Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Lü

This work presents chaos synchronization between two different chaotic systems by using active control. This technique is applied to achieve chaos synchronization for a new system and each of the dynamical systems Lorenz, Chen and Lü. Numerical simulations are also shown to verify the results.     

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].     

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.     

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.     

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.     

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.     

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.     

Attractors synthesis for a Lotka-Volterra like system

The aim of this paper is to prove numerically, via computer graphic simulations, that the synthesis algorithmprovided by Danca et al. [1] can be utilized to synthesize any attractor of a dynamical system modeling a two-predator, one prey Lotka-Volterra like system. The algorithm switches in a periodic deterministic or a random way the control parameter inside a set of a chosen values. The obtained attractor is the same with the attractor obtained for parameter value taken as averaged value of the switched control values. This simple and effective algorithm relies on a convex property induced in the set of the attractors corresponding to the chosen switching parameters. The algorithm was tested successfully on systems depending linearly on the control parameter like Lorenz, Chen, Rossler, networks and other systems.     

Impulsive control and its application to Lü''s chaotic system

In this paper, we derive some less stringent conditions for the exponential and asymptotic stability of impulsive control systems with impulses at fixed times. These conditions are then used to design an impulsive control law for Lü system, which drives the chaotic state to a zero equilibrium.     

Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria

This paper presents a new 3-D autonomous chaotic system, which is topologically non-equivalent to the original Lorenz and all Lorenz-like systems. Of particular interest is that the chaotic system can generate double-scroll chaotic attractors in a very wide parameter domain with only two stable equilibria. The existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear. Finally, the complicated dynamics are studied by virtue of theoretical analysis, numerical simulation and Lyapunov exponents spectrum. The obtained results clearly show that the chaotic system deserves further detailed investigation.     

Parameter identification for a class of nonlinear chaotic and hyperchaotic flows

Based on the stability theory, an accurate and fast parameter identification law of chaotic and hyperchaotic flows is created by selecting right initial values and suitable observers. The suitable observers are designed and used to identify any uncertain parameters of chaotic and hyperchaotic flows. Furthermore, the accuracy of parameter identification is very much improved by making use of wavelet de-noising. Theory analysis and numerical simulations of Lorenz and hyperchaotic Chen systems are implemented to verify the effectiveness and feasibility of the observers to identify the parameters.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Adaptive generalized function projective lag synchronization of different chaotic systems with fully uncertain parameters

In this paper, a novel projective synchronization scheme called adaptive generalized function projective lag synchronization (AGFPLS) is proposed. In the AGFPLS method, the states of two different chaotic systems with fully uncertain parameters are asymptotically lag synchronized up to a desired scaling function matrix. By means of the Lyapunov stability theory, an adaptive controller with corresponding parameter update rule is designed for achieving AGFPLS between two diverse chaotic systems and estimating the unknown parameters. This technique is employed to realize AGFPLS between uncertain Lü chaotic system and uncertain Liu chaotic system, and between Chen hyperchaotic system and Lorenz hyperchaotic system with fully uncertain parameters, respectively. Furthermore, AGFPLS between two different uncertain chaotic systems can still be achieved effectively with the existence of noise perturbation. The corresponding numerical simulations are performed to demonstrate the validity and robustness of the presented synchronization method.     

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.     

Design of sliding mode controller for a class of fractional-order chaotic systems

In this paper, a sliding mode control law is designed to control chaos in a class of fractional-order chaotic systems. A class of unknown fractional-order systems is introduced. Based on the sliding mode control method, the states of the fractional-order system have been stabled, even if the system with uncertainty is in the presence of external disturbance. In addition, chaos control is implemented in the fractional-order Chen system, the fractional-order Lorenz system, and the same to the fractional-order financial system by utilizing this method. Effectiveness of the proposed control scheme is illustrated through numerical simulations.     

Chaotic attractors in nonlinear dissipative systems of ordinary differential equations

A mechanism is proposed describing the formation of irregular attractors in a wide class of three-dimensional nonlinear autonomous dissipative systems of ordinary differential equations with singular cycles. The attractors of such systems, called singular attractors, lie on two-dimensional surfaces in the phase space and have no positive Lyapunov exponents. In all systems of this class the onset of chaos follows the same universal mechanism: a cascade of Feigenbaum’s period doubling bifurcations, a subharmonic cascade of Sharkovskii’s bifurcations, and eventually a homoclinic cascade. All classical chaotic systems, including Lorenz, Rössler, and Chua systems, satisfy these conditions.     

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.     

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．     

Controlling uncertain Lü system using backstepping design

In this paper, we discussed how to control Lü system with unknown parameters. Firstly we designed an observer to identify the unknown parameter of Lü system, then we used backstepping design method to control the system, and track any desired trajectory by the same way. At the same time we gave the numerical simulation for the results we had gained.     

Finite-time synchronization of uncertain unified chaotic systems based on CLF

This paper studies the problem of finite-time synchronization for the unified chaotic systems. We prove that global finite-time synchronization can be achieved for unified chaotic systems which have uncertain parameters. Simulation results for Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.