretical Results

This paper is concerned with the deterministic and the stochastic delayed Kaldor–Kalecki nonlinear business cycle models of the income. They will take into consideration the investment demand in the form suggested by Rodano. The existence of the Hopf bifurcation is studied and the direction and the local stability of the Hopf bifurcation is also taken into consideration. For the stochastic model, the dynamics of the mean values and the square mean values of the model’s variables are set. Numerical examples are given to illustrate our theoretical results.

Periodic solutions for: ẋ(t) = λf(x(t),x(t – 1))

1988 ◽  
Vol 109 (3-4) ◽  
pp. 245-260 ◽  
Author(s):  
O. Arino ◽  
R. Benkhalti
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
A Priori ◽  
Monotonicity Property ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Fundamental Assumption ◽  
Existence Of Periodic Solutions ◽  
Image Position

SynopsisWe present a new result on the existence of periodic solutions for the equation:for all positive parameters λ sufficiently large. Our fundamental assumption is the following monotonicity property: if ø ≧ ψ (ø and ψ are data) then x(ø) ≧ x(ψ). The proof consists in applying the global Hopf bifurcation theorem. The main steps are: (i) a classical estimation of the periods; (ii) an a priori estimate for the solutions along a connected branch; (iii) a transformation acting on periodic solutions.

GLOBAL EXISTENCE OF PERIODIC SOLUTIONS IN THE LINEARLY COUPLED MACKEY–GLASS SYSTEM

2011 ◽  
Vol 21 (03) ◽  
pp. 711-724 ◽  
Author(s):  
YANQIU LI ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Glass System ◽  
Positive Equilibrium ◽  
Higher Dimensional ◽  
Distribution Of Eigenvalues ◽  
Existence Of Periodic Solutions ◽  
The Stability

The dynamics of a linearly coupled Mackey–Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson's criterion for higher dimensional ordinary differential equations due to [Li & Muldowney, 1993].

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