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 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.

Mode-Locking and Chaos in a Jeffcott Rotor With Bearing Clearances

1994 ◽  
Vol 61 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Sang-Kyu Choi ◽  
Sherif T. Noah
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Mode Locking ◽  
Rotor System ◽  
Winding Number ◽  
Fixed Point Algorithm ◽  
Jeffcott Rotor ◽  
Multiple Periodic Solutions ◽  
The Stability

A complex mode-locking (or entrainment) structure underlying the nonlinear whirling phenomenon of a horizontal Jeffcott rotor with a discontinuous nonlinearity (bearing clearance) was identified. A winding number is introduced as a measure of the ratio between two frequencies involved in the aperiodic whirling motions of the rotor system considered. Utilizing the winding number map, it was revealed that the alternating periodic and quasi-periodic responses take place according to the Farey number tree. The winding number varies in the form of the so-called “Devil’s staircase” as a certain system parameter varies. From the mode-locking pattern in the parameter space of the forcing amplitude and frequency, it was observed that as the forcing amplitude increases, the size of each locking interval increases so that its growth takes place in the form of “Arnol’d tongues,” where the winding number remains a rational number. Moreover, inside each locking zone, i.e., each “Arnol’d tongue,” there exist many smaller tongues similar to the main tongue, in which a sequence of period-doubling bifurcations leading to chaos occurred. The boundaries of each locking zone was obtained using a fixed-point algorithm along with the Floquet theory for checking the stability of the periodic solutions. The winding numbers were estimated utilizing a fixed-point algorithm modified to obtain quasi-periodic responses. A jump phenomenon was also observed by tracking multiple periodic solutions for several parameters of the rotor system.

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.

STABILIZING A TWO-CELL DC-DC BUCK CONVERTER BY FIXED POINT INDUCED CONTROL

2009 ◽  
Vol 19 (06) ◽  
pp. 2043-2057 ◽  
Author(s):  
A. EL AROUDI ◽  
F. ANGULO ◽  
G. OLIVAR ◽  
B. G. M. ROBERT ◽  
M. FEKI
Keyword(s):  
Fixed Point ◽  
Numerical Simulations ◽  
Period Doubling ◽  
Buck Converter ◽  
Noisy Environment ◽  
Noise Levels ◽  
The Stability ◽  
Lyapunov Exponent Spectrum ◽  
Bifurcation Behavior ◽  
Power Electronic Converter

In this paper, we study nonlinear and bifurcation behavior of a two-cell DC-DC buck power electronic converter. The system shows nonsmooth period doubling bifurcation and chaotic phenomena in a certain zone of parameter space. This zone is located both analytically and from numerical simulations. One-dimensional, two-dimensional bifurcation diagrams and Lyapunov exponent spectrum are used to detect the different dynamic behaviors of the system. The Fixed Point Induced Control (FPIC) technique is applied to the system in order to widen the stability zone. The performance of the FPIC technique applied to the stabilization of a two-cell DC-DC buck converter is analyzed. With this technique, stabilization is achieved without changing the fixed point. The robustness in the presence of a noisy environment is checked by numerical simulations by considering different noise levels.

scholarly journals Complex Dynamics in a Nonlinear Cobweb Model for Real Estate Market

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Junhai Ma ◽  
Lingling Mu
Keyword(s):  
Real Estate ◽  
Demand Function ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Period Doubling ◽  
Real Estate Market ◽  
Delayed Feedback Control ◽  
Cobweb Model ◽  
Sensitive Dependence ◽  
The Stability

We establish a nonlinear real estate model based on cobweb theory, where the demand function and supply function are quadratic. The stability conditions of the equilibrium are discussed. We demonstrate that as some parameters varied, the stability of Nash equilibrium is lost through period-doubling bifurcation. The chaotic features are justified numerically via computing maximal Lyapunov exponents and sensitive dependence on initial conditions. The delayed feedback control (DFC) method is applied to control the chaos of system.

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

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 On the stability of set-valued functional equations with the fixed point alternative

2012 ◽  
Vol 2012 (1) ◽  
pp. 81 ◽  
Author(s):  
Hassan Kenary ◽  
Hamid Rezaei ◽  
Yousof Gheisari ◽  
Choonkil Park
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
The Stability

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.

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