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Synchronization of chaotic systems has attracted extensive concern in the past few years. In this study, we investigate a new structure of Duffing system by the variable decomposition method. Then, we analyze the state observer synchronization based on the new Duffing system. It is proved theoretically that the designed observer can keep synchronization with Duffing chaotic system in transmitter. The design is presented reasonably with the conditional Lyapunov exponents, and its effectiveness is clearly shown in simulation results.

Duffing equations are well-known nonlinear equations of applied science which are used as a powerful tool to discuss some important practical phenomena such as periodic orbit extraction, nonuniformity caused by an infinite domain, and nonlinear mechanical oscillators.

Synchronization is a very common phenomenon of nonlinear dynamic system. It shows that two moving objects can make their states converge through the medium strong or weak coupling. Since Pecora and Carroll [

For obtaining synchronization, both the structure and the parameters of target system are often supposed to be available, while the only unknown is the initial condition. As the model of a system is commonly imprecise, obviously, it is not necessarily a realistic case. Therefore, the synchronization or autosynchronization of chaotic systems with unknown parameters has been widely studied in recent years. The synchronization of Lorenz system, one of the most popular chaotic systems, is stressed, and the conditions of synchronization have been studied by various researchers [

Due to easy engineering implementation, the observer synchronization method has gained extensive attention. Observer synchronization has simple structure, fast synchronous speed, and high precision. The method can be widely used in communications and information processing.

At present, the study of Duffing equation is in the form of two-dimensional state equation [

This paper is organized as follows. In Section

There are many problems in engineering, such as packaging systems based on displacement excitation of nonlinear vibrations, pressure sensors nonlinear vibration. They can be simplified into a forced Duffing equation with cubic non-linearity, governed by

The particular form of Duffing system related to system (

When given the system initial state

Two-dimensional Duffing chaotic system phase diagram with

Given a constant

Choose the following differential equation:

From the above analysis, it follows that

When given

The simulation results of equivalent condition are the premise of system initial states

When it is assumed that

Three-dimensional Duffing chaotic system phase diagram with

Three-dimensional Duffing chaotic system phase diagram in the xoy plane projection with

Three-dimensional Duffing chaotic system phase diagram in the yoz plane projection with

Three-dimensional Duffing chaotic system phase diagram in the xoz plane projection with

Three-dimensional Duffing chaotic system phase diagram in the xoy plane projection with

Three-dimensional Duffing chaotic system phase diagram in the xoy plane projection with

Suppose that there exist

For (

There exist

At the receiving system, when the nonlinear input feedback can be measured, system (

Now define

If all eigenvalues of

By

So the state observer design problem turns into finding the right

From (

All

By (

Consider the expression

Assume

By Theorem

When

Now let

Let

In this section, we evaluate the performance of state observer synchronization used in three-dimensional Duffing system. Under the condition of interference and no interference, we simulated transceiver system synchronization.

To discuss the synchronization performance of system, we assumed coefficient of transmitting system

State observer synchronization simulation system used in three-dimensional Duffing system.

Under the condition of no interference, synchronization error is shown in Figure

The error curves driven by the outside periodic force

The curve

The curve

The curve

When there is interference, we assume adding Gaussian white noise to

The present work has studied a new three-dimensional chaotic synchronization of Duffing system. The chaotic system is different from the previous Duffing system. It contains three equations. The third equation describes nonlinear effect on the new Duffing system. This system redefined state variables, nonlinear feedback input, and driving force. Because the synchronization method based on observer does not need the initial state in the same basin of attraction, we prove observer synchronization in three-dimensional Duffing system. By adopting the method of simple linear observer, receiving system can successfully keep synchronization with transmitting chaotic system.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China under Grant no. 51277011.