In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibrium

Over the past decades, as we have seen, researchers have paid a great attention to the control of nonlinear dynamical systems exhibiting Hopf bifurcation phenomena, because the presence of bifurcation is very important in many physical, biological, and chemical nonlinear systems [

Recently, Dias et al. [

Phase trajectory in 3-D space and various projections of the chaotic attractor.

Moreover, the dynamics of the system (

(a) Time history, (b) frequency spectrum, (c) Poincaré map in

The system possesses three equilibria

In this work, we will design a control laws such that our feedback system undergoes a controllable Hopf bifurcation. To accomplish the control of Hopf bifurcation in the system (

This paper is organized as follows. In Section

This section is a review of the projection method described in [

Consider the differential equation

Suppose that

Let

The first Lyapunov coefficient

The second Lyapunov coefficient is defined by

A Hopf point

A Hopf point is called transversal if the parameter-dependent complex eigenvalues cross the imaginary axis with nonzero derivative. In a neighborhood of a transversal Hopf point with

A Hopf point of codimension 2 is a Hopf point, where

In this section, we will study the stability of

The Jacobian matrix of the system (

The equation

Then, using the notion of the previous section, the multilinear symmetric functions corresponding to

The eigenvalues of

Now, consider the family of differential equation (

Using these calculations, we prove the next theorem.

Consider the six-parameter family of differential equations (

Let the damping coefficient

When the parameters

if

if

if

if

if

Nonlinear dynamics of system (

With the analysis performed here, one can find that the Hopf bifurcation at the equilibrium

The stable domain on the parameter plane

Time history and phase diagram of system (

Time history and phase diagram of system (

Time history and phase diagram of system (

According to Routh-Hurwitz criterion, if and only if (

Let

Next, we combined with Figure

In this section, we will study the stability of

According to Dias et al. [

Then, using the notion of the previous section, the multilinear symmetric functions corresponding to

The eigenvalues of

Now, consider the family of differential equation (

Using these calculations, we prove the next theorem.

Consider the six-parameter family of differential equations (

The sign of the first Lyapunov coefficient is determined by

Define

Using these calculations, we prove the next theorem.

If

Next, we shall give a numerical example of system (

Time history and phase diagram of system (

Time history and phase diagram of system (

When the parameters

Nonlinear dynamics of system (

Nonlinear dynamics of controlled system (

Letting

Time history and phase diagram of system (

Time history and phase diagram of system (

In this section, two possible electronic circuits are given to realize (

Circuit diagram for realizing the chaotic attractor of system.

Experimental observations of the chaotic attractor in different planes.

Projection on

Projection on

Projection on

Circuit diagram for realizing the chaotic attractor of system (inside the black-dotted box is the controller).

Experimental observations of the time history and phase diagram of system (

The time history

The phase diagram

Experimental observations of the time history and phase diagram of system (

The time history

The phase diagram

Experimental observations of the time history and phase diagram of system (

The time history

The phase diagram

In this paper, we consider the problem of anticontrol of Hopf bifurcations; that is, an anticontroller for a Lorenz-like system is designed with desired location and properties by appropriate controls. By the numerical analysis, we prove that Hopf bifurcation occurs when the bifurcation parameter passes through the critical value. In particular, we designed a washout controller such that the equilibrium

The authors express their gratitude to the referee for their valuable comments on the first version of the paper. The authors also gratefully acknowledge the support from the National Natural Science Foundation (nos. 11161027 and 61364001), the Key Project of Chinese Ministry of Education (no. 212180), the Fundamental Research Funds for the Universities of Gansu Province (no. 620023), and the Natural Science Foundation of Gansu Province (no. 1010RJZA067), Government of China.