scholarly journals Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2558
Author(s):  
Sameh Askar
Keyword(s):  
Phase Plane ◽  
Global Analysis ◽  
Equilibrium Points ◽  
Complex Structure ◽  
Piecewise Smooth ◽  
Invariant Curves ◽  
Complex Dynamic ◽  
Decision Variables ◽  
Dynamic Map ◽  
Qualitative Changes

This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise smooth discrete dynamic map was constructed in order to study the complex dynamic characteristics of the game. The phase plane of the map was divided into two different regions, separated by border curve. The equilibrium points of the map, in each region on where they are defined, were calculated, and their stability conditions were investigated. Furthermore, we conducted a global analysis to investigate the complex structure of the map, such as closed invariant curves, periodic cycles, and chaotic attractors and their basins, which cause qualitative changes as some parameters are allowed to vary.

scholarly journals Nested Closed Invariant Curves in Piecewise Smooth Maps

2019 ◽  
Vol 29 (07) ◽  
pp. 1930017
Author(s):  
Viktor Avrutin ◽  
Zhanybai T. Zhusubaliyev
Keyword(s):  
Piecewise Linear ◽  
Phase Locking ◽  
Basins Of Attraction ◽  
Piecewise Smooth ◽  
Phase Portraits ◽  
Invariant Curves ◽  
Smooth Maps ◽  
Piecewise Smooth Maps ◽  
Smooth Map ◽  
Stable Invariant

The paper describes how several coexisting stable closed invariant curves embedded into each other can arise in a two-dimensional piecewise-linear normal form map. Phenomena of this type have been recently reported for a piecewise smooth map, modeling the behavior of a power electronic DC–DC converter. In the present work, we demonstrate that this type of multistability exists in a more general class of models and show how it may result from the well-known period adding bifurcation structure due to its deformation so that the phase-locking regions start to overlap. We explain how this overlapping structure is related to the appearance of coexisting stable closed invariant curves nested into each other. By means of detailed, numerically calculated phase portraits we hereafter present an example of this type of multistability. We also demonstrate that the basins of attraction of the nested stable invariant curves may be separated from each other not only by repelling closed invariant curves, as previously reported, but also by a chaotic saddle. It is suggested that the considered kind of multistability is a generic phenomenon in piecewise smooth dynamical systems.

scholarly journals Riddled Attraction Basin and Multistability in Three-Element-Based Memristive Circuit

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Quan Xu ◽  
Xiao Tan ◽  
Yunzhen Zhang ◽  
Han Bao ◽  
Yihua Hu ◽  
...  
Keyword(s):  
Limit Cycles ◽  
Phase Plane ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Model Parameters ◽  
Stable Node ◽  
Dynamical Behaviors ◽  
Attraction Basins ◽  
Memristive Circuit ◽  
Crisis Scenarios

By coupling a diode bridge-based second-order memristor and an active voltage-controlled memristor with a capacitor, a three-element-based memristive circuit is synthesized and its system model is then built. The boundedness of the three-element-based memristive circuit is theoretically proved by employing the contraction mapping principle. Besides, the stability distributions of equilibrium points are theoretically and numerically expounded in a 2D parameter plane. The results imply the memristive circuit has a zero unstable saddle focus and a pair of nonzero stable node-foci or unstable saddle-foci depending on the considered parameters. The dynamical behaviors include point attractor, period, chaos, coexisting bifurcation mode, period-doubling bifurcation route, and crisis scenarios, which are explored using some common dynamical methods. Of particular concern, riddled attraction basins and multistability are uncovered under two sets of specified model parameters nearing the tiny neighborhood of crisis scenarios by local attraction basins and phase plane plots. The riddled attraction basins with island-like structure demonstrate that their dynamical behaviors are extremely sensitive to the initial conditions, resulting in the coexistence of limit cycles with period-2 and period-6, as well as the coexistence of period-1 limit cycles and single-scroll chaotic attractors. Moreover, a feasible on-breadboard hardware circuit is manually made and the experimental measurements are executed, upon which phase plane trajectories for some discrete model parameters are captured to further confirm the numerically simulated ones.

Complex Dynamic Behavior in a Model of Electro-Optic Device with Time-Delayed Feedback

1998 ◽  
Vol 08 (06) ◽  
pp. 1347-1354 ◽  
Author(s):  
Andrew V. Shobukhov
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Investigation ◽  
Infinite Number ◽  
Strain Gauge ◽  
Fiber Optic ◽  
Equilibrium Points ◽  
Nonlinear Feedback ◽  
Complex Dynamic ◽  
Electro Optic ◽  
Time Delayed Feedback

Oscillatory and chaotic regimes arising from Andronov-Hopf bifurcation are obtained in a model of fiber-optic interferometric strain gauge with delayed nonlinear feedback as a result of analytical and numerical investigation. Complicated behavior of solutions in the case of infinite number of equilibrium points is reported.

scholarly journals Nonlinear Dynamics of Cournot Duopoly Game: When One Firm Considers Social Welfare

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
S. S. Askar ◽  
A. A. Elsadany
Keyword(s):  
Nonlinear Dynamics ◽  
Social Welfare ◽  
Fixed Points ◽  
Phase Plane ◽  
Complex Dynamics ◽  
Stability Conditions ◽  
Cournot Duopoly ◽  
Discrete Dynamic ◽  
Gradient Based ◽  
Dynamic Map

In this paper, we study the competition between two firms whose outputs are quantities. The first firm considers maximization of its profit while the second firm considers maximization of its social welfare. Adopting a gradient-based mechanism, we introduce a nonlinear discrete dynamic map which is used to describe the dynamics of this game. For this map, the fixed points are calculated and their stability conditions are analyzed. This includes investigating some attracting set and chaotic behaviors for the complex dynamics of the map. We have also investigated the types of the preimages that characterize the phase plane of the map and conclude that the game’s map is noninvertible of type Z 4 − Z 2 .

Transformations of Closed Invariant Curves and Closed-Invariant-Curve-Like Chaotic Attractors in Piecewise Smooth Systems

2021 ◽  
Vol 31 (03) ◽  
pp. 2130009
Author(s):  
Zhanybai T. Zhusubaliyev ◽  
Viktor Avrutin ◽  
Frank Bastian
Keyword(s):  
Chaotic Attractor ◽  
Small Amplitude ◽  
Homoclinic Bifurcation ◽  
Piecewise Smooth ◽  
Invariant Curve ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Invariant Curves ◽  
Piecewise Smooth Systems ◽  
Smooth Map

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

A model of a thalamic neuron

1985 ◽  
Vol 225 (1239) ◽  
pp. 161-193 ◽  
Keyword(s):  
Equilibrium Point ◽  
Neuron Model ◽  
Phase Plane ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Slow Inward Current ◽  
Stable Equilibrium Point ◽  
Thalamic Neuron ◽  
Subthreshold Region ◽  
Current Step

We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x-y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e. p.) is separated from the outer stable and unstable e. ps by two saddle points. If the system is started at rest at the middle stable e. p. it has the following complex properties: (i) A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. (ii) A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency-current relationship. (iii) Hyperpolarization from rest is followed by post-inhibitory rebound. (iv) The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x -y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.

scholarly journals Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2001
Author(s):  
Sameh S. Askar ◽  
Abdulrahman Al-Khedhairi
Keyword(s):  
Phase Plane ◽  
Complex Dynamics ◽  
Logistic Map ◽  
Fractal Dimensions ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Complex Dynamic ◽  
2D Logistic Map ◽  
Two Parameters ◽  
Periodic Cycles

In this paper, we study the complex dynamic characteristics of a simple nonlinear logistic map. The map contains two parameters that have complex influences on the map’s dynamics. Assuming different values for those parameters gives rise to strange attractors with fractal dimensions. Furthermore, some of these chaotic attractors have heteroclinic cycles due to saddle-fixed points. The basins of attraction for some periodic cycles in the phase plane are divided into three regions of rank-1 preimages. We analyze those regions and show that the map is noninvertible and includes Z0,Z2 and Z4 regions.

scholarly journals Application of metric entropy for results interpretation of composite materials mechanical tests

2017 ◽  
Vol 17 (1) ◽  
pp. 70-81 ◽  
Author(s):  
G. Garbacz ◽  
L. Kyzioł
Keyword(s):  
Phase Plane ◽  
Complex Structure ◽  
Mechanical Tests ◽  
Phase Portraits ◽  
Metric Entropy ◽  
Elastic Plastic ◽  
Plastic Composite ◽  
Wide Range ◽  
Data Points ◽  
Plastic Changes

Abstract In this paper the results of mechanical studies of the Aropol 536 composite on the epoxy-resin base are described. The aim of the studies was to measure elastic-plastic changes in the composite during its deformation. The obtained results were analyzed using Kolmogorov-Sinai metric entropy. The entropy was computed applying phase portraits reconstructed from a phase plane using delayed coordinates. Resolution of the particular experimental setup limits the number of the acquired data points, i.e., from several to tens of thousands of points and it has significant influence on accuracy of the obtained results. In conclusion, in the tested composites elastic-plastic deformations are periodic and repeat in a distinctive way in a wide range of deformations of the sample. Deformation of the elastic-plastic composite are associated with its complex structure and studies of its mechanical properties require more advanced methods such as use of Kolmogorov-Sinai metric entropy.

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