AN EMPIRICAL APPROACH TO STUDY THE STABILITY OF GENERALIZED LOGISTIC MAP IN SUPERIOR ORBIT

This paper investigates a semi-empirical approach for determining the stability of systems that can be modelled by ordinary differential equations with a time delay. This type of model is relevant to biological oscillators, machining processes, feedback control systems and models for wave propagation and reflection, where the motion of the waves themselves is considered to be outside the system model. A primary aim is to investigate the extension of empirical Floquet theory to experimental or numerical data obtained from time-delayed oscillators. More specifically, the reconstructed time series from a numerical example and an experimental milling system are examined to obtain a finite number of characteristic multipliers from the reduced order dynamics. A secondary goal of this work is to demonstrate a benefit of empirical characteristic multiplier estimation by performing system identification on a delayed oscillator. The principal results from this study are the accurate estimation of delayed oscillator characteristic multipliers and the utilization the empirical results for parametric identification of model parameters. Combining these results with previous research on an experimental milling system provides a particularly relevant result—the first approach for identifying all model parameters for stability prediction directly from the cutting process vibration history.

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LOCAL FEEDBACK CONTROL OF THE NEIMARK–SACKER BIFURCATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403006972 ◽

2003 ◽

Vol 13 (04) ◽

pp. 879-893 ◽

Cited By ~ 9

Author(s):

HASSAN YAGHOOBI ◽

EYAD H. ABED

Keyword(s):

Feedback Control ◽

Logistic Map ◽

Period Doubling ◽

Discrete Time System ◽

Feedback Controllers ◽

Local Feedback ◽

The Stability ◽

Control System Model ◽

Model Reference Adaptive ◽

Control Designs

Local bifurcation control designs have been addressed in the literature for stationary, Hopf, and period doubling bifurcations. This paper addresses the local feedback control of the Neimark–Sacker bifurcation, in which an invariant closed curve emerges from a nominal fixed point of a discrete-time system as a parameter is slowly varied. The analysis of this bifurcation is more involved than for previously considered bifurcations. The paper develops the stability and amplitude equations for the bifurcated invariant curves of the Neimark–Sacker bifurcation, and then proceeds to apply these relationships in the design of nonlinear feedbacks. The feedback controllers are applied to two examples: the delayed logistic map and a model reference adaptive control system model.

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I-superior approach to study the stability of logistic map

2010 International Conference on Mechanical and Electrical Technology ◽

10.1109/icmet.2010.5598469 ◽

2010 ◽

Cited By ~ 1

Author(s):

Mamta Rani ◽

Saurabh Goel

Keyword(s):

Logistic Map ◽

The Stability ◽

Superior Approach

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Dynamic Effects Arise Due to Consumers’ Preferences Depending on Past Choices

Entropy ◽

10.3390/e22020173 ◽

2020 ◽

Vol 22 (2) ◽

pp. 173 ◽

Cited By ~ 4

Author(s):

Sameh S. Askar ◽

A. Al-khedhairi

Keyword(s):

Numerical Simulation ◽

Fixed Points ◽

Chaotic Behavior ◽

Logistic Map ◽

Dynamic Effects ◽

One Dimensional ◽

First Case ◽

The Stability ◽

Conditions Of Stability ◽

Dynamic Duopoly

We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case considers only focusing on the action done by one player. This action reduces the game’s map to a one-dimensional map, which is the logistic map. Using analytical and numerical simulation, the stability of fixed points of this map is studied. In the second case, we focus on the actions applied by both players. The fixed points, in this case, are calculated, and their stability is discussed. The conditions of stability are provided in terms of the game’s parameters. Numerical simulation is carried out to give local and global investigations of the chaotic behavior of the game’s map. In addition, we use a statistical measure, such as entropy, to get more evidences on the regularity and predictability of time series associated with this case.

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Global Dynamics and Synchronization in a Duopoly Game with Bounded Rationality and Consumer Surplus

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419300313 ◽

2019 ◽

Vol 29 (11) ◽

pp. 1930031 ◽

Cited By ~ 9

Author(s):

Yinxia Cao ◽

Wei Zhou ◽

Tong Chu ◽

Yingxiang Chang

Keyword(s):

Bounded Rationality ◽

Equilibrium Points ◽

Logistic Map ◽

Consumer Surplus ◽

Global Dynamics ◽

Bifurcation Diagrams ◽

Fractal Structures ◽

One Dimensional ◽

Adjustment Speed ◽

The Stability

Based on the oligopoly game theory, a dynamic duopoly Cournot model with bounded rationality and consumer surplus is established. On the one hand, the type and the stability of the boundary equilibrium points and the stability conditions of the Nash equilibrium point are discussed in detail. On the other hand, the potential complex dynamics of the system is demonstrated by a set of 2D bifurcation diagrams. It is found that the bifurcation diagrams have beautiful fractal structures when the adjustment speed of production is taken as the bifurcation parameter. And it is verified that the area with scattered points in the 2D bifurcation diagrams is caused by the coexistence of multiple attractors. It is also found that there may be two, three or four coexisting attractors. It is even found the coexistence of Milnor attractor and other attractors. Moreover, the topological structure of the attracting basin and global dynamics of the system are investigated by the noninvertible map theory, using the critical curve and the transverse Lyapunov exponent. It is concluded that two different types of global bifurcations may occur. Because of the symmetry of the system, it can be concluded that the diagonal of the system is an invariant one-dimensional submanifold. And it is controlled by a one-dimensional map which is equivalent to the classical Logistic map. The bifurcation curve of the system on the adjustment speed and the weight of the consumer surplus is obtained based on the properties of the Logistic map. And the synchronization phenomenon along the invariant diagonal is discussed at the end of the paper.

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A new experimental approach to study the stability of logistic map

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2008.08.022 ◽

2009 ◽

Vol 41 (4) ◽

pp. 2062-2066 ◽

Cited By ~ 20

Author(s):

Mamta Rani ◽

Rashi Agarwal

Keyword(s):

Experimental Approach ◽

Logistic Map ◽

The Stability

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Flip and Neimark–Sacker Bifurcations in a Coupled Logistic Map System

Discrete Dynamics in Nature and Society ◽

10.1155/2020/4103606 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14

Author(s):

A. Mareno ◽

L. Q. English

Keyword(s):

Bifurcation Analysis ◽

Center Manifold ◽

Logistic Map ◽

Local Bifurcation ◽

Center Manifold Theory ◽

Strongly Coupled ◽

Local Bifurcation Analysis ◽

The Stability ◽

Two Parameters ◽

Manifold Theory

In this paper, we consider a system of strongly coupled logistic maps involving two parameters. We classify and investigate the stability of its fixed points. A local bifurcation analysis of the system using center manifold theory is undertaken and then supported by numerical computations. This reveals the existence of a flip and Neimark–Sacker bifurcations.

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