scholarly journals On the Complex and Chaotic Dynamics of Standard Logistic Sine Square Map

2021 ◽  
Vol 29 (3) ◽  
pp. 201-227
Author(s):  
Sudesh Kumari ◽  
Renu Chugh ◽  
Radu Miculescu
Keyword(s):  
Chaotic Dynamics ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Nonlinear Dynamical System ◽  
Series Representation ◽  
Nonlinear Dynamical ◽  
Stability Performance ◽  
Set Up ◽  
The Stability ◽  
One Dimensional Maps

Abstract In this article, we set up a new nonlinear dynamical system which is derived by combining logistic map and sine square map in Mann orbit (a two step feedback process) for ameliorating the stability performance of chaotic system and name it Standard Logistic Sine Square Map (SLSSM). The purpose of this paper is to study the whole dynamical behavior of the proposed map (SLSSM) through various introduced aspects consisting fixed point and stability analysis, time series representation, bifurcation diagram and Lyapunov exponent. Moreover, we show that our map is significantly superior than existing other one dimensional maps. We investigate that the chaotic and complex behavior of SLSSM can be controlled by selecting control parameters carefully. Also, the range of convergence and stability can be made to increase drastically. This new system (SLSSM) might be used to achieve better results in cryptography and to study chaos synchronization.

STABILIZATION OF CHAOTIC DYNAMICS OF ONE-DIMENSIONAL MAPS BY A CYCLIC PARAMETRIC TRANSFORMATION

1996 ◽  
Vol 06 (04) ◽  
pp. 725-735 ◽  
Author(s):  
ALEXANDER Yu. LOSKUTOV ◽  
VALERY M. TERESHKO ◽  
KONSTANTIN A. VASILIEV
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Exponential Map ◽  
One Dimensional ◽  
Stable Periodic ◽  
One Dimensional Maps ◽  
Parameter Values ◽  
Parametric Transformation

We consider one-dimensional maps, the logistic map and an exponential map, in those sets of parameter values which correspond to their chaotic dynamics. It is proven that such dynamics may be stabilized by a certain cyclic parametric transformation operating strictly within the chaotic set. The stabilization is a result of the creation of stable periodic orbits in the initially chaotic maps. The period of these stable orbits is a multiple of the period of the cyclic transformation. It is shown that stabilized behavior cannot be destroyed by a weak noise smearing of the required parameter values. The regions where the behavior stabilization takes place are numerically estimated. Periods of the created stabile periodic orbits are calculated.

Is There a Relation Between Synchronization Stability and Bifurcation Type?

2020 ◽  
Vol 30 (08) ◽  
pp. 2050123
Author(s):  
Zahra Faghani ◽  
Zhen Wang ◽  
Fatemeh Parastesh ◽  
Sajad Jafari ◽  
Matjaž Perc
Keyword(s):  
Complex Networks ◽  
Nonlinear Dynamical Systems ◽  
General Relation ◽  
Dynamical Behavior ◽  
Stability Function ◽  
Nonlinear Dynamical ◽  
Practical Applications ◽  
Stability Functions ◽  
Stability And Bifurcation ◽  
The Stability

Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

scholarly journals Stability and Bifurcation Analysis of a Three-Dimensional Recurrent Neural Network with Time Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Yingguo Li
Keyword(s):  
Neural Network ◽  
Time Delay ◽  
Recurrent Neural Network ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Nonlinear Dynamical ◽  
Bifurcation Direction ◽  
The Stability ◽  
Manifold Theory

We consider the nonlinear dynamical behavior of a three-dimensional recurrent neural network with time delay. By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Applying the nor- mal form method and center manifold theory, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution. Some numerical examples are also presented to verify the theoretical analysis.

Stabilization of Periodic Flap-Lag Dynamics in Rotor Blades

1995 ◽  
Author(s):  
John M. Schmitt ◽  
Philip V. Bayly ◽  
David A. Peters
Keyword(s):  
Nonlinear Dynamical System ◽  
Rotor Blades ◽  
Discrete Control ◽  
System Control ◽  
Transient Effects ◽  
Control Effort ◽  
Nonlinear Dynamical ◽  
Helicopter Rotor Blades ◽  
Stable Periodic ◽  
The Stability

Abstract A method to increase the stability of periodic flap-lag dynamics in helicopter rotor blades is investigated. Instability in the flap-lag dynamics of stiff-in-plane rotors can occur as forward flight speed is increased, or if significant pitch-lag coupling is present. A method originally developed to control chaos can be applied to stabilize unstable or weakly stable periodic behavior. Stabilization is achieved using small perturbations of the mean blade pitch angle. The approach, which will be referred to as periodic active control (PAC), consists of applying discrete control to the Poincaré map associated with the nonlinear dynamical system. Control effort is applied efficiently, since it does not change, but only stabilizes underlying periodic motion. Stabilization can lead to higher safe speeds, decreased transient effects, and simplified designs in helicopters.

A SYNTHESIS OF INVARIANT STRUCTURES WITH MEMORY IN TWO-DIMENSIONAL SPACE

2000 ◽  
Vol 11 (05) ◽  
pp. 853-864 ◽  
Author(s):  
ALEXANDER YUDASHKIN
Keyword(s):  
Potential Function ◽  
Dimensional Space ◽  
Nonlinear Dynamical System ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Nonlinear Dynamical ◽  
Self Assembling ◽  
Nonlinear Potential ◽  
The Stability ◽  
High Level

A problem of self-organized structure storage and retrieving is considered. The novel model of nonlinear dynamical system for multi-unit structures memorizing is proposed and its properties are investigated in this paper. The multi-unit system consists of N points in the two-dimensional Euclidean space, and its dynamics is defined by a potential function, that is translation and rotation invariant relatively points coordinates. The nonlinear potential function allows to compose attractors corresponding to the proper configurations of points, while these configurations are memorized by the system. Any initial form flows to one of the attractors independently from possible rotations and spatial shifts. The system can memorize and successfully restore up to N-2 required configurations. The characteristics of structure retrieving in the presence of beginning form distortion are considered, and the stability of method is shown even in the case of high level noise. The proposed approach could be helpful to design physical, technical and informational objects with the desired self-assembling properties.

INFLUENCE OF ENVIRONMENTAL NOISE ON THE DYNAMICS OF A REALISTIC ECOLOGICAL MODEL

2007 ◽  
Vol 07 (01) ◽  
pp. L61-L77 ◽  
Author(s):  
R. K. UPADHYAY ◽  
A. MUKHOPADHYAY ◽  
S. R. K. IYENGAR
Keyword(s):  
Ecological Model ◽  
Dynamical Behavior ◽  
Nonlinear Dynamical System ◽  
Growth Parameter ◽  
Environmental Noise ◽  
Model System ◽  
Dynamical System Theory ◽  
Chain Model ◽  
Nonlinear Dynamical ◽  
Food Chain Model

The present paper investigates the influence of environmental noise on a fairly realistic three-species food chain model based on the Leslie-Gower scheme. The self- growth parameter for the prey species is assumed to be perturbed by white noise characterized by a Gaussian distribution with mean zero and unit spectral density. Using tools borrowed from the nonlinear dynamical system theory, we study the dynamical behavior of the model system. The behavior of the stochastic system (perturbed one) is studied and the fluctuations in the populations are measured both analytically (for the linearized system) and numerically by computer simulation. Varying one of the control parameters in its range, while keeping all the others constant, we monitor the changes in the dynamical behavior of the model system, thereby fixing the regimes in which the system exhibits chaotic dynamics. Our study suggests that the trophic level (top, middle or bottom) at which a population is positioned, the amplitude of environmental noise and the population's susceptibility to environmental noise play key roles in how noise affects the population dynamics.

ENTROPY AND EXTENDED MEMORY IN DISCRETE CHAOTIC DYNAMICS

1996 ◽  
Vol 06 (04) ◽  
pp. 611-625 ◽  
Author(s):  
WERNER EBELING ◽  
JAN FREUND ◽  
KATJA RATEITSCHAK
Keyword(s):  
Phase Space ◽  
Symbolic Dynamics ◽  
Long Memory ◽  
Chaotic Dynamics ◽  
Logistic Map ◽  
Tent Map ◽  
One Dimensional ◽  
Dynamical Description ◽  
One Dimensional Maps ◽  
Special Case

We investigate simple one-dimensional maps which allow for exact solutions of their related statistical properties. In addition to the originally refined dynamical description a coarsegrained level of description based on certain partitions of the phase space is selected. The deterministic micropscopic dynamics is shifted to a stochastic symbolic dynamics. The higher order entropies are studied for the logistic map, the tent map, and the shark fin map. Markov sources of any prescribed order are constructed explicitly. In a special case, long memory tails are observed. Systems of this type may be of interest for modelling naturally ocurring phenomena.

scholarly journals Standing Swells Surveyed Showing Surprisingly Stable Solutions for the Lorenz '96 Model

2014 ◽  
Vol 24 (10) ◽  
pp. 1430027 ◽  
Author(s):  
Morgan R. Frank ◽  
Lewis Mitchell ◽  
Peter Sheridan Dodds ◽  
Christopher M. Danforth
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Standing Waves ◽  
Nonlinear Dynamical ◽  
Hidden Order ◽  
Stable Solutions ◽  
Lorenz 96

The Lorenz '96 model is an adjustable dimension system of ODEs exhibiting chaotic behavior representative of the dynamics observed in the Earth's atmosphere. In the present study, we characterize statistical properties of the chaotic dynamics while varying the degrees of freedom and the forcing. Tuning the dimensionality of the system, we find regions of parameter space with surprising stability in the form of standing waves traveling amongst the slow oscillators. The boundaries of these stable regions fluctuate regularly with the number of slow oscillators. These results demonstrate hidden order in the Lorenz '96 system, strengthening the evidence for its role as a hallmark representative of nonlinear dynamical behavior.

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 STABILITY AND STABILIZATION OF NONLINEAR DYNAMICAL SYSTEMS

2017 ◽  
Vol 20 (1) ◽  
pp. 61-70
Author(s):  
P. Sattayatham ◽  
R. Saelim ◽  
S. Sujitjorn
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Feedback Controllers ◽  
Nonlinear Dynamical ◽  
System A ◽  
Continuous Feedback ◽  
Stability And Stabilization ◽  
The Stability

Exponential and asymptotic stability for a class of nonlinear dynamical systems with uncertainties is investigated.  Based on the stability of the nominal system, a class of bounded continuous feedback controllers is constructed.  By such a class of controllers, the results guarantee exponential and asymptotic stability of uncertain nonlinear dynamical system.  A numerical example is also given to demonstrate the use of the main result.

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