scholarly journals Nonlinear dynamics and complexity in the generalized Lorenz system

2018 ◽  
Vol 94 (4) ◽  
pp. 2957-2968 ◽  
Author(s):  
Wiesław M. Macek
Keyword(s):  
Nonlinear Dynamics ◽  
Lorenz System ◽  
Generalized Lorenz System

Hopf Bifurcations, Periodic Windows and Intermittency in the Generalized Lorenz Model

2019 ◽  
Vol 29 (14) ◽  
pp. 1930042
Author(s):  
Anna Wawrzaszek ◽  
Agata Krasińska
Keyword(s):  
Lorenz System ◽  
Dimensional System ◽  
Type I ◽  
Control Parameters ◽  
System Behavior ◽  
Lorenz Model ◽  
Generalized Lorenz System ◽  
Subcritical Hopf Bifurcation ◽  
Bifurcation Structures ◽  
The Magnetic Field

In the present study, we analyze the dynamics of a four-dimensional generalized Lorenz system with one variable describing the profile of the magnetic field induced in a convected magnetized fluid. In particular, we identify the subcritical Hopf bifurcation, at which the dimension of the unstable manifold is increased or reduced by two. Moreover, the new four-dimensional system behavior depending on the control parameters is considered and bidirectional bifurcation structures are revealed. The results show the existence of several windows of nonchaotic variation (windows of order), in particular period-3 windows at the edge of which type I intermittency is observed.

scholarly journals Statistical Modeling in Nonlinear Systems

2005 ◽  
Vol 18 (16) ◽  
pp. 3388-3399 ◽  
Author(s):  
Edward P. Campbell
Keyword(s):  
Nonlinear Dynamics ◽  
Statistical Models ◽  
Lorenz System ◽  
Prediction Models ◽  
Climate Data ◽  
Bayesian Hierarchical ◽  
Alternative Approaches ◽  
The Lorenz System ◽  
Bayesian Hierarchical Methods ◽  
The Impact

Abstract The use of linear statistical methods in building climate prediction models is examined, particularly the use of anomalies. The author’s perspective is that the climate system is a nonlinear interacting system, so the impact of modeling using anomalies rather than observed data directly is considered. With reference to the Lorenz system and a simple model for regime dependence, it is shown that anomalies impair our ability to reconstruct nonlinear dynamics. Some alternative approaches in the literature that offer an attractive way forward are explored, focusing on Bayesian hierarchical methods to construct so-called physical–statistical models. The author’s view is that anomalies should be reserved in most cases as a tool for enhancing graphical representations of climate data. The exceptions are when the implicit assumptions underlying the use of anomalies are met or when an anomaly representation is physically motivated.

scholarly journals On Darboux polynomials and rational first integrals of the generalized Lorenz system

2014 ◽  
Vol 138 (3) ◽  
pp. 317-322 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Lorenz System ◽  
First Integrals ◽  
Darboux Polynomials ◽  
Generalized Lorenz System

ON A GENERALIZED LORENZ CANONICAL FORM OF CHAOTIC SYSTEMS

2002 ◽  
Vol 12 (08) ◽  
pp. 1789-1812 ◽  
Author(s):  
SERGEJ ČELIKOVSKÝ ◽  
GUANRONG CHEN
Keyword(s):  
Large Class ◽  
Canonical Form ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chen System ◽  
Scalar Parameter ◽  
Generalized Lorenz System ◽  
Special Cases ◽  
The One ◽  
Generalized Lorenz Canonical Form

This paper shows that a large class of systems, introduced in [Čelikovský & Vaněček, 1994; Vaněček & Čelikovský, 1996] as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [Čelikovský & Vaněček, 1994; Vaněček & Čelikovský, 1996], and also covers the so-called Chen system, recently introduced in [Chen & Ueta, 1999; Ueta & Chen, 2000].Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some other closely related topics are also studied and discussed in the paper.

Dynamical behavior of a generalized Lorenz system model and its simulation

Complexity ◽  
2015 ◽  
Vol 21 (S1) ◽  
pp. 99-105 ◽  
Author(s):  
Fuchen Zhang ◽  
Xiaofeng Liao ◽  
Guangyun Zhang
Keyword(s):  
Lorenz System ◽  
Dynamical Behavior ◽  
System Model ◽  
Generalized Lorenz System

Message Embedded Chaotic Masking Synchronization Scheme Based on the Generalized Lorenz System and Its Security Analysis

2016 ◽  
Vol 26 (08) ◽  
pp. 1650140 ◽  
Author(s):  
Sergej Čelikovský ◽  
Volodymyr Lynnyk
Keyword(s):  
Numerical Experiments ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Bounded Function ◽  
Security Analysis ◽  
Encryption Scheme ◽  
The Novel ◽  
Generalized Lorenz System ◽  
Synchronization Signal ◽  
Synchronization Scheme

This paper focuses on the design of the novel chaotic masking scheme via message embedded synchronization. A general class of the systems allowing the message embedded synchronization is presented here, moreover, it is shown that the generalized Lorenz system belongs to this class. Furthermore, the secure encryption scheme based on the message embedded synchronization is proposed. This scheme injects the embedded message into the dynamics of the transmitter as well, ensuring thereby synchronization with theoretically zero synchronization error. To ensure the security, the embedded message is a sum of the message and arbitrary bounded function of the internal transmitter states that is independent of the scalar synchronization signal. The hexadecimal alphabet will be used to form a ciphertext making chaotic dynamics of the transmitter even more complicated in comparison with the transmitter influenced just by the binary step-like function. All mentioned results and their security are tested and demonstrated by numerical experiments.

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