On the Convergence of Logistic Map in NOOR Orbit

Fractal and Fractional

A control scheme for finite-time stabilization of unstable orbits of the fractional difference logistic map is proposed in this paper. The presented technique is based on isolated perturbation impulses used to correct the evolution of the map’s trajectory after it deviates too far from the neighborhood of the unstable orbit, and does not require any feedback control loops. The magnitude of the control impulses is determined by means of H-rank algorithm, which helps to reveal the pseudo-manifold of non-asymptotic convergence of the fractional difference logistic map. Numerical experiments are used to illustrate the effectiveness and the feasibility of the proposed approach, which is applicable beyond the studied fractional difference logistic map.

Recurrence as behaviour in the dynamical system is also a state which happens as a result of an outcome of a point which does return to its initial state. But in dynamical system the most simple and easy way to study regime in one dimension is the logistic function. In this study we seek to understudy the recurrence as a full strong state from the nature or behaviour of the logistic function, where the periodic orbits as behaviour the logistic function is considered. A point can only be termed as recurrent if it is in its own future state. A periodic orbit returns infinitely often to each point on the orbit. And so it is clear that an orbit is recurrent when it returns repeatedly to each neighbourhood of its initial position. Recurrence as in dynamical system is a result of periodic formation which is a movement that returns back to the original state or position at a constant rate. A systematic example for each periodic point from the logistic function relative to a control paramet...

International Journal of Information Technology and Computer Science, 2020

Recently, the logistic map is studied to analyse the impact on the chaotic dynamics of various iterated logistic maps using Picard, Mann, and many more. The purpose of this paper is to explore the behavior of a multi-scale population model, i.e. modified logistic map (Mod-LM) and chosen population proportion model, i.e. extended logistic map (Ex-LM) in an I-superior orbit using a bifurcation diagram. The additional parameters of Mod-LM and Ex-LM with the three-step iteration system, increase the degree of freedom which invariably enhances the stability of both the functions. A detailed study of possible scenarios has been conducted to discover the effect of each parameter to the fixed point and its location, periodic cycle, and stability condition by examining the corresponding bifurcation diagram. The experimental result is discussed in terms of convergence point and chaotic range of the given dynamical systems.

Simple dynamical systems often involve periodic motion. Quasi-periodic or chaotic motion is frequently present in more complicated dynamical systems. However, for the most part, underneath periodic motion models chaotic motion. Chaotic attractors are nearly always present in such dissipative systems. Since their discovery in 1963 by E. Lorenz, they have been extensively studied in order to understand their nature. In the past decade, the aim of the research has been shifted to the applications for industrial mathematics. Their importance in this field is rapidly growing. Chaotic orbits embedded in chaotic attractor can be controlled allowing the possibility to control laser beams or chemical processes and improving techniques of communications. They can also produce very long sequences of numbers which can be used as efficiently as random numbers even if they have not the same nature. However, mathematical results concerning chaotic orbits are often obtained using sets of real numbers (belonging to R or R n) (e.g. the famous theorem of A. N. Sharkovskiǐ which defines which ones periods exist for continuous functions such as logistic or tent maps). O.E. Lanford III reports the results of some computer experiments on the orbit structure of the discrete maps on a finite set which arise when an expanding map of the circle is iterated "naively" on the computer. There is a huge gap between these results and the theorem of Sharkovskiǐ, due to the discrete nature of floating points used by computers. This article introduces new models of very very weakly coupled logistic and tent maps for which orbits of very long period are found. The length of these periods is far greater than one billion. We call giga-periodic orbits such orbits for which the length is greater than 10 9 and less than 10 12. Tera, and peta periodic orbits are the name of the orbits the length of which is one thousand or one million greater. The property of these models relatively to the distribution of the iterates (invariant measure) are described. They are found very useful for industrial mathematics for a variety of purposes such as generation of cryptographic keys, computer games and some classes of scientific experiments.

International Journal of Mathematics and Mathematical Sciences

In this paper, we employ the logistic map and the cubic map to locate the relaxation and the convergence to the periodic fixed point of a system, specifically, the period—1 fixed point. The study has shown that the period—1 fixed point of a logistic map as a recurrence has its convergence at a transcritical bifurcation having its power-law fit with exponent β = − 1 when α = 1 and μ = 0 . The cubic map shows its convergence to the fixed point at a pitchfork bifurcation decaying at a power law with exponent β = − 1 / 2 α = 1 and μ = 0 . However, the system shows their relaxation time at the same power law with exponents and z = − 1 .

Discrete & Continuous Dynamical Systems - B, 2017

In this paper, the dynamics of the celebrated p−periodic onedimensional logistic map is explored. A result on the global stability of the origin is provided and, under certain conditions on the parameters, the local stability condition of the p−periodic orbit is shown to imply its global stability.

Chaos, Solitons & Fractals, 2001

We rigorously study a recent algorithm due to Davidchack and Lai (DL) [Davidchack RL, Lai Y-C. Phys Rev E 1999;60(5):6172±5] for eciently locating complete sets of hyperbolic periodic orbits for chaotic maps. We give theorems concerning sucient conditions on convergence and also describing variable sized basins of attraction of initial seeds, thus pointing out a particularly attractive feature of the DL-algorithm. We also point out the true role of involutary matrices which is dierent from that implied by Schmelcher and Diakonos [Schmelcher P, Diakonos FK.

Acta Physica Polonica A, 2015

Motivated by a possibility to optimize modelling of the population evolution we postulate a generalization of the well-know logistic map. Generalized difference equation reads: x n+1 = rx p n (1 − x q n), (1) x ∈ [0, 1], (p, q) > 0, n = 0, 1, 2, ..., where the two new parameters p and q may assume any positive values. The standard logistic map thus corresponds to the case p = q = 1. For such a generalized equation we illustrate the character of the transition from regularity to chaos as a function of r for the whole spectrum of p and q parameters. As an example we consider the case for p = 1 and q = 2 both in the periodic and chaotic regime. We focus on the character of the corresponding bifurcation sequence and on the quantitative nature of the resulting attractor as well as its universal attribute (Feigenbaum constant).

Journal of Physics: Conference Series, 2019

Logistic map is one of the simplest, and at the same time the most commonly used dynamic system, which is characterized by chaos. The article presents approximations of the logistic map through its extension into the Fourier’s series. Obtained in such way dynamical systems were analyzed, among others for the Lyapunov exponent and bifurcation diagrams. Furthermore the issue of the density of the iterated variable and some applications in chaos based cryptography were commented.

Discrete Mathematics, 2014

One of the simplest polynomial recursions exhibiting chaotic behavior is the logistic map ∀n ∈ N and a ∈ (0, 4], the discretetime model of the differential growth introduced by Verhulst almost two centuries ago (Verhulst, 1838) . Despite the importance of this discrete map for the field of nonlinear science, explicit solutions are known only for the special cases a = 2 and a = 4. In this article, we propose a representation of the Verhulst logistic map in terms of a finite power series in the map's growth parameter a and initial value x 0 whose coefficients are given by the solution of a system of linear equations. Although the proposed representation cannot be viewed as a closed-form solution of the logistic map, it may help to reveal the sensitivity of the map on its initial value and, thus, could provide insights into the mathematical description of chaotic dynamics.