Chaotifying Stable Linear Complex Networks via Single Pinning Impulsive Strategy

2019 ◽  
Vol 29 (02) ◽  
pp. 1950024 ◽  
Author(s):  
Guang Ling ◽  
Zhi-Hong Guan ◽  
Jie Chen ◽  
Qiang Lai
Keyword(s):  
Lyapunov Exponent ◽  
Complex Networks ◽  
Impulsive Control ◽  
Largest Lyapunov Exponent ◽  
Continuous Systems ◽  
Linear Complex ◽  
Numerical Examples ◽  
Linear Networks ◽  
Pinning Impulsive Control ◽  
Stable Linear

Chaotification, aiming to make an original nonchaotic system chaotic, or enhancing an existing chaos, has attracted much attention recently. This paper investigates chaotification problem of stable linear complex networks, both discrete and continuous systems, via single pinning impulsive control. Under this scheme, the boundedness of these linear networks can be guaranteed, and the largest Lyapunov exponent of these controlled linear complex networks can be calculated in detail by theoretical analysis. Finally, two numerical examples are given to demonstrate the effectiveness of this strategy.

ON THE LOCAL STOCHASTIC STABILITY OF NONLINEAR COMPLEX NETWORKS

2010 ◽  
Vol 20 (01) ◽  
pp. 177-184 ◽  
Author(s):  
ZHI-LONG HUANG ◽  
ZHOU YAN ◽  
XIAO-LING JIN ◽  
GUANRONG CHEN
Keyword(s):  
Lyapunov Exponent ◽  
Theoretical Analysis ◽  
Complex Networks ◽  
Stochastic Stability ◽  
Largest Lyapunov Exponent ◽  
Stochastic Perturbations ◽  
Approximate Analytical Solutions ◽  
Coupling Strengths ◽  
The Largest Lyapunov Exponent ◽  
The One

The local stochastic stability of nonlinear complex networks is studied, subject to stochastic perturbations to the coupling strengths and stochastic parametric excitations to the nodes. The complex network is first linearized at its trivial solution and then the linearized network is reduced to N independent subsystems by using a suitable linear transformation, where N is the size of the network. The largest Lyapunov exponent for each subsystem is then calculated and all the approximate analytical solutions are evaluated for some specific cases. It is found that the largest Lyapunov exponent among all subsystems is the one associated with the subsystem that has the largest or the smallest eigenvalue of the configuration matrix of the network. Finally, an example is given to demonstrate the validity and accuracy of the theoretical analysis.

scholarly journals Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method

Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7197
Author(s):  
Artur Dabrowski ◽  
Tomasz Sagan ◽  
Volodymyr Denysenko ◽  
Marek Balcerzak ◽  
Sandra Zarychta ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Algorithms ◽  
Alternative Methods ◽  
Largest Lyapunov Exponent ◽  
Continuous Systems ◽  
Science And Engineering ◽  
Stability Of Dynamical Systems ◽  
The Largest Lyapunov Exponent ◽  
The Stability ◽  
Mathematical Operations

Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems’ stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.

Synchronization for a Class of Fractional-order Linear Complex Networks via Impulsive Control

2018 ◽  
Vol 16 (6) ◽  
pp. 2839-2844 ◽  
Author(s):  
Na Liu ◽  
Jie Fang ◽  
Wei Deng ◽  
Zhen-Jun Wu ◽  
Guo-Qiang Ding
Keyword(s):  
Complex Networks ◽  
Fractional Order ◽  
Impulsive Control ◽  
Linear Complex ◽  
Order Linear

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

The Lyapunov Exponent Variance of an Electronic Manufacturing Enterprise’s Daily Qualified Rate Time Series by Improved Small Data Sets Approach

2012 ◽  
Vol 197 ◽  
pp. 271-277
Author(s):  
Zhu Ping Gong
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Time Series ◽  
Quality System ◽  
Quality Level ◽  
Largest Lyapunov Exponent ◽  
Small Data ◽  
Data Set ◽  
Electronic Manufacturing ◽  
Small Data Set

Small data set approach is used for the estimation of Largest Lyapunov Exponent (LLE). Primarily, the mean period drawback of Small data set was corrected. On this base, the LLEs of daily qualified rate time series of HZ, an electronic manufacturing enterprise, were estimated and all positive LLEs were taken which indicate that this time series is a chaotic time series and the corresponding produce process is a chaotic process. The variance of the LLEs revealed the struggle between the divergence nature of quality system and quality control effort. LLEs showed sharp increase in getting worse quality level coincide with the company shutdown. HZ’s daily qualified rate, a chaotic time series, shows us the predictable nature of quality system in a short-run.

Study of Rehabilitation of Injured Knee Joint Applying Chaotic Theory in Human Body Motion

2007 ◽  
Vol 342-343 ◽  
pp. 581-584
Author(s):  
Byung Young Moon ◽  
Kwon Son ◽  
Jung Hong Park
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Angular Displacement ◽  
Three Dimensional ◽  
Gait Pattern ◽  
Body Motion ◽  
Largest Lyapunov Exponent ◽  
Chaos Analysis ◽  
Nonlinear Technique ◽  
Injured Knee

Gait analysis is essential to identify accurate cause and knee condition from patients who display abnormal walking. Traditional linear tools can, however, mask the true structure of motor variability, since biomechanical data from a few strides during the gait have limitation to understanding the system. Therefore, it is necessary to propose a more precise dynamic method. The chaos analysis, a nonlinear technique, focuses on understanding how variations in the gait pattern change over time. Healthy eight subjects walked on a treadmill for 100 seconds at 60 Hz. Three dimensional walking kinematic data were obtained using two cameras and KWON3D motion analyzer. The largest Lyapunov exponent from the measured knee angular displacement time series was calculated to quantify local stability. This study quantified the variability present in time series generated from gait parameter via chaos analysis. Gait pattern is found to be chaotic. The proposed Lyapunov exponent can be used in rehabilitation and diagnosis of recoverable patients.

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