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This paper advances a new single-parameter chaotic system with a simple structure, and some basic dynamic behavior of the new single-parameter system is discussed, such as equilibria, dissipativity, the existence of an attractor, Lyapunov exponent, Poincaré map, spectrum map, and bifurcation diagram. Moreover, a new fixed-time convergence theorem is proposed for general chaotic systems based on finite-time control theory, and a fixed-time controller is also put to achieve synchronization of the new chaotic system. Simulation results are presented to show the effectiveness of the theoretical results. The conclusion of the paper is useful for the nonlinear economics and engineering application.

After the first chaotic attractor was discovered by Lorenz, people began to search for new chaotic attractors with great enthusiasm. In recent years, for autonomous three-dimensional ordinary differential equations, the search for new chaotic attractors has attracted more and more attention. For example, Lorenz system [

Furthermore, research on synchronization and control of the chaos system is particularly topical for decades [

This paper is organized as follows: Section

Consider the following system:

A new chaotic attractor.

In this section, some basic characteristics of the new chaotic system are analyzed.

Let

When

Let

For

Let

Therefore, when

Similarly, when

According to equations (

For dynamical system (

It is generally true that a system is chaotic as long as at least one of its Lyapunov exponents is positive. When the initial value is (1, 1, 1), the Lyapunov exponent of system (

It implies that system (

In Figure

An apparently continuous broadband frequency spectrum.

Poincaré maps for (a)

Bifurcation diagram of system (

Considering the following chaotic system in a general form:

So, the controlled chaos system is given by

Then, the error system can be

We also assume that

For

Assume that a continuous and a positive-definite function

In the following, the new fixed-time controller based on Lemma

Assuming that hypothesis 1 holds, two chaotic systems (

Let

Then,

Obviously,

So,

Let

When

By (

If

From (

So,

When

When

The proof is completed.

In [

For the complex dynamic network satisfying Assumption

We assume that the new chaotic system is the drive system, that is,

And, the response system is

The initial conditions of the master and slave systems are (1, 2, 1) and (2, 1, 3), respectively. Through simulation calculation,

Error evolution without the controller.

Synchronization errors.

The paper has presented a new chaotic system with a single parameter and discussed some basic dynamic behavior of the new system. Base on finite-time control theory, the fixed-time synchronization criterion of the new chaotic system has also been obtained. Finally, the result of numerical simulation proves that they are feasible. In the future, we will discuss the fixed-time synchronization of general complex networks.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (61673221); the Natural Science Foundation of Jiangsu Province (BK20181418); the Natural Science Foundation of Jiangsu Higher Education Institutions (19KJB120007); the Six Talent Peaks Project in Jiangsu Province (DZXX-019).