On the dynamics and control of a new fractional difference chaotic map

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Samir Bendoukha
Keyword(s):  
Chaotic Map ◽  
Discrete Systems ◽  
Fractional Difference ◽  
Difference Map ◽  
Dynamics And Control ◽  
Generalized Hénon Map ◽  
Use Phase ◽  
The Rich ◽  
The Stability ◽  
And Control

Abstract In this paper, we propose and study a fractional Caputo-difference map based on the 2D generalized Hénon map. By means of numerical methods, we use phase plots and bifurcation diagrams to investigate the rich dynamics of the proposed map. A 1D synchronization controller is proposed similar to that of Pecora and Carrol, whereby we assume knowledge of one of the two states at the slave and replicate the second state. The stability theory of fractional discrete systems is used to guarantee the asymptotic convergence of the proposed controller and numerical simulations are employed to confirm the findings.

scholarly journals A unified approach to rigid body rotational dynamics and control

2011 ◽  
Vol 468 (2138) ◽  
pp. 395-414 ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Aaron D. Schutte
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Equations Of Motion ◽  
Fundamental Equation ◽  
Rotational Dynamics ◽  
Constrained Motion ◽  
Dynamics And Control ◽  
Common Framework ◽  
The Stability ◽  
And Control

This paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control. This is achieved in a simple systematic manner using the so-called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework. It is shown that a first application of this equation yields, in closed form, the equations of rotational dynamics, whereas a second application of the self-same equation yields two new methods for explicitly determining, in closed form, the nonlinear control torque needed to change the orientation of a rigid body. The stability of the controllers developed is analysed, and numerical examples showing the ease and efficacy of the unified methodology are provided.

scholarly journals Analysis of infectious disease transmission and prediction through SEIQR epidemic model

2021 ◽  
Vol 8 (1) ◽  
pp. 75-86
Author(s):  
Swati Tyagi ◽  
Shaifu Gupta ◽  
Syed Abbas ◽  
Krishna Pada Das ◽  
Baazaoui Riadh
Keyword(s):  
Mathematical Model ◽  
Infectious Diseases ◽  
Disease Transmission ◽  
Disease Spread ◽  
Infectious Disease Transmission ◽  
Control Measurement ◽  
Dynamics And Control ◽  
The Stability ◽  
And Control ◽  
Special Case

Abstract In literature, various mathematical models have been developed to have a better insight into the transmission dynamics and control the spread of infectious diseases. Aiming to explore more about various aspects of infectious diseases, in this work, we propose conceptual mathematical model through a SEIQR (Susceptible-Exposed-Infected-Quarantined-Recovered) mathematical model and its control measurement. We establish the positivity and boundedness of the solutions. We also compute the basic reproduction number and investigate the stability of equilibria for its epidemiological relevance. To validate the model and estimate the parameters to predict the disease spread, we consider the special case for COVID-19 to study the real cases of infected cases from [2] for Russia and India. For better insight, in addition to mathematical model, a history based LSTM model is trained to learn temporal patterns in COVID-19 time series and predict future trends. In the end, the future predictions from mathematical model and the LSTM based model are compared to generate reliable results.

High-Frequency Vibration of Leg Masses for Improving Gait Stability of Compass Walking on Slippery Downhill

2019 ◽  
Vol 31 (4) ◽  
pp. 621-628 ◽  
Author(s):  
Longchuan Li ◽  
Fumihiko Asano ◽  
Isao Tokuda ◽  
◽  
Keyword(s):  
High Frequency ◽  
Control Method ◽  
Frictional Coefficient ◽  
Biped Robot ◽  
High Frequency Oscillation ◽  
Frequency Oscillation ◽  
Dynamics And Control ◽  
Entrainment Effect ◽  
The Stability ◽  
And Control

Towards improving the stability of point-foot biped robot on slippery downhill, a novel and indirect control method is introduced in this paper using active wobbling masses attached to both legs. The whole dynamics which contains walking, sliding and wobbling, can be dominated by high-frequency oscillation via entrainment effect. Stable gaits are therefore achieved by controlling only 1% of the whole system where the original passive dynamic walking fails. First, we derive the equations of dynamics and control for this indirectly controlled biped walking on slippery downhill. Second, we numerically show the possibility of improving the stability with high-frequency oscillation. We also find the main effect of wobbling motion on walking via phase-plane plot. Third, we prove that the range of stable walking with respect to frictional coefficient can be enlarged by employing suitable high-frequency oscillation via parametric study. Our method will be further applied to more general conditions in real tasks which contain different locomotion types, where the whole dynamics could be dominated by high-frequency oscillation and the phase properties of the dynamics will be positively utilized.

Finite Element-Based Control of a Single-Link Flexible Hydraulic Manipulator

2017 ◽  
Author(s):  
Petri Mäkinen ◽  
Jouni Mattila
Keyword(s):  
Finite Element ◽  
Controller Design ◽  
Hydraulic Cylinder ◽  
Flexible Manipulator ◽  
Flexible Link ◽  
Hydraulic Actuator ◽  
Single Link ◽  
Dynamics And Control ◽  
The Stability ◽  
And Control

In this study, a stability-guaranteed, nonlinear, finite element-based control is presented for a single-link flexible manipulator with hydraulic actuation, subject to experimental validation. The strong, inherent nonlinearities of the hydraulic cylinder and fluid dynamics, coupled with flexible link dynamics, cause remarkable challenges in controlling the system effectively. In an attempt to cope with these challenges, a controller based on the Virtual Decomposition Control (VDC) approach is introduced. The VDC approach takes advantage of subsystem-dynamics-based control, enabling the handling of the dynamics and control of the hydraulic actuator and the flexible link separately, thus keeping the controller design relatively simple. The rigorous stability theory of the VDC approach guarantees the stability of the entire system. The experiments demonstrate the VDC controller’s performance in end-point control with built-in vibration dampening.

scholarly journals Complex dynamics and control investigation of a Cournot triopoly game formed based on a log-concave demand function

2017 ◽  
Vol 9 (7) ◽  
pp. 168781401770281 ◽  
Author(s):  
K Alnowibet ◽  
SS Askar ◽  
AA Elsadany
Keyword(s):  
Local Stability ◽  
Demand Function ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Phase Portraits ◽  
Control Scheme ◽  
Sensitive Dependence ◽  
Dynamics And Control ◽  
The Stability ◽  
And Control

This article investigates the dynamics of a Cournot triopoly game whose demand function is characterized by log-concavity. The game is formed using the bounded rationality approach. The existence and local stability of steady states of the game are analyzed. We find that an increase in the game parameters out of the stability region destabilizes the Cournot–Nash steady state. We confirm our obtained results using some numerical simulation. The simulation shows the consistence with the theoretical analysis and displays new and interesting dynamic behaviors, including bifurcation diagrams, phase portraits, maximal Lyapunov exponent, and sensitive dependence on initial conditions. Finally, a feedback control scheme is adopted to overcome the uncontrollable behavior of the game’s system occurred due to chaos.

scholarly journals Fractional Form of a Chaotic Map without Fixed Points: Chaos, Entropy and Control

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 720 ◽  
Author(s):  
Adel Ouannas ◽  
Xiong Wang ◽  
Amina-Aicha Khennaoui ◽  
Samir Bendoukha ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Approximate Entropy ◽  
Linearization Method ◽  
Entropy Measure ◽  
One Dimensional ◽  
Use Phase ◽  
And Control

In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method.

Sliding Mode Control of Pendulum-Driven Spherical Robots

2012 ◽  
Vol 591-593 ◽  
pp. 1519-1522 ◽  
Author(s):  
Tao Yu ◽  
Han Xu Sun ◽  
Qing Xuan Jia ◽  
Yan Heng Zhang ◽  
Wei Zhao
Keyword(s):  
Dynamic Model ◽  
Sliding Mode ◽  
Longitudinal Motion ◽  
Lyapunov Analysis ◽  
Spherical Robot ◽  
Hierarchical Sliding Mode ◽  
Dynamics And Control ◽  
Mathematical Techniques ◽  
The Stability ◽  
And Control

In this paper, the dynamics and control aspects of longitudinal motion of pendulum-driven spherical robots are investigated. A simplified dynamic model is established for a spherical robot, which is an underactuated mechanical system. Based on this dynamic model, a hierarchical sliding mode controller is proposed. The stability of the whole system is verified through Lyapunov analysis, and the asymptotic stability of the sub-sliding surfaces is proved with mathematical techniques. The validity of the proposed controller is illustrated through a simulation study.

Synchronization, Non-linear Dynamics and Control

2018 ◽  
Author(s):  
Ginestra Bianconi
Keyword(s):  
Multivariate Time Series ◽  
Network Control ◽  
Structural Constraints ◽  
Multilayer Networks ◽  
Linear Dynamics ◽  
Multiplex Network ◽  
Non Linear ◽  
Dynamics And Control ◽  
The Stability ◽  
And Control

This chapter is entirely devoted to characterizing non-linear dynamics on multilayer networks. Special attention is given to recent results on the stability of synchronization that extend the Master Stability Function approach to the multilayer networks scenario. Discontinous synchronization transitions on multiplex networks recently reported in the literature are also discussed, and their application discussed in the context of brain networks. This chapter also presents an overview of the major results regarding pattern formation in multilayer networks, and the proposed characterization of multivariate time series using multiplex visibility graphs. Finally, the chapter discusses several approaches for multiplex network control where the dynamical state of a multiplex network needs to be controlled by eternal signals placed on replica nodes satisfying some structural constraints.

scholarly journals The Fractional Form of the Tinkerbell Map Is Chaotic

2018 ◽  
Vol 8 (12) ◽  
pp. 2640 ◽  
Author(s):  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Samir Bendoukha ◽  
Thoai Vo ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Lyapunov Exponents ◽  
Stability Theory ◽  
Chaotic Map ◽  
Discrete Systems ◽  
Numerical Results ◽  
Asymptotic Convergence ◽  
Bifurcation Diagrams ◽  
Difference Form ◽  
The Stability

This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, and the asymptotic convergence of the states is established by means of the stability theory of linear fractional discrete systems. Numerical results are employed to confirm the analytical findings.

Nonlinear Dynamics and Control of an Ideal/Nonideal Load Transportation System With Periodic Coefficients

2006 ◽  
Vol 2 (1) ◽  
pp. 32-39 ◽  
Author(s):  
N. J. Peruzzi ◽  
J. M. Balthazar ◽  
B. R. Pontes ◽  
R. M. L. R. F. Brasil
Keyword(s):  
Equations Of Motion ◽  
Periodic Problem ◽  
Dc Motor ◽  
Transportation System ◽  
Stability Diagrams ◽  
Dynamics And Control ◽  
The Stability ◽  
Load Transportation ◽  
And Control ◽  
The Mathematical Model

In this paper, a loads transportation system in platforms or suspended by cables is considered. It is a monorail device and is modeled as an inverted pendulum built on a car driven by a dc motor. The governing equations of motion were derived via Lagrange’s equations. In the mathematical model we consider the interaction between the dc motor and the dynamical system, that is, we have a so called nonideal periodic problem. The problem is analyzed, qualitatively, through the comparison of the stability diagrams, numerically obtained, for several motor torque constants. Furthermore, we also analyze the problem quantitatively using the Floquet multipliers technique. Finally, we devise a control for the studied nonideal problem. The method that was used for analysis and control of this nonideal periodic system is based on the Chebyshev polynomial expansion, the Picard iterative method, and the Lyapunov-Floquet transformation (L-F transformation). We call it Sinha’s theory.

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