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We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable when

As one of the important discoveries in 21st century, chaos has been extensively investigated in many fields over the last several decades, which has been widely applied in secure communication, signal processing, radar, image processing, power system protection, flow dynamics, and so on. As is known chaos is undesirable and needs to be controlled in many practical applications. Therefore, the investigation of controlling chaos is of great significance. Many schemes have been presented to carry out chaos control [

Recently, a new Lorenz-like system has been introduced in [

In order to reveal the forming mechanism of the chaotic attractor structure, its controlled system is proposed in [

With purpose of reflecting and controlling the complex and unpredictable dynamical behavior of the model depending on the past information of the system, it is necessary to incorporate time delay into this system. The signal error of current state and past state of the continuous time system will be given distributed delay feedback to the system itself.

As compared with the former method, a chaotic model with distributed delay feedback is more general than that with discrete delay feedback [

Many studies have been made in [

Motivated by the above works, we, in this paper, add a time-delayed force

The rest of this paper is organized as follows. In Section

In this section, we will study the stability of the equilibria and the existential conditions of local Hopf bifurcations. As the Lorena-like system (

By the linear transform

The equilibrium

Suppose (

If

If

If

It is shown that if (

In this section, we will establish the explicit formulae determining the direction, the stability, and the period of these periodic solutions bifurcating from the equilibrium

Define

For

Since

Next, we employ the idea of Hassrd et al. in [

Defining

Note that

Comparing the coefficients of (

Similarly, in view of (

Next, we will seek appropriate

we have

In this section, we give numerical simulation result of system (

The bifurcation diagrams of system (

The bifurcation diagrams of system (

The chaotic attractors of system (

Behavior and phase portraits of system (

Behavior and phase portraits of system (

Behavior and phase portraits of system (

The quasi-period solution of system (

The chaotic attractor of system (

Since the original system (

In this paper, we investigate a Lorenz-like system within chaotic attractor responding to the local Hopf bifurcation and the local stability of equilibrium

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

This work was supported by National Natural Science Foundation of China (Grant no. 61261044) and the Natural Science and Technology Foundation of Henan Province (no. 15A110046).