^{1}

^{1}

^{2}

^{2}

^{1}

^{2}

A novel reduced-order adaptive sliding mode controller is developed and experimented in this paper to antisynchronize two different chaotic systems with different order. Based upon the parameters modulation and the adaptive sliding mode control techniques, we show that dynamical evolution of third-order chaotic system can be antisynchronized with the projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: firstly Lorenz (4th-order) and Lorenz (3rd-order) and secondly the hyperchaotic Lü (4th-order) and Chen (3rd-order). Theoretical analysis and numerical simulations are shown to verify the results.

Nonlinearity is ubiquitous in the natural world around us such as in chemical reactions, cardiac arrhythmias, brain neural waves, relativity, population growth rates, and atmospheric changes. The study of the chaotic nature of the nonlinear dynamical systems, although complex and challenging, paves a way to understand the laws of nature. An understanding of the disorder and oscillating nature of these systems and their instability leads to many useful real-world science and engineering applications. Until some decades, the only systems that could be understood were those that were believed to be linear or systems that follow predictable patterns and arrangements. However, the advent of high-speed computers has provided mathematicians some considerable accessibility in the analysis of nonlinear systems and the chaos theory is one of the many milestones. Lately, the theory of chaos has become one of the most noteworthy and undoubtedly valuable subjects of research due to its wide and numerous applications. There is a need to understand and to have a reasonable control over the unforeseen natural events and phenomena. “A very small cause which escapes our notice determines a considerable effect that we cannot fail to see…even if the case that the natural laws had no longer secret for us…we could only know the initial situation approximately…It may happen that small differences in initial conditions produce very great ones in the final phenomena.” [

The work of Pecora and Carroll [

Antisynchronization (AS) or antiphase synchronization (APS), an extended scope of synchronization, is a phenomenon that the state vectors of the synchronized systems have the same amplitude but opposite signs as those of the driving system. There are many methods discussed in the literature on the synchronization and antisynchronization of dynamical systems of equal and unequal orders. Synchronization of a slave system with projections of a master system is dealt with in the reduced-order synchronization. However, it is important to make a distinction here that the problem of the reduced-order synchronization differs from the partial synchronization where the latter is mainly for coupling of two chaotic systems which have an equal order. The main characteristic feature of the reduced-order synchronization is that the order of the slave system is less than the master system.

In this paper, we address the reduced-order antisynchronization of chaotic systems via adaptive sliding mode controller. A great deal of research has already been undertaken mainly on the synchronization and the antisynchronization between two chaotic systems with the same order. However, due to the complex nature of the chaotic dynamical systems, a thorough understanding of the antisynchronization between two chaotic systems of unequal order is vital as they have much wider applications. This gave us the motivation to introduce this work.

Consider a chaotic system described by the following nonlinear differential equation:

We assume that systems (

To gain reduced order antisynchronization between the master and slave system is to basically design the controller

The sliding mode control method of antisynchronization involves two major stages: (1) choosing a suitable switching surface for the desired sliding motion and (2) designing the sliding mode controller that brings any orbit in phase space to the switching surface and then achieves the antisynchronization of the chaotic systems even in the presence of parameter and disturbance uncertainties. This is precisely why this method of antisynchronization is considered to be robust under uncertainties and external disturbances.

The sliding surface can be defined as follows:

In what follows, the appropriate sliding mode controller will be designed according to the sliding mode control theory. Choosing the controller

The following theorem contains the necessary conditions for the stability of error system in (

Considering that adaptive sliding mode control input law in (

To check the stability of the controlled system, one can consider the following Lyapunov candidate function:

Since

The hyperchaotic Lorenz system [

Typical dynamical behavior of hyperchaotic Lorenz system. (a) Projection in

Typical dynamical behavior of three-dimentsional Lorenz system. (a) Projection in

Typical dynamical behavior of hyperchaotic Lü system. (a) Projection in

Typical dynamical behavior of three-dimensional Chen system. (a) Projection in

In order to observe reduced order antisynchronization behavior between chaotic systems via adaptive sliding mode control, we consider two examples. The first one is hyperchaotic Lorenz system in (

For the hyperchaotic Lorenz system, the master system is considered to be the projections in the direction of

The error dynamics is represented by

Since

State trajectories of drive system (

In the manner, similar to the previous example, for the hyperchaotic Lü system, the master system is considered to be the projections in the direction of

Since

State trajectories of drive system (

The novelty of our technique in solving reduced-order antisynchronization problem is demonstrated and proved using rigorous analytical and numerical procedures to antisynchronize two uncertain chaotic systems. The antisynchronization of the dynamical evolution of a 3rd-order chaotic system was realized with the canonical projection of a 4th-order chaotic system even though their parameters were unknown. This was based upon the parameters modulation and the adaptive sliding mode control techniques. The proven techniques were applied to the two examples: Lorenz (4th-order) with Lorenz (3rd-order) and hyperchaotic Lü (4th-order) with Chen (3rd-order). The theoretical analyses and numerical simulations have verified and supported our assumptions. The scope for the applications of antisynchronization of two chaotic systems with different orders is much wide ranging.

This work is financially supported by the Universiti Kebangsaan Malaysia Grant: UKM-DLP-2011-016.