scholarly journals Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates

2008 ◽  
Vol 2008 ◽  
pp. 1-23 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Stability Region ◽  
Extreme Values ◽  
Period Doubling ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Bifurcation Curve ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We explore a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods marketα, and the degree of capital mobilityβ. We determine by a numerical grid search method the stability region in parameter space and find that flexible rates cause enhanced stability of equilibrium with respect to variations of the parameters. We identify the Hopf-Neimark bifurcation curve and the flip bifurcation curve, and find that the period doubling cascades which leads to chaos is the dominant behavior of the system outside the stability region, persisting to large values ofβ. Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate is also briefly discussed. Finally, we explore a special model variation incorporating the “wealth effect” which is found to behave similarly to the basic model, contrary to the model of fixed exchange rates in which incorporation of this effect causes an entirely different behavior.

scholarly journals Numerical Exploration of Kaldorian Interregional Macrodynamics: Enhanced Stability and Predominance of Period Doubling under Flexible Exchange Rates

2010 ◽  
Vol 2010 ◽  
pp. 1-29 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Vassilis Kalantonis ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Government Expenditure ◽  
Period Doubling ◽  
Numerical Approach ◽  
Capital Movement ◽  
Dimensional Parameter ◽  
Stability Of Equilibrium ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We present a discrete two-regional Kaldorian macrodynamic model with flexible exchange rates and explore numerically the stability of equilibrium and the possibility of generation of business cycles. We use a grid search method in two-dimensional parameter subspaces, and coefficient criteria for the flip and Hopf bifurcation curves, to determine the stability region and its boundary curves in several parameter ranges. The model is characterized by enhanced stability of equilibrium, while its predominant asymptotic behavior when equilibrium is unstable is period doubling. Cycles are scarce and short-lived in parameter space, occurring at large values of the degree of capital movementβ. By contrast to the corresponding fixed exchange rates system, for cycles to occur sufficient amount of trade is requiredtogetherwith high levels of capital movement. Rapid changes in exchange rate expectations and decreased government expenditure are factors contributing to the creation of interregional cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating period doubling or cycles, and their development into chaotic attractors, are given. The paper illustrates the feasibility and effectiveness of the numerical approach for dynamical systems of moderately high dimensionality and several parameters.

scholarly journals Numerical Exploration of Kaldorian Macrodynamics: Hopf-Neimark Bifurcations and Business Cycles with Fixed Exchange Rates

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Panagiotis Markellos
Keyword(s):  
Business Cycles ◽  
Exchange Rates ◽  
Parameter Space ◽  
Open Economy ◽  
Capital Mobility ◽  
Complex Dynamics ◽  
Model Parameters ◽  
Bifurcation Curve ◽  
Fixed Exchange Rates ◽  
The Stability

We explore numerically a three-dimensional discrete-time Kaldorian macrodynamic model in an open economy with fixed exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods marketα, and the degree of capital mobilityβon the stability of equilibrium and on the existence of business cycles. We determine the stability region in the parameter space and find that increase ofαdestabilizes the equilibrium more quickly than increase ofβ. We determine the Hopf-Neimark bifurcation curve along which business cycles are generated, and discuss briefly the occurrence of Arnold tongues. Bifurcation and Lyapunov exponent diagrams are computed providing information on the emergence, persistence, and amplitude of the cycles and illustrating the complex dynamics involved. Examples of cycles and other attractors are presented. Finally, we discuss a two-dimensional variation of the model related to a “wealth effect,” called model 2, and show that in this case,αdoes not destabilize the equilibrium more quickly thanβ, and that a Hopf-Neimark bifurcation curve does not exist in the parameter space, therefore model 2 does not produce cycles.

The Stability of Flexible Exchange Rates--The Canadian Experience

1969 ◽  
Vol 2 (2) ◽  
pp. 324 ◽  
Author(s):  
R. G. Penner ◽  
G. Hartley Mellish ◽  
Robert G. Hawkins
Keyword(s):  
Exchange Rates ◽  
Canadian Experience ◽  
The Stability ◽  
Flexible Exchange Rates

scholarly journals Bifurcation and Chaos in a Price Game of Irrigation Water in a Coastal Irrigation District

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Baogui Xin ◽  
Yuting Li
Keyword(s):  
Irrigation Water ◽  
Period Doubling ◽  
Irrigation District ◽  
Continuation Methods ◽  
Flip Bifurcation ◽  
Codimension Two ◽  
Price Game ◽  
Matlab Package ◽  
Test Algorithm ◽  
The Stability

We propose a price game model of irrigation water in a coastal irrigation district. Then, we discuss the stability and codimension-two period-doubling (flip) bifurcation. Then, the MATLAB package Cl_MatContM is employed to illustrate its numerical bifurcations-based continuation methods. Lastly, the 0-1 test algorithm is used to compute the median value of correlation coefficient which can indicate whether the underlying dynamics is regular or chaotic.

Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting

2016 ◽  
Vol 10 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Boshan Chen ◽  
Jiejie Chen
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Stage Structure ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of [Formula: see text], by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.

Vibration Frequencies in High-Speed Milling Processes or a Positive Answer to Davies, Pratt, Dutterer and Burns

2004 ◽  
Vol 126 (3) ◽  
pp. 481-487 ◽  
Author(s):  
T. Insperger ◽  
G. Ste´pa´n
Keyword(s):  
High Speed ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Vibration Frequencies ◽  
High Speed Milling ◽  
Machine Tool Chatter ◽  
Special Cases ◽  
Universal Approach ◽  
The Stability ◽  
Excitation Model

The stability charts of high-speed milling are constructed. New unstable regions and vibration frequencies are identified. These are related to flip bifurcation, i.e. period doubling vibrations occur apart of the conventional self-excited vibrations well-known for turning or low-speed milling with multiple active teeth. The Semi-Discretization method is applied for the delayed parametric excitation model of milling providing the connection of the two existing and experimentally verified results of machine tool chatter research. The two extreme models in question, that is, the traditional autonomous delayed model of time-independent turning, and the recently introduced discrete map model of time-dependent highly interrupted machining, are both involved as special cases in the universal approach presented in this study.

Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics, and Experiments

2004 ◽  
Vol 127 (2) ◽  
pp. 197-203 ◽  
Author(s):  
Gabor Stepan ◽  
Robert Szalai ◽  
Brian P. Mann ◽  
Philip V. Bayly ◽  
Tamas Insperger ◽  
...  
Keyword(s):  
Small Parameter ◽  
High Speed ◽  
Period Doubling ◽  
Hopf Bifurcations ◽  
Cutting Condition ◽  
Flip Bifurcation ◽  
Machining Parameters ◽  
High Speed Milling ◽  
The Stability ◽  
Discrete Mathematical Model

High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model, playing an essential role in machine tool vibrations, breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self-excited vibrations or Hopf bifurcations. The present work investigates the nonlinear vibrations in the case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in the case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.

scholarly journals Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

2020 ◽  
Vol 24 (3) ◽  
pp. 137-151
Author(s):  
Z. T. Zhusubaliyev ◽  
D. S. Kuzmina ◽  
O. O. Yanochkina
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Stability Region ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Continuous Map ◽  
The Stability ◽  
Smooth Map

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

scholarly journals Delay Cournot Duopoly Game with Gradient Adjustment: Berezowski Transition from a Discrete Model to a Continuous Model

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 32
Author(s):  
Akio Matsumoto ◽  
Ferenc Szidarovszky ◽  
Keiko Nakayama
Keyword(s):  
Stability Region ◽  
Period Doubling ◽  
Continuous Model ◽  
Delay Model ◽  
Chaotic Oscillations ◽  
Cournot Duopoly ◽  
Numerical Studies ◽  
Inertia Coefficient ◽  
The Stability ◽  
Theoretical Results

This paper investigates the asymptotical behavior of the equilibrium of linear classical duopolies by reconsidering the two-delay model with two different positive delays. In a two-dimensional analysis, the stability switching curves were first analytically determined. Numerical studies verified and illustrated the theoretical results. In the sensitivity analysis it was demonstrated that the inertia coefficient has a twofold effect: enlarges the stability region as well as simplifies the complicated dynamics with period-halving cascade. In contrary, the adjustment speed contracts the stability region and complicates simple dynamics with period-doubling bifurcation. In addition, for various values of τ1 and τ2, a wide variety of dynamics appears ranging from simple cycle via a Hopf bifurcation to chaotic oscillations.

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