A Novel Fixed Point Feedback Approach Studying the Dynamical Behaviors of Standard Logistic Map

2019 ◽  
Vol 29 (01) ◽  
pp. 1950010 ◽  
Author(s):  
Ashish ◽  
Jinde Cao
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Traffic Control ◽  
Logistic Map ◽  
Secure Communications ◽  
Iterative Technique ◽  
Control Parameters ◽  
Dynamical Behaviors ◽  
One Step ◽  
Dynamical Properties

The standard logistic map is one of the oldest and simplest systems which has found a celebrated place in the dynamical systems and in different vital applications of science like image encryption in cryptography, secure communications and traffic control models. Generally, the dynamical systems are characterized by one or more control parameters that determine the dynamical behaviors of the system. Traditionally, the discrete logistic map allows only one parameter [Formula: see text] to determine its complete behavior. This study takes one step forward, using the superior fixed point iterative technique to study the dynamical properties of the discrete logistic map. The proposed technique provides an extra degree of freedom on control parameters that renders superior dynamical properties and may increase the performance of many applications. Analytical analysis as well as numerical simulations are presented to show the effectiveness, flexibility and efficiency of new method.

scholarly journals Bifurcation Analysis of Logistic Map using Four Step Feedback Procedure

2019 ◽  
Vol 9 (1) ◽  
pp. 704-707
Keyword(s):  
Fixed Point ◽  
Chaos Theory ◽  
Analytical Study ◽  
Logistic Map ◽  
Growth Parameter ◽  
Control Parameters ◽  
Chaotic Dynamical System ◽  
Point Analysis ◽  
Dynamical Properties ◽  
Superior Orbit

The logistic map occupies a renowned place in the dynamics of chaos theory and in diverse areas of science. Picard orbit and Superior orbit (Mann orbit) have been used to control this discrete chaotic dynamical system. In this article, we further extend the analytical study of logistic map using a four step feedback procedure (SP orbit). The dynamical properties such as fixed point, range of convergence and stability, periodicity and chaos of the logistic map have been investigated. These properties are illustrated experimentally by adopting dynamical techniques like fixed point analysis and bifurcation plot. Using this approach, one can easily control the chaotic system and make the system stable for higher values of population growth parameter r by selecting the control parameters carefully.

DIFFUSION, INTERMITTENCY, AND NOISE-SUSTAINED METASTABLE CHAOS IN THE LORENZ EQUATIONS: EFFECTS OF NOISE ON MULTISTABILITY

2008 ◽  
Vol 18 (06) ◽  
pp. 1749-1758 ◽  
Author(s):  
WEN-WEN TUNG ◽  
JING HU ◽  
JIANBO GAO ◽  
VINCENT A. BILLOCK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Stable Fixed Point ◽  
Lorenz Equations ◽  
Periodic Forcing ◽  
Parameter Region ◽  
Interesting Phenomenon ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors

Multistability is an interesting phenomenon of nonlinear dynamical systems. To gain insights into the effects of noise on multistability, we consider the parameter region of the Lorenz equations that admits two stable fixed point attractors, two unstable periodic solutions, and a metastable chaotic "attractor". Depending on the values of the parameters, we observe and characterize three interesting dynamical behaviors: (i) noise induces oscillatory motions with a well-defined period, a phenomenon similar to stochastic resonance but without a weak periodic forcing; (ii) noise annihilates the two stable fixed point solutions, leaving the originally transient metastable chaos the only observable; and (iii) noise induces hopping between one of the fixed point solutions and the metastable chaos, a three-state intermittency phenomenon.

scholarly journals Repelling invariant curves in planar discrete dynamical systems

1994 ◽  
Vol 49 (3) ◽  
pp. 469-481 ◽  
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Continuous Dependence ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Invariant Curves ◽  
Dynamical Properties ◽  
Attracting Fixed Point

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

scholarly journals On a method of estimating chaos control parameters from time series

2018 ◽  
Vol XXI (2) ◽  
pp. 19-28
Author(s):  
Deleanu D.
Keyword(s):  
Time Series ◽  
Fixed Point ◽  
Chaos Control ◽  
Logistic Map ◽  
Control Parameters ◽  
Simple Method ◽  
Short Time Series ◽  
Measured Time ◽  
One Dimensional Maps ◽  
Short Time

The algorithm of Ott, Grebogi and Yorke (OGY) is recognized for its efficiency in controlling chaotic dynamical systems, even if the system’s equations are not known and the input data are provided by measured time series in experimental settings. Recently, Santos and Graves (SG) proposed a simple method for estimating the chaos control parameters required by OGY algorithm and applied it to the logistic map. Using only two time series of 100 values, they obtained approximate results for the fixed point case within 2 % of the analytical ones. Although the outputs refer only to a particular case, their conclusion seems to be that the method works as well as in general. To check this statement, we performed a large amount of numerical simulations on different one – dimensional maps. With slight different nuances, our findings were the same so we only presented in the paper the logistic map case. We have noticed that the use of only two short time series implies high risks in a reasonable estimate of the location of the fixed points and of the two control parameters (especially of the second). For large enough number of time series (three or five sets of 400 values each, in the paper) the results provided by numerical simulation approximated the theoretical ones within the limit of a few percent at most. The role played by each method parameter, as the radius for a close encounter of the fixed point or the number of the series and their lengths, is also investigated.

Noninvertible Piecewise Linear Maps Applied to Chaos Synchronization and Secure Communications

1997 ◽  
Vol 07 (07) ◽  
pp. 1617-1634 ◽  
Author(s):  
G. Millerioux ◽  
C. Mira
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Piecewise Linear ◽  
A Priori ◽  
Secure Communications ◽  
Technological Parameters ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Necessary And Sufficient

Recently, it was demonstrated that two chaotic dynamical systems can synchronize each other, leading to interesting applications as secure communications. We propose in this paper a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters make them good candidates. The classical concept of controllability in the control theory is presented and used in order to choose and predict the number of appropriate variables to be transmitted for synchronization. A necessary and sufficient condition of chaotic synchronization is established without computing numerical quantities, introducing a state affinity structure of chaotic systems which provides an a priori establishment of synchronization.

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