scholarly journals Coupled Discrete Fractional-Order Logistic Maps

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2204
Author(s):  
Marius-F. Danca ◽  
Michal Fečkan ◽  
Nikolay Kuznetsov ◽  
Guanrong Chen
Keyword(s):  
Numerical Integration ◽  
Fractional Order ◽  
Chaotic Dynamics ◽  
Chaotic Map ◽  
Hidden Attractors ◽  
Fractional Difference ◽  
Coexistence Of Attractors ◽  
Dynamical Properties

This paper studies a system of coupled discrete fractional-order logistic maps, modeled by Caputo’s delta fractional difference, regarding its numerical integration and chaotic dynamics. Some interesting new dynamical properties and unusual phenomena from this coupled chaotic-map system are revealed. Moreover, the coexistence of attractors, a necessary ingredient of the existence of hidden attractors, is proved and analyzed.

scholarly journals Coexisting Hidden and Self-Excited Attractors in an Economic Model of Integer or Fractional Order

2021 ◽  
Vol 31 (04) ◽  
pp. 2150062
Author(s):  
Marius-F. Danca
Keyword(s):  
Economic System ◽  
Fractional Order ◽  
Economic Model ◽  
Foreign Capital ◽  
Capital Inflow ◽  
Hidden Attractors ◽  
Coexisting Attractors ◽  
Integer Order ◽  
Coexistence Of Attractors ◽  
System Variables

In this paper, the dynamics of an economic system with foreign financing, of integer or fractional order, are analyzed. The symmetry of the system determines the existence of two pairs of coexisting attractors. The integer-order version of the system proves to have several combinations of coexisting hidden attractors with self-excited attractors. Because one of the system variables represents the foreign capital inflow, the presence of hidden attractors could be of real interest in economic models. The fractional-order variant presents another interesting coexistence of attractors in the fractional-order space.

scholarly journals Discrete fractional order two-point boundary value problem with some relevant physical applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
R. Dhineshbabu ◽  
S. Rashid ◽  
M. Rehman
Keyword(s):  
Boundary Value Problem ◽  
Fractional Order ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Difference Operators ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Rassias Stability

Abstract The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann–Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.

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.

Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis, chaos control and synchronisation in its fractional-order form

Pramana ◽  
2018 ◽  
Vol 91 (1) ◽  
Author(s):  
Victor Kamdoum Tamba ◽  
Sifeu Takougang Kingni ◽  
Gaetan Fautso Kuiate ◽  
Hilaire Bertrand Fotsin ◽  
Pierre Kisito Talla
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Van Der Pol ◽  
Order Form ◽  
Coexistence Of Attractors

Analysis of a 4-D Hyperchaotic Fractional-Order Memristive System with Hidden Attractors

2017 ◽  
pp. 207-235 ◽  
Author(s):  
Christos Volos ◽  
V.-T. Pham ◽  
E. Zambrano-Serrano ◽  
J. M. Munoz-Pacheco ◽  
Sundarapandian Vaidyanathan ◽  
...  
Keyword(s):  
Fractional Order ◽  
Hidden Attractors ◽  
Memristive System

scholarly journals Finite Time Synchronization for Fractional Order Sprott C Systems with Hidden Attractors

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Cui Yan ◽  
He Hongjun ◽  
Lu Chenhui ◽  
Sun Guan
Keyword(s):  
Fractional Order ◽  
Finite Time ◽  
Chaotic Systems ◽  
Time Synchronization ◽  
Engineering Practice ◽  
Fractional Order Systems ◽  
Spectral Entropy ◽  
Hidden Attractors ◽  
Finite Time Stability ◽  
Peculiar Phenomenon

Fractional order systems have a wider range of applications. Hidden attractors are a peculiar phenomenon in nonlinear systems. In this paper, we construct a fractional-order chaotic system with hidden attractors based on the Sprott C system. According to the Adomain decomposition method, we numerically simulate from several algorithms and study the dynamic characteristics of the system through bifurcation diagram, phase diagram, spectral entropy, and C0 complexity. The results of spectral entropy and C0 complexity simulations show that the system is highly complex. In order to apply such research results to engineering practice, for such fractional-order chaotic systems with hidden attractors, we design a controller to synchronize according to the finite-time stability theory. The simulation results show that the synchronization time is short and the robustness is stable. This paper lays the foundation for the study of fractional order systems with hidden attractors.

scholarly journals Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
Mary Jacintha ◽  
Abdullah Özbekler
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Fractional Operators ◽  
Damping Term ◽  
Fractional Difference ◽  
Riccati Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Positive Integers

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.

scholarly journals A Multidimensional Hyperjerk Oscillator: Dynamics Analysis, Analogue and Embedded Systems Implementation, and Its Application as a Cryptosystem

Sensors ◽  
2019 ◽  
Vol 20 (1) ◽  
pp. 83 ◽  
Author(s):  
Tsafack Nestor ◽  
Nkapkop De Dieu ◽  
Kengne Jacques ◽  
Effa Yves ◽  
Abdullah Iliyasu ◽  
...  
Keyword(s):  
Embedded System ◽  
Image Encryption ◽  
Dynamics Analysis ◽  
Entropy Analysis ◽  
Hidden Attractors ◽  
Systems Implementation ◽  
Substitution Process ◽  
Simple Implementation ◽  
Dynamical Properties ◽  
Image Encryption Algorithm

A lightweight image encryption algorithm is presented based on chaos induction via a 5-dimensional hyperjerk oscillator (5DHO) network. First, the dynamics of our 5DHO network is investigated and shown to exhibit up to five coexisting hidden attractors in the state space that depend exclusively on the system’s initial values. Further, a simple implementation of the circuit was used to validate its ability to exhibit chaotic dynamical properties. Second, an Arduino UNO platform is used to confirm the usability of our oscillator in embedded system implementation. Finally, an efficient image encryption application is executed using the proposed chaotic networks based on the use of permutation-substitution sequences. The superior qualities of the proposed strategy are traced to the dynamic set of keys used in the substitution process which heralds the generation of the final ciphered image. Based on the average results obtained from the entropy analysis (7.9976), NPCR values (99.62), UACI tests (33.69) and encryption execution time for 512 × 512 images (0.1141 s), the proposed algorithm is adjudged to be fast and robust to differential and statistical attacks relative to similar approaches.

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