On the generalized Lorenz canonical form

This short note is to briefly introduce the new notion of generalized Lorenz canonical form (GLCF), which contains the classical Lorenz system and the newly discovered Chen system as two extreme cases, along with infinitely many chaotic systems in between. It also points out that some recently reported chaotic systems are special cases of the GLCF.     

Synchronizing chaotic systems using control based on a special matrix structure and extending to fractional chaotic systems

This work presents a direct approach to design stabilizing controller based on a special matrix structure to synchronize chaotic systems and extends the approach to synchronize fractional chaotic systems. With this method, chaos synchronization is implemented in Lorenz chaotic systems with known parameters and the same to Lorenz chaotic systems with unknown parameters. Especially, fractional Lorenz chaotic system with unknown parameters is synchronized by fractional Chen chaotic system too. Numerical simulations confirm the effectiveness of the method proposed.     

Chaos control and synchronization for a special generalized Lorenz canonical system – The SM system

This paper presents some simple feedback control laws to study global stabilization and global synchronization for a special chaotic system described in the generalized Lorenz canonical form (GLCF) when τ = −1 (which, for convenience, we call Shimizu–Morioka system, or simply SM system). For an arbitrarily given equilibrium point, a simple feedback controller is designed to globally, exponentially stabilize the system, and reach globally exponent synchronization for two such systems. Based on the system’s coefficients and the structure of the system, simple feedback control laws and corresponding Lyapunov functions are constructed. Because all conditions are obtained explicitly in terms of algebraic expressions, they are easy to be implemented and applied to real problems. Numerical simulation results are presented to verify the theoretical predictions.     

通过同步实现从混沌到有序的转变

本文旨在研究连续的混沌系统是否存在“混沌＋混沌=有序”的现象．证明了两个双向耦合的连续混沌系统在一些情况下可产生有序的动力学行为．作为例子，通过选取适当的耦合参数使Lorenz系统以及Chen和Lee引入的混沌系统同步，进而对同步系统的动力学行为进行了理论分析和数值模拟．结果表明，逐渐改变参数，系统实现了从混沌到有序的过渡．     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

A universal unfolding of the Lorenz system

A universal unfolding of the Lorenz system is derived and studied in this paper. Both rigorous theoretical analysis and numerical simulations show that the Lorenz system, the Chen system, and the Lü system belong to the same universal unfolding. Therefore, they all have similar dynamical behaviors in the sense that if the Lorenz system has limit cycles produced from a Hopf bifurcation for a certain set of parameter values, then the other two systems also have limit cycles from the same set of parameter values; and if the Lorenz, Chen, and Lü systems are chaotic for some parameter values (for example, some typical parameter values), respectively, then the homotopic system for the Lorenz system and the Chen system, and the homotopic system for these three systems, are all chaotic within the entire domain of these homotopic parameters.     

Evolving chaos: Identifying new attractors of the generalised Lorenz family

In a recent paper, we presented an intelligent evolutionary search technique through genetic programming (GP) for finding new analytical expressions of nonlinear dynamical systems, similar to the classical Lorenz attractor's which also exhibit chaotic behaviour in the phase space. In this paper, we extend our previous finding to explore yet another gallery of new chaotic attractors which are derived from the original Lorenz system of equations. Compared to the previous exploration with sinusoidal type transcendental nonlinearity, here we focus on only cross-product and higher-power type nonlinearities in the three state equations. We here report over 150 different structures of chaotic attractors along with their one set of parameter values, phase space dynamics and the Largest Lyapunov Exponents (LLE). The expressions of these new Lorenz-like nonlinear dynamical systems have been automatically evolved through multi-gene genetic programming (MGGP). In the past two decades, there have been many claims of designing new chaotic attractors as an incremental extension of the Lorenz family. We provide here a large family of chaotic systems whose structure closely resemble the original Lorenz system but with drastically different phase space dynamics. This advances the state of the art knowledge of discovering new chaotic systems which can find application in many real-world problems. This work may also find its archival value in future in the domain of new chaotic system discovery.     

Function projective synchronization for fractional-order chaotic systems

This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between the fractional-order Lorenz systems with different orders, and achieve synchronization between the fractional-order Lorenz system and fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

On the study of globally exponentially attractive set of a general chaotic system

In this paper, we prove that there exists globally exponential attractive and positive invariant set for a general chaotic system, which does not belong to the known Lorenz system, or the Chen system, or the Lorenz family. We show that all the solution orbits of the chaotic system are ultimately bounded with exponential convergent rates and the convergent rates are explicitly estimated. The method given in this paper can be applied to study other chaotic systems.     

Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters

In this paper, we investigate the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results.     

Generating a new chaotic attractor by feedback controlling method

A new chaotic system is found by feedback controlling method in this paper. According to the definition of the generalized Lorenz system, the new chaotic system does not belong to generalized Lorenz systems. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map. The particular interest is that this novel system can generate two one‐scroll and one two‐scroll chaotic attractors with the variation of a single parameter. The obtained results show clearly that the system is a new chaotic system and deserves a further detailed investigation. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability analysis of impulsive control systems

This paper studies impulsive control systems. Several stability criteria are established by employing the method of Lyapunov functions. These criteria may be used for impulsive feedback control design. As an application, impulsive control of the Lorenz chaotic system is discussed. Numerical experiments are carried out for the control of the Lorenz system. It is shown that small and frequent impulses need to be used in order to stabilize the Lorenz system.     

Generalized synchronization of continuous chaotic system

Sufficient condition for the generalized synchronization of continuous chaotic system with a kind of nonlinear transformation is derived. The method is illustrated by applications to Lorenz and Duffing chaotic systems and the simulation results demonstrate the effectiveness of the proposed theorem.     

Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller

This paper examines chaos control of two four-dimensional chaotic systems, namely: the Lorenz–Stenflo (LS) system that models low-frequency short-wavelength gravity waves and a new four-dimensional chaotic system (Qi systems), containing three cross products. The control analysis is based on recursive backstepping design technique and it is shown to be effective for the 4D systems considered. Numerical simulations are also presented.     

Controlling the Lorenz system: combining global and local schemes

The Lorenz equations are one of the best-known and analyzed systems exhibiting chaotic behavior. In this paper, a new control scheme for the Lorenz system combining local and global techniques is introduced. This scheme is based on a feedback law which is only applied in a bounded state space region of control (SSRC). The SSRC is determined by the enclosure of the Lorenz attractor.     

Multi-scroll and multi-wing chaotic attractor generated with Julia process fractal

This paper proposes a general multi-scrolls and multi-wings attractor systems while combining in one hand, Julia’s process with Lorenz’s attractor and in another hand Julia’s process with two coupled Lorenz systems. We have also studied the combination of Julia’s process with the chaotic attractor system proposed in [1].     

Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control

Based on the stability theory of fractional order systems, this paper analyses the synchronization conditions of the fractional order chaotic systems with activation feedback method. And the synchronization of commensurate order hyperchaotic Lorenz system of the base order 0.98 is implemented based on this method. Numerical simulations show the effectiveness of this method in a class of fractional order chaotic systems.     

Some new less conservative criteria for impulsive synchronization of a hyperchaotic Lorenz system based on small impulsive signals

In this Letter the issue of impulsive Synchronization of a hyperchaotic Lorenz system is developed. We propose an impulsive synchronization scheme of the hyperchaotic Lorenz system including chaotic systems. Some new and sufficient conditions on varying impulsive distances are established in order to guarantee the synchronizability of the systems using the synchronization method. In particular, some simple conditions are derived for synchronizing the systems by equal impulsive distances. The boundaries of the stable regions are also estimated. Simulation results show the proposed synchronization method to be effective.     

Connecting the Lorenz and Chen systems via nonlinear control

The Lü system is a new chaotic system, which connects the Lorenz system and the Chen system and represents the transition from one to the other. In this letter, based on the concept of nonresonant parametric perturbations, further detailed analysis about the forming mechanism and its compound structure for the chaotic Lü system are offered. The obtained results clearly reveal the intermediate chaotic system has another novel forming mechanism: the compression and pull forming mechanism, which provides an enlighten insight about the relationship of its vibration “mode” and the two-scroll “base” chaotic attractor. Then motivated by this novel forming mechanism, by adding a simple nonlinear term to the Lü system, its role as a joint function is revisited. With the gradual tuning the parameter of the nonlinear controller, the transition from the canonical Lorenz attractor to the Chen attractor through the Lü attractor is revived. The scheme herein goes beyond the traditional framework for studying the Lorenz-like systems, which can be very helpful in generating and analyzing of all similar and closely related chaotic systems.     

Dynamical analysis and numerical simulation of a new Lorenz-type chaotic system

In this paper, a new 3D autonomous Lorenz-type chaotic system is modelled based on the condition that the system may generate chaos whereas it has only stable or non-hyperbolic equilibrium points. This system also includes some well-known Lorenz-like systems as its special cases, such as the diffusionless Lorenz system, the Burke-Shaw system and some other systems found. Although the new chaotic system is similar to other Lorenz-type systems in algebraic structure, they are topologically non-equivalent. This interesting fact motivates one to further investigate its dynamical behaviours, such as the number and the stability of equilibrium points, Hopf bifurcation and its direction, Poincaré maps, Lyapunov exponents and dissipativity, etc. Given numerical simulations not only verify the corresponding theoretically analytical results, but also demonstrate that this system possesses abundant and complex dynamical properties, which need further attention.