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.

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.

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 Stability of High-Speed Milling

2000 ◽  
Author(s):  
Tamás Insperger ◽  
Gábor Stépán
Keyword(s):  
Parametric Excitation ◽  
Stability Criterion ◽  
High Speed ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Vibration Frequencies ◽  
High Speed Milling ◽  
Low Speed ◽  
The Stability ◽  
Excitation Model

Abstract The stability charts of high-speed milling are constructed. Non-conventional 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 typical for turning or low-speed milling with multiple active teeth. A new stability criterion is proposed and applied for the delayed parametric excitation model of milling.

Turning Dynamics: Part 1 — Experimental Analysis

2012 ◽  
Author(s):  
Eric B. Halfmann ◽  
C. Steve Suh ◽  
N. P. Hung
Keyword(s):  
Instantaneous Frequency ◽  
Period Doubling ◽  
Displacement Sensor ◽  
Laser Displacement Sensor ◽  
Turning Process ◽  
Time Frequency ◽  
Spectral Components ◽  
Tool Vibrations ◽  
Exact Moment ◽  
The Stability

The workpiece and tool vibrations in a lathe are experimentally studied to establish improved understanding of cutting dynamics that would support efforts in exceeding the current limits of the turning process. A Keyence laser displacement sensor is employed to monitor the workpiece and tool vibrations during chatter-free and chatter cutting. A procedure is developed that utilizes instantaneous frequency (IF) to identify the modes related to measurement noise and those innate of the cutting process. Instantaneous frequency is shown to thoroughly characterize the underlying turning dynamics and identify the exact moment in time when chatter fully developed. That IF provides the needed resolution for identifying the onset of chatter suggests that the stability of the process should be monitored in the time-frequency domain to effectively detect and characterize machining instability. It is determined that for the cutting tests performed chatters of the workpiece and tool are associated with the changing of the spectral components and more specifically period-doubling bifurcation. The analysis presented provides a view of the underlying dynamics of the lathe process which has not been experimentally observed before.

scholarly journals A unique vadose zone model for shallow aquifers: the Hetao irrigation district, China

2019 ◽  
Vol 23 (7) ◽  
pp. 3097-3115 ◽  
Author(s):  
Zhongyi Liu ◽  
Xingwang Wang ◽  
Zailin Huo ◽  
Tammo Siert Steenhuis
Keyword(s):  
Field Experiment ◽  
Irrigation Water ◽  
Yellow River ◽  
Shallow Groundwater ◽  
Groundwater Table ◽  
Irrigation District ◽  
Zone Model ◽  
Calibration And Validation ◽  
Irrigation Water Use ◽  
Hetao Irrigation District

Abstract. Rapid population growth is increasing pressure on the world water resources. Agriculture will require crops to be grown with less water. This is especially the case for the closed Yellow River basin, necessitating a better understanding of the fate of irrigation water in the soil. In this paper, we report on a field experiment and develop a physically based model for the shallow groundwater in the Hetao irrigation district in Inner Mongolia, in the arid middle reaches of the Yellow River. Unlike other approaches, this model recognizes that field capacity is reached when the matric potential is equal to the height above the groundwater table and not by a limiting soil conductivity. The field experiment was carried out in 2016 and 2017. Daily moisture contents at five depths in the top 90 cm and groundwater table depths were measured in two fields with a corn crop. The data collected were used for model calibration and validation. The calibration and validation results show that the model-simulated soil moisture and groundwater depth fitted well. The model can be used in areas with shallow groundwater to optimize irrigation water use and minimize tailwater losses.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

scholarly journals Further Investigations on the Dynamics and Multistability Coexisted in a Memory-Based Cobweb Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
S. S. Askar
Keyword(s):  
Fixed Point ◽  
Equilibrium Point ◽  
Profit Maximization ◽  
Period Doubling ◽  
Speed Of Adjustment ◽  
Discrete Dynamic ◽  
Cobweb Model ◽  
Numerical Studies ◽  
Market Clearing Price ◽  
The Stability

Based on a nonlinear demand function and a market-clearing price, a cobweb model is introduced in this paper. A gradient mechanism that depends on the marginal profit is adopted to form the 1D discrete dynamic cobweb map. Analytical studies show that the map possesses four fixed points and only one attains the profit maximization. The stability/instability conditions for this fixed point are calculated and numerically studied. The numerical studies provide some insights about the cobweb map and confirm that this fixed point can be destabilized due to period-doubling bifurcation. The second part of the paper discusses the memory factor on the stabilization of the map’s equilibrium point. A gradient mechanism that depends on the marginal profit in the past two time steps is adopted to incorporate memory in the model. Hence, a 2D discrete dynamic map is constructed. Through theoretical and numerical investigations, we show that the equilibrium point of the 2D map becomes unstable due to two types of bifurcations that are Neimark–Sacker and flip bifurcations. Furthermore, the influence of the speed of adjustment parameter on the map’s equilibrium is analyzed via numerical experiments.

scholarly journals MICROMORPHOLOGICAL DIAGNOSIS OF THE STABILITY OF CHERNOZEMS UNDER IRRIGATION

2014 ◽  
pp. 52-73 ◽  
Author(s):  
T. V. Tursina
Keyword(s):  
Irrigation Water ◽  
Ionic Composition ◽  
Soil Porosity ◽  
Long Period ◽  
The Stability ◽  
Anthropogenic Evolution

Chernozems under irrigation for a long period of time have been studied using the micromorphological methods. The soil porosity, the amount of microaggregates, biogeneity, humus microforms and the presence or formation of optically oriented clay were taken as the basic indices for estimating the stability of chernozems to irrigation during 30-50 years. The different ionic composition of the irrigation water serves as evidence of varying anthropogenic evolution of chernozems.

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