scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

scholarly journals Qualitative Analysis of Class of Fractional-Order Chaotic System via Bifurcation and Lyapunov Exponents Notions

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Ndolane Sene
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Chaotic System ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Fractional Chaotic System ◽  
The Stability ◽  
Predictor Corrector

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

scholarly journals CHAOS CONTROL AND SYNCHRONIZATION USING SYNERGETIC CONTROLLER WITH FRACTIONAL AND LINEAR EXTENDED MANIFOLD

2016 ◽  
Vol 17 (1) ◽  
pp. 115-126
Author(s):  
Morteza Pourmehdi ◽  
Abolfazl Ranjbar Noei ◽  
Jalil Sadati
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Order System ◽  
State Variables ◽  
System Quality ◽  
Control Effort ◽  
Lower Amplitude ◽  
The Stability ◽  
Fractional Order Controller ◽  
Control And Synchronization

In this manuscript, for the first time, a fractional-order manifold in a synergetic approach using a fractional order controller is introduced. Furtheremore, in the synergetic theory a macro variable is expended into a linear combination of state variables. An aim is to increase the convergence rate as well as time response of the whole closed loop system. Quality of the proposed controller is investigated to control and synchronize a nonlinear chaotic Coullet system in comparison with an integer order manifold synergetic controller. The stability of the proposed controller is proven using the Lyapunov method. In this regard stabilizing control effort is yielded. Simulation result confirm convergence of states towards zero. This is achieved through a control effort with fewer oscillations and lower amplitude of signls which confirm feasibility of the control effort in practice.KEYWORDS:  synergetic control theory; fractional order system; synchronization; nonlinear chaotic Coullet system; chaos control

scholarly journals On Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Ndolane Sene ◽  
Ameth Ndiaye
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Order Derivative ◽  
Order System ◽  
Fractional Order Systems ◽  
Fractional Order Derivative ◽  
The Stability

In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.

Quaternionic Neural Networks

2009 ◽  
pp. 411-439 ◽  
Author(s):  
Teijiro Isokawa ◽  
Nobuyuki Matsui ◽  
Haruhiko Nishimura
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Network Models ◽  
Recurrent Network ◽  
Affine Transformations ◽  
Neural Network Models ◽  
Fundamental Properties ◽  
Modern Physics ◽  
The Stability

Quaternions are a class of hypercomplex number systems, a four-dimensional extension of imaginary numbers, which are extensively used in various fields such as modern physics and computer graphics. Although the number of applications of neural networks employing quaternions is comparatively less than that of complex-valued neural networks, it has been increasing recently. In this chapter, the authors describe two types of quaternionic neural network models. One type is a multilayer perceptron based on 3D geometrical affine transformations by quaternions. The operations that can be performed in this network are translation, dilatation, and spatial rotation in three-dimensional space. Several examples are provided in order to demonstrate the utility of this network. The other type is a Hopfield-type recurrent network whose parameters are directly encoded into quaternions. The stability of this network is demonstrated by proving that the energy decreases monotonically with respect to the change in neuron states. The fundamental properties of this network are presented through the network with three neurons.

NEURAL ADAPTIVE CONTROL OF NONLINEAR MULTIVARIABLE SYSTEMS WITH APPLICATION TO A CLASS OF INVERTED PENDULUMS

2002 ◽  
Vol 12 (05) ◽  
pp. 411-424
Author(s):  
SHOULING HE
Keyword(s):  
Neural Networks ◽  
Control System ◽  
Control Algorithm ◽  
Learning Algorithm ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Base Point ◽  
Upper Body ◽  
Model Parameters ◽  
State Variables

In this paper multilayer neural networks (MNNs) are used to control the balancing of a class of inverted pendulums. Unlike normal inverted pendulums, the pendulum discussed here has two degrees of rotational freedom and the base-point moves randomly in three-dimensional space. The goal is to apply control torques to keep the pendulum in a prescribed position in spite of the random movement at the base-point. Since the inclusion of the base-point motion leads to a non-autonomous dynamic system with time-varying parametric excitation, the design of the control system is a challenging task. A feedback control algorithm is proposed that utilizes a set of neural networks to compensate for the effect of the system's nonlinearities. The weight parameters of neural networks updated on-line, according to a learning algorithm that guarantees the Lyapunov stability of the control system. Furthermore, since the base-point movement is considered unmeasurable, a neural inverse model is employed to estimate it from only measured state variables. The estimate is then utilized within the main control algorithm to produce compensating control signals. The examination of the proposed control system, through simulations, demonstrates the promise of the methodology and exhibits positive aspects, which cannot be achieved by the previously developed techniques on the same problem. These aspects include fast, yet well-maintained damped responses with reasonable control torques and no requirement for knowledge of the model or the model parameters. The work presented here can benefit practical problems such as the study of stable locomotion of human upper body and bipedal robots.

Dynamical Analysis and Circuit Simulation of a New Fractional-Order Hyperchaotic System and Its Discretization

2016 ◽  
Vol 26 (13) ◽  
pp. 1650222 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsonbaty ◽  
A. A. Elsadany ◽  
A. E. Matouk
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Control Technique ◽  
Order System ◽  
Qualitative Behavior ◽  
Hyperchaotic System

This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh–Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.

scholarly journals Stability and Multistability of a Bounded Rational Mixed Duopoly Model with Homogeneous Product

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Wei Zhou ◽  
Na Zhao ◽  
Tong Chu ◽  
Ying-Xiang Chang
Keyword(s):  
Joint Venture ◽  
Fractal Structure ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Mixed Duopoly ◽  
Formation Mechanisms ◽  
Multiple Attractors ◽  
Homogeneous Product ◽  
The Stability

In this paper, a mixed duopoly dynamic model with bounded rationality is built, where a public-private joint venture and a private enterprise produce homogeneous products and compete in the same market. The purpose of this research is to study the stability and the multistability of the established model. The local stability of all the equilibrium points is discussed by using Jury condition, and the stability region of the Nash equilibrium point has been given. A special fractal structure called “hub of periodicity” has been found in the two-parameter space by numerical simulation. In addition, the phenomena of multistability (also called coexistence of multiple attractors) are also studied using basins of attraction and 1-D bifurcation diagrams with adiabatic initial conditions. We find that there are two different coexistences of multiple attractors. And, the fractal structure of the attracting basin is also analyzed, and the formation mechanisms of “holes” and “contact” bifurcation have been revealed. At last, the long-term profits of the enterprises are studied. We find that some enterprises can even make more profits under a chaotic circumstance.

scholarly journals UNIQUENESS OF BUBBLE-FREE SOLUTION IN LINEAR RATIONAL EXPECTATIONS MODELS

2003 ◽  
Vol 7 (2) ◽  
pp. 171-191 ◽  
Author(s):  
Gabriel Desgranges ◽  
Stéphane Gauthier
Keyword(s):  
Rational Expectations ◽  
Initial Conditions ◽  
Free Solution ◽  
State Variables ◽  
Minimal Set ◽  
Current State ◽  
Stable Path ◽  
Market Fundamentals ◽  
The Stability ◽  
Natural Test

One usually identifies bubble solutions to linear rational expectations models by extra components (irrelevant lags) arising in addition to market fundamentals. Although there are still many solutions relying on a minimal set of state variables, i.e., relating in equilibrium the current state of the economic system to as many lags as initial conditions, there is a conventional wisdom that the bubble-free (fundamentals) solution should be unique. This paper examines the existence of endogenous stochastic sunspot fluctuations close to solutions relying on a minimal set of state variables, which provides a natural test for identifying bubble and bubble-free solutions. It turns out that only one solution is locally immune to sunspots, independently of the stability properties of the perfect-foresight dynamics. In the standard saddle-point configuration for these dynamics, this solution corresponds to the so-called saddle stable path.

scholarly journals Sensitivity, Equilibria, and Lyapunov Stability Analysis in Droop’s Nonlinear Differential Equation System for Batch Operation Mode of Microalgae Culture Systems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2192
Author(s):  
Abraham Guzmán-Palomino ◽  
Luciano Aguilera-Vázquez ◽  
Héctor Hernández-Escoto ◽  
Pedro Martin García-Vite
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Value Added ◽  
Biofuel Production ◽  
State Variables ◽  
Minimal Cell ◽  
Cell Quota ◽  
Depth Analysis ◽  
Number Of Equilibria

Microalgae-based biomass has been extensively studied because of its potential to produce several important biochemicals, such as lipids, proteins, carbohydrates, and pigments, for the manufacturing of value-added products, such as vitamins, bioactive compounds, and antioxidants, as well as for its applications in carbon dioxide sequestration, amongst others. There is also increasing interest in microalgae as renewable feedstock for biofuel production, inspiring a new focus on future biorefineries. This paper is dedicated to an in-depth analysis of the equilibria, stability, and sensitivity of a microalgal growth model developed by Droop (1974) for nutrient-limited batch cultivation. Two equilibrium points were found: the long-term biomass production equilibrium was found to be stable, whereas the equilibrium in the absence of biomass was found to be unstable. Simulations of estimated parameters and initial conditions using literature data were performed to relate the found results to a physical context. In conclusion, an examination of the found equilibria showed that the system does not have isolated fixed points but rather has an infinite number of equilibria, depending on the values of the minimal cell quota and initial conditions of the state variables of the model. The numerical solutions of the sensitivity functions indicate that the model outputs were more sensitive, in particular, to variations in the parameters of the half saturation constant and minimal cell quota than to variations in the maximum inorganic nutrient absorption rate and maximum growth rate.

scholarly journals Image Encryption Algorithm Based on a Novel Improper Fractional-Order Attractor and a Wavelet Function Map

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Jian-feng Zhao ◽  
Shu-ying Wang ◽  
Li-tao Zhang ◽  
Xiao-yan Wang
Keyword(s):  
Image Encryption ◽  
Fractional Order ◽  
Three Dimensional ◽  
Wavelet Function ◽  
Encryption Algorithm ◽  
Discrete Wavelet ◽  
State Variables ◽  
Secret Key ◽  
Initial State ◽  
Image Encryption Algorithm

This paper presents a three-dimensional autonomous chaotic system with high fraction dimension. It is noted that the nonlinear characteristic of the improper fractional-order chaos is interesting. Based on the continuous chaos and the discrete wavelet function map, an image encryption algorithm is put forward. The key space is formed by the initial state variables, parameters, and orders of the system. Every pixel value is included in secret key, so as to improve antiattack capability of the algorithm. The obtained simulation results and extensive security analyses demonstrate the high level of security of the algorithm and show its robustness against various types of attacks.

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