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A nonlinear control is proposed for the exponential stabilization and synchronization of chaotic behaviour in a model of Bose-Einstein condensate (BEC). The active control technique is designed based on Lyapunov stability theory and Routh-Hurwitz criteria. The control design approach in both cases guarantees the stability of the controlled states. Whereas the synchronization of two identical BEC in their chaotic states can be realized using the scheme; a suitable controller is also capable of driving the otherwise chaotic oscillation to a stable state which could be expected in practice. The effectiveness of this technique is theoretically and numerically demonstrated.

Chaotic behaviour is a well-known phenomenon in physics, engineering, biology, and many other scientific disciplines. Recently, it has received much attention [

The control and synchronization of chaotic systems are challenging problems, since a chaotic system is extremely sensitive to small perturbations. Notwithstanding, the possibility of control and synchronization of chaotic systems under certain conditions has been established [

Progress in the research activities on chaos synchronization and stabilization has given birth to various methods such as OGY method [

In this light, various linear and nonlinear techniques have been proposed for achieving control and synchronization of a wide variety of physical systems. Examples of these abounds. For instance, in ratchet systems where directed transports play prominent role, synchronization has been proposed as a mechanism for controlling directed transports [

In this paper, first, we modified the active control technique presented by Lei et al. [

The Bose-Einstein condensates which we study here has been observed in trapped gases of rubidium, sodium, and lithium [

Recently, vibrational state inversion of a Bose Einstein condensate with optimal control and state tomography was considered in [

In this paper, we propose a nonlinear control scheme to realize both chaotic synchronization and exponential stabilization of BEC. In the next section, we formulate the control theories and in Section

Let us consider a driver chaotic system having the same form as in our previous paper [

The dynamic equation of synchronization error,

Our goal is to design a suitable control input

Then, the error dynamical system (

If there exists the feedback matrix

The proof of Theorem

Consider the controlled chaotic system given by (

If there exists the feedback matrix

Here, we consider the BEC system described in [

Inserting

It is convenient to recast equation (

According to the general theory of the Duffing equation, underlying equation (

Solving (

The period-1 attractor (single well) of the BEC with

The period-1 attractor (double well) of the BEC with

The chaotic attractor of the BEC.

The time series of the chaotic attractor of the BEC.

In this section, the method proposed in Section

According to Lyapunov stability theory and Routh-Hurwitz criteria, if

The 4th-order Runge-Kutta algorithm was used to obtain the numerical solutions of (

With the form of (

The simulation results when the controller (

The asymptotic convergence of the synchronization error system to zero for identical BEC system.

The

The

Here we propose a recursive active control-to-control the evolution of chaotic state in the BEC system. The goal of the control technique is to drive the BEC system from its current chaotic state

The controlled state of the BEC system.

In this paper, the synchronization between two-coupled Bose-Einstein Condensates in their chaotic states have been investigated based on a technique derived from nonlinear control theory. The nonlinear control is obtained using the active control and the controller is chosen such that a single control input is sufficient to guarantee global stability of the synchronized state. The systematic design approach of the control input does not require the calculation of the Lyapunov exponents; hence, it is simple and convenient. Besides the synchronization of two chaotic BEC systems, the method was also used to obtain a nonlinear control that is capable of driving the otherwise chaotic oscillation to a stable equilibrium state which could be expected in applications. The effectiveness and feasibility of this technique are theoretically and numerically demonstrated. We believe that such nonlinear control would find applications in the experimental control of spatial chaos in BEC. It is likely that nonlinear interactions are necessary for the stability of continuously pumped atom lasers as described in [

According to Haine et al. [

The research work of U. E. Vincent is supported by the Royal Society of London through the Newton International Fellowship Alumni Scheme. The author acknowledges useful discussions with Peter McClintock.

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