Generalized Lorenz System Research Articles

ULTIMATE BOUND ESTIMATION OF A CLASS OF HIGH DIMENSIONAL QUADRATIC AUTONOMOUS DYNAMICAL SYSTEMS

This paper aims to propose a unified approach for the ultimate bound estimation of a class of High Dimensional Quadratic Autonomous Dynamical Systems (HDQADS). Using the proposed method and the optimization idea, a sufficient condition is then given for estimating the ultimate bounds of a class of HDQADS. To validate the above sufficient condition, this paper further investigates the ultimate bound estimation of a hyperchaotic system, a 6D and a 9D chaotic system, separately. Moreover, the ultimate bounds for a general Lorenz system, a low-order atmospheric circulation model, and a new 3D chaotic system are also discussed in detail. In particular, it should be pointed out that a unified and accurate ultimate bound estimation is attained for the generalized Lorenz system and it includes several well-known results as its special cases. Some numerical simulations are also given to verify and visualize the corresponding theoretical results.

Generating Grid Multiwing Chaotic Attractors by Constructing Heteroclinic Loops Into Switching Systems

Over the last two decades, multiwing chaos generation has seen promising advances and becomes an active research field today. It is well known that there is a gap between theoretical design and engineering applications in multiwing chaos generation. That is, most theoretical designs of multiwing chaotic attractors with mathematical proofs or numerical verification have rather complex expressions; however, most engineering applications of multiwing chaotic attractors without theoretical supports have simple expressions. To bridge the gap between theoretical design and engineering applications in multiwing chaos generation, this paper introduces a novel practical approach for generating grid multiwing butterfly chaotic attractors from the multipiecewise Lu system by constructing heteroclinic loops. It should be particularly pointed out that the designed multiwing chaotic attractors exhibit typical heteroclinic chaos from the heteroclinic Shil'nikov theorem and also have clear potential engineering applications. The proposed method can be easily extended to the generalized Lorenz system family.

Optimal Fuzzy Synchronization Of Generalized Lorenz Chaotic Systems

In this article two identical generalized Lorenz systems have been synchronized by a fuzzy controller based on mamdani approach and stability of the proposed scheme has been established by the Lyapunov stability theorem. Controller parameters have been optimized by the genetic algorithm. Effectiveness of proposed method has been demonstrated through computer simulation.

The Logistic-Unified hybrid chaotic system

A Logistic-Unified hybrid chaotic system is generated. In this system, following the random changing of the state variable values of the Logistic map, the parameter values of the Unified system can be modulated randomly, and the Logistic-Unified hybrid chaotic system can be switched between the generalized Lorenz system, Lü system and generalized Chen system randomly. The extremely complicated chaotic signal is generated via the Logistic-Unified hybrid chaotic system. The Logistic-Unified hybrid chaotic system is realized based on digital signal processing (DSP). Hardware experiments and software simulation are completely consistent, and the results demonstrate the validily of the theoretical analysis.

Bidirectionally coupled synchronization of the generalized Lorenz systems

Wu, Chen, and Cai (2007) investigated chaos synchronization of two identical generalized Lorenz systems unidirectionally coupled by a linear state error feedback controller. However, bidirectional coupling in real life such as complex dynamical networks is more universal. This paper provides a unified method for analyzing chaos synchronization of two bidirectionally coupled generalized Lorenz systems. Some sufficient synchronization conditions for some special coupling matrices (diagonal matrices, so-called dislocated coupling matrices, and so on) are derived through rigorously mathematical theory. In particular, for the classical Lorenz system, the authors obtain synchronization criteria which only depend upon its parameters using new estimation of the ultimate bounds of Lorenz system (Chaos, Solitons, and Fractals, 2005). The criteria are then applied to four typical generalized Lorenz systems in the numerical simulations for verification.

A Combined Input-state Feedback Linearization Scheme and Independent Component Analysis Filter for the Control of Chaotic Systems with Significant Measurement Noise

Chaotic motion is an undesirable phenomenon in many engineering applications since it leads to a significant degradation of the system performance and restricts the feasible operational range. Therefore, the problem of controlling or suppressing chaos has attracted considerable attention in the literature. However, most chaos control schemes utilize a state feedback signal, and are therefore sensitive to measurement noise. Therefore, a requirement exists for filtering systems capable of separating the measurement noise from the chaotic signal in order to improve the performance of the controlled system. However, the chaotic signal and the measurement noise are both broadband signals, and thus the measurement noise can not be filtered using a low-pass filter since this causes a distortion of the chaotic signal. Accordingly, this paper presents a control scheme for chaotic dynamic systems with significant measurement noise featuring an input-state linearization scheme and a filter based upon an independent component analysis algorithm. The feasibility and effectiveness of the proposed approach are demonstrated by way of numerical simulations using a general Lorenz system for illustration purposes.

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.

Bifurcation analysis of some forced Lu systems

Bifurcation behaviour of a forced Lu system is analyzed as the system parameter c and a forcing parameter F are varied. The Lu system belongs to a family of generalized Lorenz system. Members of this family are known to exhibit different types of chaotic attractors. Some of these attractors have been named Lorenz type L, Lu or Transition type T, Chen type T and Transverse 8 Type S. These different types of chaotic attractors are visually distinct when the parameters are widely separated. However, there is a need for identifying the precise point where transition from one type of chaotic attractor to another takes place. We identified signatures in the return map, which could be used for determining the point of transition and classifying the different types of chaotic attractors. These signatures helped to identify the point in coordinate space associated with such transitions. We find that such transitions take place when a chaotic attractor comes very close to a one-dimensional manifold on which the time derivatives of two of the variables is zero. We also find that just before coming to this point in coordinate space associated with the transition, the trajectory had approached, very closely, the equilibrium point at the origin.

Generalized Lorenz Equation Derived from Thermal Convection of Viscoelastic Fluids in a Loop

A new generalized Lorenz system is presented based on the thermal convection of Oldroyd-B fluids in a circular loop. Two non-dimensional parameters De1 (a measure of the fluid relaxation) and De2 (a measure of the fluid retardation) appear in the equation. Then we study this generalized Lorenz equation numerically and find that the values of De1 and De2 can greatly influence the behavior of the solution. The fluid relaxation (De1) is found to precipitate the onset of periodic solution (limit cycle) in the system and impedes the onset of chaos while the fluid retardation (De2) tends to delay the onset of the periodic solution and precipitate the onset of chaos in the system.

Robust Adaptive Control of a Class of Nonlinear Systems and Its Applications

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper addresses a global robust adaptive control problem for a class of uncertain nonlinear systems by output feedback control. The problem will be solved by the internal model design method. As our problem formulation includes the chaotic control and synchronization problem of some typical nonlinear systems as special cases, a direct application of our main result will lead to the solution of some interesting control problems such as the global disturbance rejection of the FitzHugh–Nagumo system and the robust output synchronization of the generalized Lorenz system and the harmonic system. </para>

Adaptive fuzzy bilinear feedback control design for synchronization of TS fuzzy bilinear generalized Lorenz system with uncertain parameters

In this Letter, we propose an adaptive fuzzy bilinear feedback control (FBFC) design for synchronization of Takagi–Sugeno (TS) fuzzy bilinear generalized Lorenz system (FBGLS) with uncertain parameters. The generalized Lorenz system (GLS) can be described to TS FBGLS. We design an adaptive synchronization scheme of the response system based on TS FBGLS, feedback control scheme and Lyapunov theory. Lyapunov theory is employed to guarantee the stability of error dynamic system and to derive the adaptive laws to estimate unknown parameters. Numerical example is given to demonstrate the validity of our proposed adaptive FBFC approach with comparative results for synchronization.

Spectrum Analysis and Circuit Implementation of a New 3D Chaotic System with Novel Chaotic Attractors

The new autonomous system with only three equilibrium points is introduced. This system does not belong to the generalized Lorenz systems. The novel attractors are observed over a large range of parameters, which have rarely been reported in previous work. As an important component in chaotic signal generators, a physical circuit has been designed. The experimental results are in agreement with numerical simulations. More significantly, spectral analysis shows that the system has an extremely broad frequency spectral bandwidth in 0–131.6 Hz, without investigating any possible electronic techniques, which is more desirable for secure communications.

Estimating the bound for the generalized Lorenz system

A chaotic system is bounded, and its trajectory is confined to a certain region which is called the chaotic attractor. No matter how unstable the interior of the system is, the trajectory never exceeds the chaotic attractor. In the present paper, the sphere bound of the generalized Lorenz system is given, based on the Lyapunov function and the Lagrange multiplier method. Furthermore, we show the actual parameters and perform numerical simulations.

A new chaotic attractor from general Lorenz system family and its electronic experimental implementation

This article introduces a novel three-dimensional continuous autonomous chaotic system with six terms and two quadratic nonlinearities. The new system contains two variational parameters and exhibits Lorenz-like attractors in numerical simulations and experimental measurements. The basic dynamical properties of the new system are analyzed by means of equilibrium points, eigenvalue structures, and Lyapunov exponents. The new system examined in Matlab-Simulink^{\textregistered} and Orcad-PSpice^{\textregistered}. An electronic circuit realization of the proposed system is presented using analog electronic elements such as capacitors, resistors, operational amplifiers and multipliers. The behaviour of the realized system is evaluated with computer simulations.

Coexistence of anti-phase and complete synchronization in the generalized Lorenz system

This paper investigates a class of new synchronization phenomenon. Some control strategy is established to guarantee the coexistence of anti-phase and complete synchronization in the generalized Lorenz system. The efficiency of the control scheme is revealed by some illustrative simulations.

Adaptive fuzzy bilinear observer based synchronization design for generalized Lorenz system

This Letter proposes an adaptive fuzzy bilinear observer (FBO) based synchronization design for generalized Lorenz system (GLS). The GLS can be described to TS fuzzy bilinear generalized Lorenz model (FBGLM) with their states immeasurable and their parameters unknown. We design an adaptive FBO based on TS FBGLM for synchronization. Lyapunov theory is employed to guarantee the stability of error dynamic system via linear matrix equalities (LMIs) and to derive the adaptive laws to estimate unknown parameters. Numerical example is given to demonstrate the validity of our proposed adaptive FBO approach for synchronization.

Chaos in the Fractional Order Generalized Lorenz Canonical Form

A fractional-order generalized Lorenz system is constructed and numerically investigated. Chaotic behavior existing in the fractional-order generalized Lorenz system is found. The numerical simulations and interesting figures are performed.

GENERALIZED SYNCHRONIZATION OF DIFFERENT DIMENSIONAL CHAOTIC SYSTEMS BASED ON PARAMETER IDENTIFICATION

This paper investigates the generalized synchronization issue for two different dimensional chaotic systems with unknown parameters. Based on Lyapunov stability theory and adaptive control theory, an adaptive controller is derived to achieve the generalized synchronization whether the dimension of drive system is greater than the one of the response system or not. Meanwhile, corresponding parameter updating laws can be obtained so as to exactly identify uncertain parameters. This technique has been successfully applied to two examples, the generalized synchronization of hyperchaotic Rössler system and chaotic Lorenz system, chaotic Chen system and generalized Lorenz system. Numerical simulations are finally shown to illustrate the effectiveness of the proposed approach.

MODIFIED GENERALIZED LORENZ SYSTEM AND FOLDED CHAOTIC ATTRACTORS

A modified generalized Lorenz system in a canonical form extended from the generalized Lorenz system is proposed in this paper. This novel system has a folded factor and can display complex 2-scroll folded attractors and 1-scroll folded attractors at different parameter values. Three typical normal forms, called Lorenz-like, Chen-like and Lü-like chaotic system respectively, of three-dimensional quadratic autonomous chaotic systems are derived, and their dynamical behaviors are further investigated by employing Lyapunov exponent spectrum, bifurcation diagram, Poincaré mapping and phase portrait, etc. Of particular interest is the fact that the folded factor makes Chen-like and Lü-like chaotic systems exhibit complicated nonlinear dynamical phenomena.

Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation

The chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to the circuit unit, an electronic circuit is designed to realize a 3.8-order generalized Lorenz chaotic system. Furthermore, synchronization between two fractional-order systems is achieved by utilizing a single-variable feedback method. Circuit experiment simulation results verify the effectiveness of the proposed scheme.