On a representation of the Verhulst logistic map

Electronics and Communications in Japan (Part I: Communications), 1985

In an attempt to discover the effect of recurrence on the topological dynamics, a nonlinear function whose regime of periodicity amounts to recurrence was considered, thus the logistic function. This research seeks to study the logistic function as to how it really behaves. In the field of dynamics most especially this function in terms of discrete form has been studied. Logistic equation as a model based on population growth was initially originated by the famous Pierre-Francios Verhulst. It is a continuous form written as í µí±‘í µí±¥ í µí±‘í µí±¡ = í µí±Ÿ(í µí±¥ − í µí±¥ 2), which depend on time. It can be restructured from the continuous form into a discrete form known as logistic function, written as; í µí±¥ í µí±›+1 = í µí±Ÿí µí±¥ í µí±› (1 − í µí±¥ í µí±›), with í µí±› = 0,1,2,3 … … ,í µí±¥ í µí±› is

2021

We present a new difference equation with two parameters c ∈ [0, 1] and A ∈ . This equation is equivalent to the logistic mapping if c = 1 and the Morishita mapping if c = 0, which are the well-known chaotic and integrable mappings, respectively. We first consider the case A = 4 and investigate the time evolution by changing the parameter c ∈ [0, 1]. We next change both two parameters A ∈ [3, 4] and c ∈ [0, 1] and present the corresponding 3D bifurcation diagram.

Technological Forecasting and Social Change, 1990

Some comments are made on a recent paper by Gordon and Greenspan [l]. Two interesting variants of the logistic difference equation in the context of innovation diffusion, the first with a superimposed periodic force term and the second with an additional lag of one generation, with their new features are discussed. Gordon and Greenspan [I] have recently reviewed in the context of technological forecasting the complex and interesting behavior of the logistic difference equation X ffl = AX(1-Xl (1) as an exemplar of nonlinear systems and drawn attention to the fact that this one equation can converge to equilibrium, oscillate stably, diverge unstably, or exhibit persistent chaos within predetermined boundaries depending upon the value of the control parameter A. They have also drawn attention to the extreme sensitivity of the response of the system to small changes in the initial conditions. The logistic eq. (1) has in fact been studied in great detail by many authors [2]. The equation assumes that the value of X at any instant (t+ 1) depends only on its value at the previous instant t. A is a constant that controls the behavior of the system, and X (which is normalized to lie between 0 and 1) could represent, for example, the species population, or the fraction of people who have adopted a new technology, or the acquired stock of a new product. This paper is intended to be read in conjunction with the paper of Gordon and Greenspan, and is written with two objectives in mind: first, to indicate the kinds of problems that may be encountered if we use the logistic difference equation for technological and social forecasting without worrying about the magnitude of the growth

International Journal of Theoretical Physics, 1990

We study the onset of chaos in a logistic map whose parameter is modulated nonlinearly. The bifurcation pattern with respect to a parameter/z is obtained and the critical value of ~ is seen to be 0.89, where periodicity just ends. Further evidence for this regime is obtained from the analysis of the intermittency pattern. The stability in the different ranges of p, under repeated iteration is exhibited by the values of Lyapunov exponents. Beyond /~ = 0.89, the largest Lyapunov exponent becomes positive and the situation turns out to be unstable. Confirmation comes from a functional analysis of the stable and unstable manifolds which touch at ~ = 0.89.

Symmetry

In this paper, we study the complex dynamic characteristics of a simple nonlinear logistic map. The map contains two parameters that have complex influences on the map’s dynamics. Assuming different values for those parameters gives rise to strange attractors with fractal dimensions. Furthermore, some of these chaotic attractors have heteroclinic cycles due to saddle-fixed points. The basins of attraction for some periodic cycles in the phase plane are divided into three regions of rank-1 preimages. We analyze those regions and show that the map is noninvertible and includes Z0,Z2 and Z4 regions.

2014

This paper serves as an introduction to the analysis of chaotic systems, with techniques being developed by working through two famous examples. The rst is the logistic map, a rst-order discrete dynamical system, and the second is the Lorenz system, a three-dimensional system of dierential equations.

Communications in Nonlinear Science and Numerical Simulation, 2012

An algebraic approach based on the rank of a sequence is proposed for the exploration of the onset of chaos in discrete nonlinear dynamical systems. The rank of the partial solution is identified and a special technique based on Hankel matrices is used to decompose the solution into algebraic primitives comprising roots of the modified characteristic equation. The distribution of roots describes the dynamical complexity of a solution and is used to explore properties of the nonlinear system and the onset of chaos. (M. Ragulskis), zenonas.navickas@ktu.lt (Z. Navickas), rita.palivonaite@ktu.lt (R. Palivonaite), mantas. landauskas@ktu.lt (M. Landauskas).

International Journal of Bifurcation and Chaos, 2005

The standard map x′ = x + y′, y′ = y + (K/2π) sin (2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ΔK, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theore...

International Journal of Computer Applications

In this paper the study of rigorous basic dynamical facts on bifurcation and chaos for discrete models in time dynamics and introduce a generalized logistic map and its dynamical behavior with tent and Henon Map has recognized.Different discrete curves have been developed and more general biological logistic curve are studied. Review and compare several such maps and analysis properties of those maps on the applications of bifurcation and chaos. Discuss the concept of chaos and bifurcations in the discrete time dynamical tent maps and generalized logistic growth models as time dynamical attractor.