scholarly journals Complex Dynamical Behaviors in a Bertrand Game with Service Factor and Differentiated Products

2021 ◽  
Author(s):  
Wei Zhou ◽  
Hui Li
Keyword(s):  
Basin Of Attraction ◽  
Spillover Effect ◽  
Dynamic Game ◽  
Phase Portraits ◽  
Stable Operation ◽  
Small Range ◽  
Differentiated Products ◽  
Flip Bifurcation ◽  
Service Factor ◽  
Bertrand Game

Abstract In this paper, taking the factor of service level provided by the manufacturers into consideration, a static duopolistic Bertrand game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic game model of duopoly Bertrand with boundedly rational is established with using the gradient mechanism. By using numerical simulation tools, there are two paths for the system to drop into chaos, that is, flip bifurcation and Neimark-Sacker bifurcation. The symmetric structures can be found from two-parameter bifurcation diagrams. Saddle-homoclinic bifurcation also can be observed from the evolution process of phase portraits. In addition, the emergence of intermittent chaos implies that the established system has the capability of self-regulating, where PM-I intermittency, PM-III intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated. At last, it can be found that the values of product differentiation degree and service spillover effect are not the bigger the better. Keeping these two parameters in a relatively small range will be conducive to the long-term stable operation of the two manufacturers.

Various Attractors, Coexisting Attractors and Antimonotonicity in a Simple Fourth-Order Memristive Twin-T Oscillator

2018 ◽  
Vol 28 (04) ◽  
pp. 1850050 ◽  
Author(s):  
Ling Zhou ◽  
Chunhua Wang ◽  
Xin Zhang ◽  
Wei Yao
Keyword(s):  
Fourth Order ◽  
Chaotic Systems ◽  
Basin Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Dynamical Behaviors ◽  
Random Bit Generator ◽  
In Series ◽  
Dynamical Features

By replacing the resistor in a Twin-T network with a generalized flux-controlled memristor, this paper proposes a simple fourth-order memristive Twin-T oscillator. Rich dynamical behaviors can be observed in the dynamical system. The most striking feature is that this system has various periodic orbits and various chaotic attractors generated by adjusting parameter [Formula: see text]. At the same time, coexisting attractors and antimonotonicity are also detected (especially, two full Feigenbaum remerging trees in series are observed in such autonomous chaotic systems). Their dynamical features are analyzed by phase portraits, Lyapunov exponents, bifurcation diagrams and basin of attraction. Moreover, hardware experiments on a breadboard are carried out. Experimental measurements are in accordance with the simulation results. Finally, a multi-channel random bit generator is designed for encryption applications. Numerical results illustrate the usefulness of the random bit generator.

Numerical Investigation of Nonlinear Dynamics of a Pneumatic Artificial Muscle With Hard Excitation

2020 ◽  
Vol 15 (4) ◽  
Author(s):  
Bhaben Kalita ◽  
Santosha K. Dwivedy
Keyword(s):  
Nonlinear Dynamics ◽  
Resonance Condition ◽  
Basin Of Attraction ◽  
Artificial Muscle ◽  
Phase Portraits ◽  
System Response ◽  
Pneumatic Artificial Muscle ◽  
Superharmonic Resonance ◽  
System Parameters ◽  
Hard Excitation

Abstract In this work, a numerical analysis has been carried out to study the nonlinear dynamics of a system with pneumatic artificial muscle (PAM). The system is modeled as a single degree-of-freedom system and the governing nonlinear equation of motion has been derived to study the various responses of the system. The system is subjected to hard excitation and hence the subharmonic and superharmonic resonance conditions have been studied. The second-order method of multiple scales (MMS) has been used to find the response, stability, and bifurcations of the system. The effect of various system parameters on the system response has been studied using time response, phase portraits, and basin of attraction. In these responses, while the saddle node bifurcation is found in both super and subharmonic resonance conditions, the Hopf bifurcation is found only in superharmonic resonance condition. By changing different system parameters, it has been shown that the response with three periods leads to chaotic response for superharmonic resonance condition. This study will find applications in the design of PAM actuators.

scholarly journals Complexity Analysis of Pricing in a Multichannel Supply Chain with Spillovers from Online to Offline Sales

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Fengxia Mai ◽  
Jianxiong Zhang ◽  
Rui Yang ◽  
Xiaojie Sun
Keyword(s):  
Supply Chain ◽  
Dynamic Pricing ◽  
Complexity Analysis ◽  
Spillover Effect ◽  
Dynamic Game ◽  
Period Doubling ◽  
Dynamic Adjustment ◽  
Adjustment Strategy ◽  
Adjustment Speed ◽  
The Stability

In recent years, many manufacturers have been selling their products to online consumers through e-tailers by adopting reselling mode and agency selling mode simultaneously. The sales from the online channels inevitably incur spillover effect to the traditional offline channels. This paper develops a dynamic pricing game model on the basis of a long-term gradient adjustment mechanism for a multichannel supply chain that consists of a manufacturer and an e-tailer and focuses on examining the impacts of spillover effect, agency fee, and adjustment speed on the stability and complexity of the dynamic game system. The results show that both a greater spillover effect and a higher agency fee can make the dynamic game system more stable, and a higher adjustment speed can destabilize the dynamic game system through period doubling bifurcation. Furthermore, it is interesting to find that the destabilization of the game system benefits the e-tailer and the supply chain while having little influence on the manufacturer, and thus the dynamic adjustment strategy may improve the supply chain efficiency.

scholarly journals Robustness of Supercavitating Vehicles Based on Multistability Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi ◽  
Jun Guan
Keyword(s):  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Small Range ◽  
Practical Application ◽  
Stable States ◽  
Supercavitating Vehicles ◽  
Control Gain ◽  
Stable Periodic ◽  
Setting Parameters

Supercavity can increase speed of underwater vehicles greatly. However, external interferences always lead to instability of vehicles. This paper focuses on robustness of supercavitating vehicles. Based on a 4-dimensional dynamic model, the existence of multistability is verified in supercavitating system through simulation, and the robustness of vehicles varying with parameters is analyzed by basins of attraction. Results of the research disclose that the supercavitating system has three stable states in some regions of parameters space, namely, stable, periodic, and chaotic states, while in other regions it has various multistability, such as coexistence of two types of stable equilibrium points, coexistence of a limit cycle with a chaotic attractor, and coexistence of 1-periodic cycle with 2-periodic cycle. Provided that cavitation number varies within a small range, with increase of the feedback control gain of fin deflection angle, size of basin of attraction becomes smaller and robustness of the system becomes weaker. In practical application, robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initial launching conditions.

scholarly journals Bifurcation Analysis and Chaos Control in a Discrete-Time Parasite-Host Model

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Xueli Chen ◽  
Lishun Ren
Keyword(s):  
Discrete Time ◽  
Control Method ◽  
Phase Portraits ◽  
Flip Bifurcation ◽  
Step Size ◽  
Host System ◽  
Center Manifold Theorem ◽  
Unstable Fixed Point ◽  
Existence And Stability ◽  
Feedback Control Method

A discrete-time parasite-host system with bifurcation is investigated in detail in this paper. The existence and stability of nonnegative fixed points are explored and the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. And we also prove the chaos in the sense of Marotto. The numerical simulations not only illustrate the consistence with the theoretical analysis, but also exhibit other complex dynamical behaviors, such as bifurcation diagrams, Maximum Lyapunov exponents, and phase portraits. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period orbits, attracting invariant cycles and chaotic attractors of the discrete-time parasite-host system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point by using the feedback control method.

scholarly journals Nonlinear Actuation of Casimir Oscillators toward Chaos: Comparison of Topological Insulators and Metals

Universe ◽  
2021 ◽  
Vol 7 (5) ◽  
pp. 123
Author(s):  
Fatemeh Tajik ◽  
Zahra Babamahdi ◽  
Mehdi Sedighi ◽  
George Palasantzas
Keyword(s):  
Casimir Force ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Surface Conductance ◽  
Stable Operation ◽  
Casimir Forces ◽  
Driven Systems ◽  
Operation Conditions ◽  
Increased Risk

In the current study, we explore the sensitivity of the actuation dynamics of electromechanical systems on novel materials, e.g., Bi2Se3, which is a well-known 3D Topological Insulator (TI), and compare their response to metallic conductors, e.g., Au, that are currently used in devices. Bifurcation and phase portraits analysis in conservative systems suggest that the strong difference between the conduction states of Bi2Se3 and Au yields sufficiently weaker Casimir force to enhance stable operation. Furthermore, for nonconservative driven systems, the Melnikov function and Poincare portrait analysis probed the occurrence of chaotic behavior leading to increased risk for stiction. It was found that the presence of the TI enhanced stable operation against chaotic behavior over a significantly wider range of operation conditions in comparison to typical metallic conductors. Therefore, the use of TIs can allow sufficient surface conductance to apply electrostatic compensation of residual contact potentials and, at the same time, to yield sufficiently weak Casimir forces favoring long-term stable actuation dynamics against chaotic behavior.

On the modeling of some triodes-based nonlinear oscillators with complex dynamics: case of the Van der Pol oscillator

2021 ◽  
Author(s):  
Joakim Vianney Ngamsa Tegnitsap ◽  
Merlin Brice Saatsa Tsefack ◽  
Elie Bertrand Megam Ngouonkadi ◽  
Hilaire Bertrand Fotsin
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponent ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Oscillator ◽  
Phase Portraits ◽  
Van Der Pol ◽  
Physical Consideration

Abstract In this work, the dynamic of the triode-based Van der Pol oscillator coupled to a linear circuit is investigated (Triode-based VDPCL oscillator). Towards this end, we present a mathematical model of the triode, chosen from among the many different ones present in literature. The dynamical behavior of the system is investigated using classical tools such as two-parameter Lyapunov exponent, one-parameter bifurcation diagram associated with the graph of largest Lyapunov exponent, phase portraits, and time series. Numerical simulations reveal rather rich and complex phenomena including chaos, transient chaos, the coexistence of solutions, crisis, period-doubling followed by reverse period-doubling sequences (bubbles), and bursting oscillation. The coexistence of attractors is illustrated by the phase portraits and the cross-section of the basin of attraction. Such triode-based nonlinear oscillators can find their applications in many areas where ultra-high frequencies and high powers are demanded (radio, radar detection, satellites communication, etc.) since triode can work with these performances and are often used in the aforementioned areas. In contrast to some recent work on triode-based oscillators, LTSPICE simulations, based on real physical consideration of the triode, are carried out in order to validate the theoretical results obtained in this paper as well as the mathematical model adopted for the triode.

scholarly journals Dynamics Analysis of Game and Chaotic Control in the Chinese Fixed Broadband Telecom Market

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jia Liu ◽  
Guoliang Liu ◽  
Na Li ◽  
Hongliang Xu
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Control Method ◽  
Chaotic Behavior ◽  
Spillover Effect ◽  
Dynamic Game ◽  
Nash Equilibrium Point ◽  
Heterogeneous Players ◽  
Existence And Stability ◽  
Feedback Control Method

This paper considers a dynamic duopoly Cournot model based on nonlinear cost functions. The model with heterogeneous players and the spillover effect is applied to the Chinese fixed broadband telecom market. We have studied its dynamic game process. The existence and stability of the Nash equilibrium of the system have been discussed. Simulations are used to show the complex dynamical behaviors of the system. The results illustrate that altering the relevant parameters of system can affect the stability of the Nash equilibrium point and cause chaos to occur. With the use of the delay feedback control method, the chaotic behavior of the model has been stabilized at the Nash equilibrium point. The analysis and results will be of great importance for the Chinese fixed broadband telecom market.

BIFURCATION AND CHAOS IN THE TINKERBELL MAP

2011 ◽  
Vol 21 (11) ◽  
pp. 3137-3156 ◽  
Author(s):  
SHAOLIANG YUAN ◽  
TAO JIANG ◽  
ZHUJUN JING
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Phase Portraits ◽  
Transient Chaos ◽  
Flip Bifurcation ◽  
Maximum Lyapunov Exponent ◽  
Invariant Circle ◽  
Dynamical Behaviors ◽  
Fold Bifurcation ◽  
Period Doubling Bifurcations

In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.

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