scholarly journals Chaotic Map with No Fixed Points: Entropy, Implementation and Control

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 279 ◽  
Author(s):  
Van Huynh ◽  
Adel Ouannas ◽  
Xiong Wang ◽  
Viet-Thanh Pham ◽  
Xuan Nguyen ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Experimental Observation ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Return Map ◽  
Control Schemes ◽  
And Control

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map.

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.

Two Simplest Quadratic Chaotic Maps Without Equilibrium

2018 ◽  
Vol 28 (12) ◽  
pp. 1850144 ◽  
Author(s):  
Shirin Panahi ◽  
Julien C. Sprott ◽  
Sajad Jafari
Keyword(s):  
Dynamical System ◽  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic Maps ◽  
Basin Of Attraction ◽  
Hidden Attractor ◽  
Return Map ◽  
Right Hand ◽  
Dynamical Properties ◽  
The Right

Two simple chaotic maps without equilibria are proposed in this paper. All nonlinearities are quadratic and the functions of the right-hand side of the equations are continuous. The procedure of their design is explained and their dynamical properties such as return map, bifurcation diagram, Lyapunov exponents, and basin of attraction are investigated. These maps belong to the hidden attractor category which is a newly introduced category of dynamical system.

scholarly journals Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. Q. Khan ◽  
M. B. Javaid
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Chaotic Behavior ◽  
Discrete Analogue ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Time Model ◽  
Local Dynamics

AbstractThe local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model.

scholarly journals Dynamics of the Modified n-Degree Lorenz System

2019 ◽  
Vol 4 (2) ◽  
pp. 315-330 ◽  
Author(s):  
Sk. Sarif Hassan ◽  
Moole Parameswar Reddy ◽  
Ranjeet Kumar Rout
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Lyapunov Exponents ◽  
Limit Cycle ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Lorenz Model ◽  
Periodic Behavior ◽  
Chaotic Model

AbstractThe Lorenz model is one of the most studied dynamical systems. Chaotic dynamics of several modified models of the classical Lorenz system are studied. In this article, a new chaotic model is introduced and studied computationally. By finding the fixed points, the eigenvalues of the Jacobian, and the Lyapunov exponents. Transition from convergence behavior to the periodic behavior (limit cycle) are observed by varying the degree of the system. Also transiting from periodic behavior to the chaotic behavior are seen by changing the degree of the system.

On the Integrability and Chaotic Behavior of an Ecological Model

1997 ◽  
Vol 07 (02) ◽  
pp. 463-468 ◽  
Author(s):  
M. P. Joy
Keyword(s):  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Food Chain ◽  
Ecological Model ◽  
Chaotic Behavior ◽  
Painlevé Analysis ◽  
Chain Model ◽  
Numerical Techniques ◽  
Food Chain Model ◽  
Painleve Analysis

A three-species food chain model is studied analytically as well as numerically. Integrability of the model is studied using Painlevé analysis while chaotic behavior is studied using numerical techniques, such as calculation of Lyapunov exponents, plotting the bifurcation diagram and phase plots. We correct and critically comment on the wrong results reported recently on this ecological model, in a paper by Rai "1995".

scholarly journals On Two-Dimensional Fractional Chaotic Maps with Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 756 ◽  
Author(s):  
Fatima Hadjabi ◽  
Adel Ouannas ◽  
Nabil Shawagfeh ◽  
Amina-Aicha Khennaoui ◽  
Giuseppe Grassi
Keyword(s):  
Fixed Points ◽  
Lyapunov Exponents ◽  
Chaotic Maps ◽  
Chaotic Behavior ◽  
Closed Curve ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Coexisting Attractors

In this paper, we propose two new two-dimensional chaotic maps with closed curve fixed points. The chaotic behavior of the two maps is analyzed by the 0–1 test, and explored numerically using Lyapunov exponents and bifurcation diagrams. It has been found that chaos exists in both fractional maps. In addition, result shows that the proposed fractional maps shows the property of coexisting attractors.

scholarly journals PPF-Dependent Fixed Point Results for Multi-Valued ϕ-F-Contractions in Banach Spaces and Applications

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1375 ◽  
Author(s):  
Mohammed M. M. Jaradat ◽  
Babak Mohammadi ◽  
Vahid Parvaneh ◽  
Hassen Aydi ◽  
Zead Mustafa
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theory ◽  
Real Life ◽  
Point Theory ◽  
Control Engineering ◽  
Life Problems ◽  
Ulam Theorem ◽  
Delay Differential ◽  
And Control

The solutions for many real life problems is obtained by interpreting the given problem mathematically in the form of f ( x ) = x . One of such examples is that of the famous Borsuk–Ulam theorem, in which using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this paper, we initiate ϕ − F -contractions and study the existence of PPF-dependent fixed points (fixed points for mappings having variant domains and ranges) for these related mappings in the Razumikhin class. Our theorems extend and improve the results of Hammad and De La Sen [Mathematics, 2019, 7, 52]. As applications of our PPF dependent fixed point results, we study the existence of solutions for delay differential equations (DDEs) which have numerous applications in population dynamics, bioscience problems and control engineering.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

Biohybrid robots are synthetic biology systems

2018 ◽  
Author(s):  
Joseph Ayers
Keyword(s):  
Nervous System ◽  
Synthetic Biology ◽  
Behavioral Control ◽  
Eukaryotic Cells ◽  
Chemical Senses ◽  
Biomimetic Robots ◽  
Control Schemes ◽  
Induced Pluripotent ◽  
And Control ◽  
And Robotics

This chapter describes how synthetic biology and organic electronics can integrate neurobiology and robotics to form a basis for biohybrid robots and synthetic neuroethology. Biomimetic robots capture the performance advantages of animal models by mimicking the behavioral control schemes evolved in nature, based on modularized devices that capture the biomechanics and control principles of the nervous system. However, current robots are blind to chemical senses, difficult to miniaturize, and require chemical batteries. These obstacles can be overcome by integration of living engineered cells. Synthetic biology seeks to build devices and systems from fungible gene parts (gene systems coding different proteins) integrated into a chassis (induced pluripotent eukaryotic cells, yeast, or bacteria) to produce devices with properties not found in nature. Biohybrid robots are examples of such systems (interacting sets of devices). A nascent literature describes genes that can mediate organ levels of organization. Such capabilities, applied to biohybrid systems, portend truly biological robots guided, controlled, and actuated solely by life processes.

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