scholarly journals The Dynamics of a Four-Step Feedback Procedure to Control Chaos

2021 ◽  
Author(s):  
Jose S. Cánovas
Keyword(s):  
Topological Entropy ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Simple Complex ◽  
Step Procedure ◽  
Parrondo’S Paradox

Abstract In this paper we make a description of the dynamics of a four-step procedure to control the dynamics of the logistic map. Some massive calculations are made for computing the topological entropy with prescribed accuracy. This provides us the parameter regions where the model has a complicated dynamical behavior. Our computations also show the dynamic Parrondo's paradox ``simple+simple=complex'', which should be taking into account to avoid undesirable dynamics.

scholarly journals On α-Limit Sets in Lorenz Maps

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1153
Author(s):  
Łukasz Cholewa ◽  
Piotr Oprocha
Keyword(s):  
Topological Entropy ◽  
Dynamical Behavior ◽  
Unimodal Maps ◽  
Limit Sets ◽  
Lorenz Maps ◽  
Provided Examples

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.

Polynomial Entropy of the Logistic Map

2021 ◽  
Vol 58 (2) ◽  
pp. 206-215
Author(s):  
Milan Perić
Keyword(s):  
Topological Entropy ◽  
Logistic Map ◽  
Low Complexity ◽  
Almost All

We study the polynomial entropy of the logistic map depending on a parameter, and we calculate it for almost all values of the parameter. We show that polynomial entropy distinguishes systems with a low complexity (i.e. for which the topological entropy vanishes).

Applicable Image Security Based on New Hyperchaotic System

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2290
Author(s):  
Jingya Wang ◽  
Xianhua Song ◽  
Huiqiang Wang ◽  
Ahmed A. Abd El-Latif
Keyword(s):  
Image Encryption ◽  
Complexity Analysis ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Hyperchaotic System ◽  
Two Dimensional ◽  
Analysis Methods ◽  
Lyapunov Index ◽  
And Diffusion ◽  
New Hyperchaotic System

Hyperchaotic systems are widely applied in the cryptography domain on account of their more complex dynamical behavior. In view of this, the greatest contribution of this paper is that a two-dimensional Sine coupling Logistic modulated Sine (2D-SCLMS) system is proposed based on Logistic map and Sine map. By a series of analyses, including Lyapunov index (LE), 0–1 test, two complexity analysis methods, and two entropy analysis methods, it is concluded that the new 2D-SCLMS map is hyperchaotic with a wider range of chaos and more complex randomness. The new system combined with two-dimensional Logistic-Sine Coupling Mapping (2D-LSCM) is further applied to an image encryption application. SHA-384 is used to generate the initial values and parameters of the two chaotic systems. Symmetric keys are generated during this operation, which can be applied to the proposed image encryption and decryption algorithms. The encryption process and the decryption process of the new image encryption approaches mainly include pixel scrambling, exclusive NOR (Xnor), and diffusion operations. Multiple experiments illustrate that this scheme has higher security and lower time complexity.

scholarly journals Computation of the topological entropy in chaotic biophysical bursting models for excitable cells

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Jorge Duarte ◽  
Luís Silva ◽  
J. Sousa Ramos
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Excitable Cells ◽  
Physiological Models ◽  
Nonlinear Dynamical ◽  
Chaotic Bursting ◽  
Oscillatory State ◽  
One Dimensional Maps

One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described. In the present paper, we study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics on the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.

scholarly journals A New Cipher Based on Feistel Structure and Chaotic Maps

2019 ◽  
Vol 16 (1(Suppl.)) ◽  
pp. 0270
Author(s):  
Al-Bahrani Et al.
Keyword(s):  
Chaotic Maps ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Data Encryption ◽  
Lookup Table ◽  
Secret Key ◽  
Text Length ◽  
Sequence Generator ◽  
And Diffusion ◽  
Confusion And Diffusion

Chaotic systems have been proved to be useful and effective for cryptography. Through this work, a new Feistel cipher depend upon chaos systems and Feistel network structure with dynamic secret key size according to the message size have been proposed. Compared with the classical traditional ciphers like Feistel-based structure ciphers, Data Encryption Standards (DES), is the common example of Feistel-based ciphers, the process of confusion and diffusion, will contains the dynamical permutation choice boxes, dynamical substitution choice boxes, which will be generated once and hence, considered static,             While using chaotic maps, in the suggested system, called Chaotic-based Proposed Feistel Cipher System (CPFCS), we made the confusion and diffusion in dynamical behavior based on Standard and Lorenz maps. The first is used for substitution, and the second one for permutation operations .A proposed cryptographic system uses the same work (the same way) for both enciphering and deciphering. The proposed cipher operates on more than 500 bytes (4000-bit) readable text blocks by six round computing. Within the basic operator of the cipher, i.e., in the function of the round F, a dynamical lookup table 2D standard map system is used to enhance the complexity and diffusion of the unreadable text. Also, a 3D Logistic map used for key sequence generator and chaos based dynamical Initial Permutation (dynamical IP) are used to increase the diffusion and confusion. Three different image sizes and three different text length were implemented in CPFCS.  The results of the proposed system and security tests improve the applicability of PFCS in the data protection and security.

scholarly journals Development of mandelbrot set for the logistic map with two parameters in the complex plane

2021 ◽  
Vol 6 (3 (114)) ◽  
pp. 47-56
Author(s):  
Wasan Saad Ahmed ◽  
Saad Qasim Abbas ◽  
Muntadher Khamees ◽  
Mustafa Musa Jaber
Keyword(s):  
Fixed Points ◽  
Critical Point ◽  
Complex Plane ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Iteration Function ◽  
Two Parameters ◽  
Positive Integers

In this paper, the study of the dynamical behavior of logistic map has been disused with representing fractals graphics of map, the logistic map depends on two parameters and works in the complex plane, the map defined by f(z,α,β)=αz(1–z)β. where  and  are complex numbers, and β is a positive integers number, the visualization method used in this work to generate fractals of the map and to inspect the relation between the value of β and the shape of the map, this visualization analysis showed also that, as the value of β increasing, as the number of humps in the function also increasing, and it demonstrate that is true also for the function’s first iteration , f2(x0)=f(f(x0)) and the second iteration , f3(x0)=f(f2(x0)), beside that , the visualization technique showed that the number of humps in that fractal is less than the ones in the second iteration of the original function ,the study of the critical points and their properties of the logistic map also discussed it, whereas finding the fixed point led to find the critical point of the function f, in addition , it haven proven for the set of all pointsα∈C and β∈N, the iteration function f(f(z) has an attractive fixed points, and belongs to the region specified by the disc |1–β(α–1)|<1. Also, The discussion of the Mandelbrot set of the function defined by the f(f(z)) examined in complex plans using the path principle, such that the path of the critical point z=z0 is restricted, finally, it has proven that the Mandelbrot set f(z,α,β) contains all the attractive fixed points and all the complex numbers  in which α≤(1/β+1) (1/β+1) and the region containing the attractive fixed points for f2(z,α,β) was identified

scholarly journals A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level

2020 ◽  
Vol 16 (4) ◽  
pp. 625-635
Author(s):  
J.G. Damasceno ◽  
◽  
J.A.G. Miranda ◽  
L.G. Perona ◽  
◽  
...  
Keyword(s):  
Energy Level ◽  
Topological Entropy ◽  
Dynamical Behavior ◽  
High Energy ◽  
Lagrangian Systems ◽  
High Energy Level ◽  
Positive Topological Entropy ◽  
Mather Theory ◽  
Closed Orbits ◽  
Lagrangian Flow

In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $E_{L}^{-1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale property in $E_{L}^{-1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_{L}^{-1}(c)$). The proof requires the use of well-known results from Aubry – Mather theory.

On the Fundamental Issues in Nonstationary Dynamics and Chaos

1993 ◽  
Author(s):  
Ross M. Evan-Iwanowski ◽  
Chu Ho Lu ◽  
Germain L. Ostiguy
Keyword(s):  
Duffing Oscillator ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Period Doubling ◽  
Codimension One ◽  
Extended Value ◽  
Limit Motion ◽  
Short Time ◽  
Nonstationary Dynamics ◽  
Considerable Complexity

Abstract In nonstationary (NS) systems some control parameters (CPs) have the following forms: CP(t) = CP0 + ψ(t), where CP0 = const, and ψ(t) are arbitrary functions of t. Other arbitrary functions which play a pivotal role in the NS systems are the parameter functions ϕ(CP) = 0, CP = {CP1, CP2...CPn}. While the functions ψ(t) determine the time directions of the NS dynamical behavior, the functions ϕ(CP) = 0 determine the paths for the CPs to follow. The NS processes are permanently transient due to the functions ψ(t) and/or ϕ(CP) = 0, and for that reason, they can be extremely complex. Clearly then, it is essential to address the fundamental problems of cohesion and definitness of these processes. Using select examples, these issues have been studied in this presentation and have been resolved in positive. Specifically, the following have been demonstrated: (1) convergence (definitivness) of the NS logistic map and the softening Duffing oscillator to an NS limit motion (2) the appearance of a sequence of similar attractors for different NS bifurcations (3) the effects of different parameter paths, ϕ(CP) = 0, in the period doubling region of the Duffing oscillator (4) the effects of linear and cyclic paths in transition through the Ueda bifurcation regions. The results obtained show considerable complexity of the NS dynamic and chaotic responses (5) for exponential ψ(t), the Lorenz “weather” three-term model exhibit a periodic “window” in the chaotic range for an extended value of t. (6) the effects of different ψ(t) in the typical codimension one bifurcations (7) the ST chaos may be created or annihilated by injection of NS inputs (8) an efficient and fast stabilization, i.e., reduction of ST vibration to near the static equilibrium in a short time, can be accomplished by NS changes of the parameters of the system.

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