scholarly journals On Dynamics of a Fractional-Order Discrete System with Only One Nonlinear Term and without Fixed Points

Electronics ◽  
2020 ◽  
Vol 9 (12) ◽  
pp. 2179
Author(s):  
Amina-Aicha Khennaoui ◽  
Adel Ouannas ◽  
Shaher Momani ◽  
Iqbal M. Batiha ◽  
Zohir Dibi ◽  
...  
Keyword(s):  
Fixed Points ◽  
Fractional Order ◽  
Difference Equations ◽  
Discrete System ◽  
Nonlinear Term ◽  
Approximate Entropy ◽  
Discrete Systems ◽  
Largest Lyapunov Exponent ◽  
Hidden Attractors ◽  
Chaotic Modes

Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called “self-excited attractors” has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called “hidden attractors”, which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system.

Hidden attractors in a new fractional–order discrete system: Chaos, complexity, entropy, and control

2020 ◽  
Vol 29 (5) ◽  
pp. 050504 ◽  
Author(s):  
Adel Ouannas ◽  
Amina Aicha Khennaoui ◽  
Shaher Momani ◽  
Viet-Thanh Pham ◽  
Reyad El-Khazali
Keyword(s):  
Fractional Order ◽  
Discrete System ◽  
Hidden Attractors ◽  
And Control

Dynamics of Some Discretized Fractional-Order Differential Equations

2020 ◽  
pp. 58-114
Author(s):  
Sanaa Moussa Salman ◽  
Ahmed M. A. El-Sayed
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Fixed Points ◽  
Fractional Order ◽  
Difference Equations ◽  
Discretization Method ◽  
Piecewise Constant Arguments ◽  
Piecewise Constant ◽  
Fractional Order Differential Equations ◽  
Discretization Process

This chapter deals with fractional-order differential equations and their discretization. First of all, a discretization process for discretizing ordinary differential equations with piecewise constant arguments is presented. Secondly, a discretization method is proposed for discretizing fractional-order differential equations. Stability of fixed points of the discretized equations are investigated. Numerical simulations are carried out to show the dynamic behavior of the resulting difference equations such as bifurcation and chaos.

NUMERICAL COMPUTATIONS OF CONNECTING ORBITS IN DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1281-1293 ◽  
Author(s):  
FENGSHAN BAI ◽  
GABRIEL J. LORD ◽  
ALASTAIR SPENCE
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete System ◽  
Numerical Technique ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Connecting Orbits

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in nonautonomous and autonomous systems of ordinary differential equations. First, the existence and computation of connecting orbits between fixed points in discrete dynamical systems is discussed; then it is shown that the problem of finding connections between equilibria and periodic solutions in continuous systems may be reduced to finding connections between fixed points in a discrete system. Implementation of the method is considered: the choice of a linear solver is discussed and phase conditions are suggested for the discrete system. The paper concludes with some numerical examples: connections for equilibria and periodic orbits are computed for discrete systems and for nonautonomous and autonomous systems, including systems arising from the discretization of a partial differential equation.

scholarly journals The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles

2019 ◽  
Vol 17 (1) ◽  
pp. 1014-1024
Author(s):  
Hong Yan Xu ◽  
Xiu Min Zheng
Keyword(s):  
Fixed Points ◽  
Difference Equations ◽  
Painlevé Equations ◽  
Meromorphic Solutions ◽  
Painleve Equations ◽  
Zeros And Poles

Abstract The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q-difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q-difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.

scholarly journals Hidden Attractors in Discrete Dynamical Systems

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 616
Author(s):  
Marek Berezowski ◽  
Marcin Lawnik
Keyword(s):  
Dynamical Systems ◽  
Chaos Theory ◽  
Scientific Literature ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Hidden Attractors ◽  
Negative Effects ◽  
Chemical Reactors ◽  
Points Of View

Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.

scholarly journals On Solutions of Fractional Order Boundary Value Problems with Integral Boundary Conditions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Hussein A. H. Salem ◽  
Mieczysław Cichoń
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Point Of View ◽  
Integral Boundary ◽  
Special Cases

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

Hidden Extreme Multistability, Antimonotonicity and Offset Boosting Control in a Novel Fractional-Order Hyperchaotic System Without Equilibrium

2018 ◽  
Vol 28 (13) ◽  
pp. 1850167 ◽  
Author(s):  
Sen Zhang ◽  
Yicheng Zeng ◽  
Zhijun Li ◽  
Chengyi Zhou
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
State Feedback Control ◽  
Hyperchaotic System ◽  
Hidden Attractors ◽  
System Parameters ◽  
Offset Boosting ◽  
Extreme Multistability ◽  
Hidden Extreme Multistability ◽  
New System

Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil’nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system.

scholarly journals Iterative approximation of fixed points of generalized weak Presic type k-step iterative method for a class of operators

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 713-724 ◽  
Author(s):  
Mujahid Abbas ◽  
Dejan Ilic ◽  
Talat Nazir
Keyword(s):  
Iterative Method ◽  
Fixed Points ◽  
Difference Equations ◽  
Global Attractivity ◽  
Iterative Approximation ◽  
Type K ◽  
Matrix Difference Equations

In this paper, we study the convergence of the generalized weak Presic type k-step iterative method for a class of operators f:Xk ? X satisfying Presic type contractive conditions. We also obtain the global attractivity results for a class of matrix difference equations.

A COMPUTATIONAL FRAMEWORK TO DISCRIMINATE DIFFERENT ANESTHESIA STATES FROM EEG SIGNAL

2018 ◽  
Vol 30 (03) ◽  
pp. 1850020 ◽  
Author(s):  
Seyyed Abed Hosseini
Keyword(s):  
General Anesthesia ◽  
Rbf Neural Network ◽  
Principal Component ◽  
Approximate Entropy ◽  
Largest Lyapunov Exponent ◽  
Eeg Signal ◽  
Wavelet Coefficients ◽  
Sparse Principal Component Analysis ◽  
Beta Band ◽  
Computational Framework

This paper develops a computational framework to classify different anesthesia states, including awake, moderate anesthesia, and general anesthesia, using electroencephalography (EEG) signal. The proposed framework presents data gathering; preprocessing; appropriate selection of window length by genetic algorithm (GA); feature extraction by approximate entropy (ApEn), Petrosian fractal dimension (PFD), Hurst exponent (HE), largest Lyapunov exponent (LLE), Lempel-Ziv complexity (LZC), correlation dimension (CD), and Daubechies wavelet coefficients; feature normalization; feature selection by non-negative sparse principal component analysis (NSPCA); and classification by radial basis function (RBF) neural network. Because of the small number of samples, a five-fold cross-validation approach is used to validate the results. A GA is used to select that by observing an interval of 2.7[Formula: see text]s for further assessment. This paper assessed superior features, such as LZC, ApEn, PFD, HE, the mean value of wavelet coefficients for the beta band, and LLE. The results indicate that the proposed framework can classify different anesthesia states, including awake, moderate anesthesia, and general anesthesia, with an accuracy of 92.07%, 96.18%, and 93.42%, respectively. Therefore, the proposed framework can discriminate different anesthesia states with an average accuracy of 93.89%. Finally, the proposed framework provided a facilitative representation of the brain’s behavior in different states of anesthesia.

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