© Wuhan University 2025

0 Introduction

In nature, there are numerous chaotic phenomena. With the continuous progress of scientific research, many new chaotic systems are gradually being discovered and studied^[1-6]. However, many studies have mainly focused on quadratic polynomial chaotic systems. The main characteristics of this type of system are that the differential equations are relatively simple, making it easy to get the bifurcation characteristics of the systems. There is relatively little research on chaotic systems with infinite equilibrium points. Due to the infinite equilibrium points of these systems, they exhibit complex bifurcation characteristics.

Bifurcation is an important dynamic characteristic of chaotic systems. The analysis and control of bifurcation have been one of the important directions in nonlinear dynamics research in recent years^[7-13]. Bifurcation control mainly involves Hopf bifurcation^[14-16], pitchfork bifurcation^[17-18], Neimark-Sackeretc bifurcation^[19-21], period-doubling bifurcation^[22-24]. The purpose of these bifurcation controls is to delay the occurrence of bifurcations, to control the system into a stable state, and to change the amplitude of the limit cycle.

Anti-control of bifurcation generates bifurcation at a predetermined critical value by adjusting control parameters when the bifurcation is beneficial. The anti-control of bifurcation is not a process of eliminating bifurcation. It is a process of generating bifurcation. By generating bifurcations at designated locations, the bifurcation of the system can be delayed or advanced. The stability range of the system has also been changed, expanding the stability range of the system. There are many research results on bifurcation control, mainly including Neimark-Sackeretc bifurcation^[25], period-doubling bifurcation^[26], and Hopf bifurcation^[27-35]. Among them, Hopf bifurcation has become a hot research topic in bifurcation control. The Hopf bifurcation characteristics of the system are altered by the anti-control method. The system may generate limit cycles in the expected oscillatory behavior^[27]. The nonlinear and linear controllers are set up for the 4D Qi system, and the anti-control of Hopf bifurcation at the zero equilibrium point is studied. When the control parameters of the nonlinear controller are changed, the system amplitude is altered. When the control parameters of the linear controller are changed, the system state transitions from unstable to stable^[29]. Using the feedback controller of polynomial functions, anti-control of Hopf bifurcation of discrete maps is carried out at the desired position, without increasing the system dimension or changing the equilibrium solution^[30]. A nonlinear controller is set up for the Shimizu-Morioka system. Using central flow and paradigm theory, the correspondence between the control parameters and stability is obtained. By adjusting control parameters, it is possible to achieve Hopf bifurcation at a specified position and change the stability. The amplitude of the limit cycle can be controlled within the expected range^[31]. The Hopf bifurcation characteristics of a 4D chaotic system with infinite equilibria are analyzed at the zero equilibrium point. Using the state feedback control method, a nonlinear controller is set up for the system to achieve stability changes^[35]. Although the Hopf bifurcation of chaotic systems has a wide range of applications in mechanical engineering, bioengineering, civil engineering, and other fields, there is not much research by scholars on the anti-control of Hopf bifurcation, especially for the chaotic system with infinite equilibria. In this paper, a 4D chaotic system with infinite equilibria is proposed. According to the Routh-Hurwitz criterion and high-dimensional Hopf bifurcation theory, the Hopf bifurcation characteristics of the system are analyzed. The existence of hidden attractors in the system is verified. A hybrid controller is set up for the system, which includes both linear and nonlinear controllers. Using the state feedback control method, we adjust the control parameters in the controller to achieve a change in system stability and achieve the goal of Hopf bifurcation inverse control.

This work is organized as follows. Section 1 proposes a 4D chaotic system with infinite equilibria. The Hopf bifurcation characteristics of the system are analyzed. Section 2 analyzes the symmetry and coexistence of the hidden attractors of the system. An improved state feedback control method is adopted for the anti-control of Hopf bifurcation. The bifurcation characteristic is changed in Section 3. Finally, Section 4 concludes with some discussions.

1 A 4D Chaotic System with Infinite Equilibria

A new chaotic system with infinite equilibria is proposed, which is as follows:

$\{\begin{array}{l}\dot{x}=az,\\ \dot{y}=b\mathrm{s}\mathrm{i}\mathrm{n}\left[c\left(x-y\right)\right],\\ \dot{z}=d-xy,\\ \dot{u}=e-yu,\end{array}$(1)

where $x,y,z,u$ are the state variables, and $a,b,c,d,e$ are the real numbers.

When $abc\ne \mathrm{0},d>\mathrm{0},e\ne \mathrm{0}$, we get

$\{\begin{array}{l}x-y=\frac{k\mathrm{\pi }}{c},\\ xy=d,\\ z=\mathrm{0},\\ u=\frac{e}{y},\end{array}$(2)

where $k\in \mathbb{Z}$, $\mathbb{Z}$ is the set of all integers.

Let

$x-y=p,$(3)

where $p=\frac{k\mathrm{\pi }}{c}$.

According to (2) and (3), we can obtain

$y^{\mathrm{2}}+py-d=\mathrm{0}.$(4)

Due to $d>\mathrm{0}$, therefore,

$\mathrm{\Delta }=p^{\mathrm{2}}+\mathrm{4}d>\mathrm{0}$(5)

Solve for variables $x$ and $y$,

$\{\begin{array}{l}x=\frac{p\pm \sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\\ y=\frac{-p\pm \sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}.\end{array}$(6)

Then, the equilibrium points of the system (1) are as follows:

$E_{\mathrm{1}}=\left(\frac{p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\mathrm{0},\frac{e\left(p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}\right)}{\mathrm{2}d}\right),E_{\mathrm{2}}=\left(\frac{p-\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\frac{-p-\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\mathrm{0},\frac{e\left(p-\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}\right)}{\mathrm{2}d}\right).$

Due to $p=\frac{k\mathrm{\pi }}{c}$, $k\in \mathbb{Z}$, therefore, the equilibrium points of system (1) are infinite.

Next, we analyze the Hopf bifurcation characteristics. The Jacobian matrix of the system (1) at the equilibrium point $E_{\mathrm{1}}$ is

${\mathit{J}}_{\mathrm{1}}=\left[\begin{array}{cccc}\mathrm{0}& \mathrm{0}& a& \mathrm{0}\\ bc\mathrm{c}\mathrm{o}\mathrm{s}\left[c\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)\right]& -bc\mathrm{c}\mathrm{o}\mathrm{s}\left[c\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)\right]& \mathrm{0}& \mathrm{0}\\ -y_{E_{\mathrm{1}}}^{\mathrm{*}}& -x_{E_{\mathrm{1}}}^{\mathrm{*}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -u_{E_{\mathrm{1}}}^{\mathrm{*}}& \mathrm{0}& -y_{E_{\mathrm{1}}}^{\mathrm{*}}\end{array}\right].$(7)

The characteristic equation of the Jacobian matrix is

${\lambda }^{\mathrm{4}}+q_{\mathrm{1}}{\lambda }^{\mathrm{3}}+q_{\mathrm{2}}{\lambda }^{\mathrm{2}}+q_{\mathrm{3}}\lambda +q_{\mathrm{4}}=\mathrm{0},$(8)

where

$\{\begin{array}{l}{q}_{\mathrm{1}}=bc\mathrm{c}\mathrm{o}\mathrm{s}\left[c\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)\right]+y_{E_{\mathrm{1}}}^{\mathrm{*}},\\ q_{\mathrm{2}}=\mathrm{10}{y}_{E_{\mathrm{1}}}^{\mathrm{*}}+bc\mathrm{c}\mathrm{o}\mathrm{s}\left[c\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)\right]y_{E_{\mathrm{1}}}^{\mathrm{*}},\\ q_{\mathrm{3}}=\mathrm{10}bc\mathrm{c}\mathrm{o}\mathrm{s}\left[c\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)\right]\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}+y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)+\mathrm{10}{\left(y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)}^{\mathrm{2}},\\ q_{\mathrm{4}}=\mathrm{10}bc\mathrm{c}\mathrm{o}\mathrm{s}\left[c\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)\right]y_{E_{\mathrm{1}}}^{\mathrm{*}}\left(x_{E_{\mathrm{1}}}^{\mathrm{*}}+y_{E_{\mathrm{1}}}^{\mathrm{*}}\right).\end{array}$

According to the Routh-Hurwitz criterion, the coefficients of (8) is

${\mathrm{\Delta }}_n=\left|\begin{array}{cccc}{q}_{\mathrm{1}}& \mathrm{1}& \mathrm{0}& \mathrm{0}\\ q_{\mathrm{3}}& q_{\mathrm{2}}& q_{\mathrm{1}}& \mathrm{1}\\ \mathrm{0}& q_{\mathrm{4}}& q_{\mathrm{3}}& q_{\mathrm{2}}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& q_{\mathrm{4}}\end{array}\right|$(9)

If equation (8) satisfies the following conditions, it has a pair of pure virtual roots and two roots with negative real parts. Then the system (1) is stable at the equilibrium point $E_{\mathrm{1}}$.

$\{\begin{array}{c}\begin{array}{l}{q}_i>\mathrm{0},i=\mathrm{1,2},\mathrm{3,4},\\ {\mathrm{\Delta }}_j>\mathrm{0},j=\mathrm{3,2},\mathrm{1}.\end{array}\end{array}$(10)

When the parameters $a=\mathrm{10},b=\mathrm{3},c=\mathrm{10},d=\mathrm{20},e=\mathrm{25},k=\mathrm{10}$, the eigenvalues of the system characteristic equation (8) are ${\lambda }_{\mathrm{1}}=-\mathrm{31.813},{\lambda }_{\mathrm{2}}=-\mathrm{3.169},{\lambda }_{\mathrm{3}}=\mathrm{0.906}-\mathrm{9.411}\mathrm{i},{\lambda }_{\mathrm{4}}=\mathrm{0.906}+\mathrm{9.411}\mathrm{i}$, and the system is in an unstable state at the equilibrium point $E_{\mathrm{1}}$, as shown in Fig. 1.

Fig. 1 Waveform plot and phase portrait of the system (1) at $\mathit{a}=\mathrm{10},\mathit{b}=\mathrm{3},\mathit{c}=\mathrm{10},\mathit{d}=\mathrm{20},\mathit{e}=\mathrm{25},\mathit{k}=\mathrm{10}$

2 Hidden Attractors with Symmetric Coexistence

Due to the infinite number of equilibrium points in the system (1), the attractors generated by the system (1) are hidden attractors.

The states and parameters of the system undergo two transformations.

$\{\begin{array}{l}{T}_{\mathrm{1}}:\left(x,y,z,u,a,b,c,d,e\right)\to \left(-x,-y,z,u,-a,b,c,d,e\right),\\ T_{\mathrm{2}}:\left(x,y,z,u,a,b,c,d,e\right)\to \left(-x,-y,z,u,-a,-b,-c,d,e\right).\end{array}$(11)

Therefore, the system (1) has symmetry. Namely, if the system is chaotic under a set of parameters $a,b,c,d,e$ and initial state $\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}},u_{\mathrm{0}}\right)$, then the system is also chaotic under a set of parameters $-a,b,c,d,e$ (or $-a,-b,-c,d,e$) and initial state $\left(-x_{\mathrm{0}},-y_{\mathrm{0}},z_{\mathrm{0}},u_{\mathrm{0}}\right)$.

Symmetric hidden attractors are coexistence. 1) The projection of coexisting hidden attractors on plane $XOY$ is symmetric about the origin. 2) The projection of coexisting hidden attractors on plane $XOZ$ is symmetric about the Z-axis. 3) The projection of coexisting hidden attractors on plane $YOZ$ is symmetric about the Z-axis.

When $a=\mathrm{10},b=\mathrm{3},c=\mathrm{10},d=\mathrm{20},e=\mathrm{25}$ and $\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}},u_{\mathrm{0}}\right)=\left(-\mathrm{1,3},-\mathrm{1},-\mathrm{1}\right)$, the system generates hidden attractors, as shown on the right side of Fig. 2.

Fig. 2 Hidden attractors symmetry

When $a=-\mathrm{10},b=\mathrm{3},c=\mathrm{10},d=\mathrm{20},e=\mathrm{25}$ (or $a=-\mathrm{10},b=-\mathrm{3},c=-\mathrm{10},d=\mathrm{20},e=\mathrm{25}$) and $\left(x_{\mathrm{0}},y_{\mathrm{0}},z_{\mathrm{0}},u_{\mathrm{0}}\right)=\left(\mathrm{1},-\mathrm{3,1},-\mathrm{1}\right)$, the system generates hidden attractors, as shown on the left side of Fig. 2.

3 Anti-Control of Hopf Bifurcation

3.1 State Feedback Control Method

We propose an improved dynamic feedback control method that does not involve complex computational processes and can effectively achieve Hopf bifurcation anti-control of chaotic systems.

The nonlinear system is designed as follows:

$\dot{\mathit{x}}=f\left(\mathit{x},\mu \right)$(12)

where $f:{\mathbb{R}}^{n+\mathrm{1}}\to {\mathbb{R}}^n,\mathit{x}\in {\mathbb{R}}^n,\mu \in \mathbb{R}$, $\mathit{x}$ is the state vector, $\mu$ is the bifurcation parameter.

Assuming the equilibrium point of the system is $E_i=\left(x_{i\mathrm{1}}^e,x_{i\mathrm{2}}^e,\cdots ,x_{ij}^e,\cdots ,x_{in}^e\right),\mathrm{1}\le i\le n,\mathrm{1}\le j\le n$. The Jacobian matrix of the system at the equilibrium point $E_i$ is defined as

$\mathit{A}\left(\mu \right)=D_x\left(E_i,\mu \right)$(13)

The characteristic equation of (13) has a pair of conjugate complex roots with the Hopf bifurcation theorem.

$\lambda \left(\mu \right)=\alpha \left(\mu \right)\pm \mathrm{i}\omega \left(\mu \right)$(14)

where $\omega \left({\mu }_i\right)>\mathrm{0},\alpha \left({\mu }_i\right)=\mathrm{0},{\alpha }^{\text{'}}\left({\mu }_i\right)\ne \mathrm{0}$, and the remaining $n-\mathrm{2}$ characteristic roots of Eq. (13) have negative real parts, then Hopf bifurcation occurs at the parameter ${\mu }_i$ of the system (12).

The characteristic equation of (13) is rewritten as

${\lambda }^n+a_{\mathrm{1}}{\lambda }^{n-\mathrm{1}}+a_{\mathrm{2}}{\lambda }^{n-\mathrm{2}}+\cdots +a_{n-\mathrm{1}}\lambda +a_n=\mathrm{0}$(15)

where $a_i\left(i=\mathrm{1,2},\cdots ,n\right)$ is the expression of $\mu$. According to the Routh-Hurwitz criterion, the coefficients of (15) are defined as

${\mathrm{\Delta }}_n=\left|\begin{array}{cccccc}{a}_{\mathrm{1}}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ a_{\mathrm{3}}& a_{\mathrm{2}}& a_{\mathrm{1}}& \mathrm{1}& \cdots & \mathrm{0}\\ a_{\mathrm{5}}& a_{\mathrm{4}}& a_{\mathrm{3}}& a_{\mathrm{2}}& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ a_{\mathrm{2}m-\mathrm{1}}& a_{\mathrm{2}m-\mathrm{2}}& a_{\mathrm{2}m-\mathrm{3}}& a_{\mathrm{2}m-\mathrm{4}}& \cdots & a_m\end{array}\right|$(16)

where $m=\mathrm{1,2},\cdots ,n$, and $a_i=\mathrm{0}$ if $i>n$.

The following lemmas have been proposed.

Lemma 1^[36] If and only if $a_i>\mathrm{0}\left(i=\mathrm{1,2},\cdots ,n\right)$, ${\mathrm{\Delta }}_j>\mathrm{0}\left(j=n-\mathrm{1},n-\mathrm{3},n-\mathrm{5},\cdots \right)$, all eigenvalues of the equation have negative real parts.

Lemma 2^[36] The equation has a pair of conjugate pure imaginary roots. If and only if $a_i>\mathrm{0}$, ${\mathrm{\Delta }}_j>\mathrm{0}\left(j=n-\mathrm{1},n-\mathrm{3},n-\mathrm{5},\cdots \right)$ and ${\mathrm{\Delta }}_{n-\mathrm{1}}=\mathrm{0}$, the real parts of the other roots are negative.

For the chaotic system, the purpose of Hopf bifurcation control is that the critical value of the Hopf bifurcation can occur at any specified position. A hybrid controller is designed as (17):

$\{\begin{array}{l}{u}_j\left(x_j,y_j\right)=k_{\mathrm{1}j}\left(x_j-x_{ij}^e\right)+k_{\mathrm{2}j}{\left(x_i-x_{ij}^e\right)}^{\mathrm{3}}-k_{\mathrm{3}j}y{}_j{}^{},\\ \dot{\mathit{y}}=u_j\left(x_j,y_j\right).\end{array}$(17)

where $y\in {\mathbb{R}}^m\left(\mathrm{1}\le m\le n\right)$. $\mathit{y}$ is a newly added state variable. The control parameters of the controller are as follows.

${\mathit{k}}_{\mathrm{1}j}=\left[k_{\mathrm{11}},k_{\mathrm{12}},\cdots ,k_{\mathrm{1}m}\right],{\mathit{k}}_{\mathrm{2}j}=\left[k_{\mathrm{21}},k_{\mathrm{22}},\cdots ,k_{\mathrm{2}m}\right],{\mathit{k}}_{\mathrm{3}j}=\left[k_{\mathrm{31}},k_{\mathrm{32}},\cdots ,k_{\mathrm{3}m}\right].$

$\mathit{u}(\mathit{x},\mathit{y})$ is the hybrid controller that is set up for the system, which is continuously differentiable for $\mathit{x}$ and $\mathit{y}$. $x_{ij}^e$ is the value of each state variable of the system at the equilibrium point. The system (1) becomes a controlled system, and its expression is as follows:

$\{\begin{array}{l}\dot{\mathit{x}}=\mathit{f}\left(\mathit{x},\mathit{y}\right)+\mathit{u}\left(\mathit{x},\mathit{y}\right),\\ \dot{\mathit{y}}=\mathit{h}\left(\mathit{x},\mathit{y}\right),\end{array}$(18)

where $\mathit{u}\left(\mathit{x},\mathit{y}\right)=[u_{\mathrm{1}}\left(x_{\mathrm{1}},y_{\mathrm{1}}\right),\cdots ,u_m{\left(x_m,y_m\right),\mathrm{0},\cdots ,\mathrm{0}]}_{}^{\mathrm{T}}$, $\mathit{h}\left(\mathit{x},\mathit{y}\right)=[u_{\mathrm{1}}\left(x_{\mathrm{1}},y_{\mathrm{1}}\right),\cdots ,u_m{\left(x_m,y_m\right)]}^{\mathrm{T}}$.

Furthermore, it can be concluded that the dimension of the controlled system (18) has changed, increasing from $n$ dimensions to $n+m$ dimensions, but the structures of the equilibria point of the controlled system (18) have not changed. Normally, only one controller $u_p\left(\mathrm{1}\le p\le j\right)$ is set up to the system (1). When the control parameter $k_{\mathrm{1}p}\ne \mathrm{0}$, $k_{\mathrm{2}p}\ne \mathrm{0}$ and $k_{\mathrm{3}p}\ne \mathrm{0}$, the controller (17) can generate many controllers that meet the requirements.

Let $z{}_j{}^{}=k_{\mathrm{1}j}\left(x_j-x_{ij}^e\right)+k_{\mathrm{2}j}{\left(x_i-x_{ij}^e\right)}^{\mathrm{3}}$ as a function of $x_j$. According to the Laplace transform, (19) and (20) are obtained as follows:

$u_j\left(s\right)=z_j\left(s\right)-k_{\mathrm{3}j}{y}_j\left(s\right)$(19)

$z_j\left(s\right)=\left(s+k_{\mathrm{3}j}\right)y_j\left(s\right)$(20)

The expression of the transfer functions is

$G{}_j{}^{}\left(s\right)=\frac{u_j\left(s\right)}{z_j\left(s\right)}=\frac{s}{s+k_{\mathrm{3}j}}$(21)

where $k_{\mathrm{3}j}$ is a positive real number.

Next, the control parameters of the controller need to be calculated. We define the controlled system as follows:

$\dot{\mathit{X}}=\mathit{F}\left(\mathit{X},\mu \right)$(22)

where $\dot{\mathit{X}}={[\mathit{x},\mathit{y}]}^{\mathrm{T}}$, $\mathit{F}={[\mathit{f}\left(\mathit{x},\mathit{y}\right)+\mathit{u}\left(\mathit{x},\mathit{y}\right),\mathit{h}\left(\mathit{x},\mathit{y}\right)]}^{\mathrm{T}}$.

The Jacobian matrix of the controlled system (22) is

${\mathit{J}}_C\left(\mathit{X},\mu \right)=\frac{\partial \mathit{F}\left(\mathit{X},\mu \right)}{\partial \mathit{X}}=\left[\begin{array}{cc}\mathit{J}\left(x\right)+{\mathit{P}}_{\mathrm{1}}\left(x\right)& {\mathit{P}}_{\mathrm{2}}\left(x\right)\\ {\mathit{P}}_{\mathrm{3}}\left(x\right)& {\mathit{P}}_{\mathrm{4}}\left(x\right)\end{array}\right]$(23)

where $\mathit{J}(\mathit{x})=\frac{\partial \mathit{f}\left(\mathit{x},\mu \right)}{\partial \mathit{x}}$, ${\mathit{P}}_{\mathrm{1}}(\mathit{x})=\frac{\partial \mathit{u}\left(\mathit{x},\mathit{y}\right)}{\partial \mathit{x}}$, ${\mathit{P}}_{\mathrm{2}}(\mathit{x})=\frac{\partial \mathit{u}\left(\mathit{x},\mathit{y}\right)}{\partial \mathit{y}}$, ${\mathit{P}}_{\mathrm{3}}(\mathit{x})=\frac{\partial \mathit{h}\left(\mathit{x},\mathit{y}\right)}{\partial \mathit{x}}$, ${\mathit{P}}_{\mathrm{4}}(\mathit{x})=\frac{\partial \mathit{h}\left(\mathit{x},\mathit{y}\right)}{\partial \mathit{y}}$.

Let

$\mathit{M}=\left[\begin{array}{ccc}{k}_{\mathrm{11}}+\mathrm{3}{k}_{\mathrm{21}}{\left(x_{\mathrm{1}}-X_{i\mathrm{1}}^e\right)}^{\mathrm{2}}& \cdots & \mathrm{0}\\ ⋮& \ddots & ⋮\\ \mathrm{0}& \cdots & k_{\mathrm{1}m}+\mathrm{3}{k}_{\mathrm{2}m}{\left(x_m-X_{im}^e\right)}^{\mathrm{2}}\end{array}\right],\mathit{N}=\left[\begin{array}{ccc}{k}_{\mathrm{31}}& \cdots & \mathrm{0}\\ ⋮& \ddots & ⋮\\ \mathrm{0}& \cdots & k{}_{\mathrm{3}m}{}^{}\end{array}\right].$

Then

${\mathit{P}}_{\mathrm{1}}(x)=\left[\begin{array}{cc}\mathit{M}& {\mathrm{0}}_{(n-m)\times (n-m)}\\ {\mathrm{0}}_{(n-m)\times (n-m)}& {\mathrm{0}}_{(n-m)\times (n-m)}\end{array}\right],{\mathit{P}}_{\mathrm{2}}(x)=\left[\begin{array}{c}\mathit{N}\\ {\mathrm{0}}_{(n-m)\times (n-m)}\end{array}\right],{\mathit{P}}_{\mathrm{3}}(x)=\left[\begin{array}{cc}\mathit{M}& {\mathrm{0}}_{(n-m)\times (n-m)}\end{array}\right],{\mathit{P}}_{\mathrm{4}}(x)=\mathit{N}.$

Therefore, the characteristic equation of the controlled system (22) at the equilibrium point $E_i$ is defined as

$Q\left(\lambda ,{\mu }_e\right)=q_{\mathrm{0}}\left({\mu }_e\right){\lambda }^{n+m}+q_{\mathrm{1}}\left({\mu }_e\right){\lambda }^{n+m-\mathrm{1}}+\cdots +q_{n+m}\left({\mu }_e\right)$(24)

The Routh-Hurwitz determinant is constructed from the coefficients of the characteristic (24). Its form is as follows:

${\mathit{H}}_{n+m}=\left[\begin{array}{cccc}{q}_{\mathrm{1}}\left({\mu }_e\right)& q_{\mathrm{0}}\left({\mu }_e\right)& \cdots & \mathrm{0}\\ q_{\mathrm{3}}\left({\mu }_e\right)& q_{\mathrm{2}}\left({\mu }_e\right)& \cdots & \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮\\ q_{\mathrm{2}(n+m)-\mathrm{1}}\left({\mu }_e\right)& q_{\mathrm{2}\left(n+m\right)-\mathrm{2}}\left({\mu }_e\right)& \cdots & {q_n}_{+m}\left({\mu }_e\right)\end{array}\right].$

According to Lemma 1 and Lemma 2, we can obtain Theorem 1. The values of controller parameters are determined by the following formula.

${\mathit{H}}_{n+m-\mathrm{1}}\left({\mu }_e\right)=\mathrm{0}$

Theorem 1   When the real coefficient equation satisfies $q_i\left({\mu }_e\right)>\mathrm{0}\left(i=\mathrm{0,1},\mathrm{2},\cdots ,n\right)$, $H_j\left({\mu }_e\right)>\mathrm{0}\left(j=n+m-\mathrm{3},n+m-\mathrm{5},\cdots \right)$, and ${\mathit{H}}_{n+m-\mathrm{1}}\left({\mu }_e\right)=\mathrm{0}$, the system has a pair of conjugate pure imaginary roots, and the real parts of other roots are negative, then the controlled system generates Hopf bifurcation.

3.2 Anti-Control of Hopf Bifurcation

In order to achieve the goal of anti-control of Hopf bifurcation, a nonlinear controller is set up for the system (1). The controlled system is expressed as follows:

$\{\begin{array}{l}\dot{\mathit{x}}=az,\\ \dot{\mathit{y}}=b\mathrm{s}\mathrm{i}\mathrm{n}\left[c\left(x-y\right)\right]+k_{\mathrm{1}}\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)+k_{\mathrm{2}}{\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)}^{\mathrm{3}}-k_{\mathrm{3}}v,\\ \dot{\mathit{z}}=d-xy,\\ \dot{\mathit{u}}=e-yu,\\ \dot{\mathit{v}}=k_{\mathrm{1}}\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)+k_{\mathrm{2}}{\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)}^{\mathrm{3}}-k_{\mathrm{3}}v,\end{array}$(25)

where $v$ is the newly added system state variable. The structure of the equilibrium point $E_{\mathrm{1}}^{\text{'}}$ of the controlled system is not changed. $E_{\mathrm{1}}^{\text{'}}$ is still $\left(\frac{p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\mathrm{0},\frac{e\left(p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}\right)}{\mathrm{2}d},\mathrm{0}\right)$.

When $k=\mathrm{1}$, we can obtain the following matrices with the state feedback control method.

$\mathit{J}\left(E_{\mathrm{1}}\right)=\left[\begin{array}{cccc}\mathrm{0}& \mathrm{0}& a& \mathrm{0}\\ -bc& bc& \mathrm{0}& \mathrm{0}\\ -\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}& -\frac{p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\frac{e\left(p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}\right)}{\mathrm{2}d}& \mathrm{0}& -\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}\end{array}\right],{\mathit{P}}_{\mathrm{1}}(x)=\left[\begin{array}{cccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& k_{\mathrm{1}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right],{\mathit{P}}_{\mathrm{2}}(x)=\left[\begin{array}{c}\mathrm{0}\\ k_{\mathrm{3}}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right],$

${\mathit{P}}_{\mathrm{3}}(x)=\left[\begin{array}{cccc}\mathrm{0}& k_{\mathrm{1}}& \mathrm{0}& \mathrm{0}\end{array}\right],{\mathit{P}}_{\mathrm{4}}(x)=k_{\mathrm{3}}.$

Using the anti-control of the Hopf bifurcation method, the Jacobi matrix ${\mathit{J}}_C\left(\mathit{X},\mu \right)$ of the controlled system (25) is

${\mathit{J}}_C\left(E_{\mathrm{1}}^{\text{'}}\right)=\left[\begin{array}{lllll}\mathrm{0}& \mathrm{0}& a& \mathrm{0}& \mathrm{0}\\ -bc& k_{\mathrm{1}}+bc& \mathrm{0}& \mathrm{0}& -k_{\mathrm{3}}\\ -\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}& -\frac{p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\frac{e\left(p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}\right)}{\mathrm{2}d}& \mathrm{0}& -\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}& \mathrm{0}\\ \mathrm{0}& k_{\mathrm{1}}& \mathrm{0}& \mathrm{0}& -k_{\mathrm{3}}\end{array}\right].$

The characteristic equation of the characteristic matrix is obtained.

${\lambda }^{\mathrm{5}}+p_{\mathrm{1}}{\lambda }^{\mathrm{4}}+p_{\mathrm{2}}{\lambda }^{\mathrm{3}}+p_{\mathrm{3}}{\lambda }^{\mathrm{2}}+p_{\mathrm{4}}\lambda +p_{\mathrm{5}}=\mathrm{0},$(26)

where

$\{\begin{array}{l}{p}_{\mathrm{1}}=-bc-k_{\mathrm{1}}+k_{\mathrm{3}}+\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}},\\ p_{\mathrm{2}}=-bc{k}_{\mathrm{3}}+\left(a-bc+k_{\mathrm{3}}-k_{\mathrm{1}}\right)\left(\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}\right),\\ p_{\mathrm{3}}=-abc\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}+\left(a{k}_{\mathrm{3}}-a{k}_{\mathrm{1}}-bc{k}_{\mathrm{3}}+\frac{-ap+a\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}\right)\left(\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}\right),\\ p_{\mathrm{4}}=-abc{k}_{\mathrm{3}}\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}-\mathrm{2}abcd-a\left(bc+k_{\mathrm{1}}-k_{\mathrm{3}}\right){\left(\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}\right)}^{\mathrm{2}},\\ p_{\mathrm{5}}=-abc{k}_{\mathrm{3}}\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}\left(\frac{-p+\sqrt[]{p^{\mathrm{2}}+\mathrm{4}d}}{\mathrm{2}}\right).\end{array}$

The Routh-Hurwitz determinant of the coefficients in equation (26) is constructed as follows:

${\mathrm{\Delta }}_{n+m}=\left|\begin{array}{ccccc}{p}_{\mathrm{1}}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ p_{\mathrm{3}}& p_{\mathrm{2}}& p_{\mathrm{1}}& \mathrm{1}& \mathrm{0}\\ p_{\mathrm{5}}& p_{\mathrm{4}}& p_{\mathrm{3}}& p_{\mathrm{2}}& p_{\mathrm{1}}\\ \mathrm{0}& \mathrm{0}& p_{\mathrm{5}}& p_{\mathrm{4}}& p_{\mathrm{3}}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& p_{\mathrm{5}}\end{array}\right|$

Since ${\mathrm{\Delta }}_{\mathrm{4}}=\mathrm{0}$, the control parameters of the controlled system (25) can be expressed as (27):

$p_{\mathrm{1}}{p}_{\mathrm{2}}{p}_{\mathrm{3}}{p}_{\mathrm{4}}+p_{\mathrm{3}}^{\mathrm{2}}{p}_{\mathrm{4}}-p_{\mathrm{1}}^{\mathrm{2}}{p}_{\mathrm{4}}^{\mathrm{1}}-p_{\mathrm{1}}{p}_{\mathrm{2}}^{\mathrm{2}}{p}_{\mathrm{5}}+p_{\mathrm{2}}{p}_{\mathrm{3}}{p}_{\mathrm{5}}+\mathrm{2}{p}_{\mathrm{1}}{p}_{\mathrm{4}}{p}_{\mathrm{5}}-p_{\mathrm{5}}^{\mathrm{2}}=\mathrm{0}$(27)

Eq. (27) contains the system's real parameters $a,b,c,d$ and control parameters $k_{\mathrm{1}},k_{\mathrm{3}}$. The relationship between the control parameters $k_{\mathrm{1}}$ and $k_{\mathrm{3}}$ can be obtained by extrapolating from Eq. (27). It can be seen that the critical value of Hopf bifurcation is determined by the control parameters of the linear controller, and is independent of the control parameters of the nonlinear control. Therefore, under the given system real parameters $a,b,c,d$, and the value of any parameter between control parameters $k_{\mathrm{1}}$ and $k_{\mathrm{3}}$, another control parameter value can be determined. The anti-control of the Hopf bifurcation of the system is realized. The anti-control of Hopf bifurcation is achieved.

We will provide an example to achieve anti-control of Hopf bifurcation in a 4D hyperchaotic system with infinite equilibrium points and coexisting attractors, and demonstrate the results.

We set $a=\mathrm{10},b=-\mathrm{3},c=-\mathrm{10},d=\mathrm{20},e=\mathrm{25}$. The controlled system (25) can be expressed as (28).

$\{\begin{array}{l}\dot{x}=\mathrm{10}z,\\ \dot{y}=-\mathrm{3}\mathrm{s}\mathrm{i}\mathrm{n}\left[-\mathrm{10}\left(x-y\right)\right]+k_{\mathrm{1}}\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)+k_{\mathrm{2}}{\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)}^{\mathrm{3}}-k_{\mathrm{3}}v,\\ \dot{z}=\mathrm{20}-xy,\\ \dot{u}=\mathrm{25}-yu,\\ \dot{v}=k_{\mathrm{1}}\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)+k_{\mathrm{2}}{\left(y-y_{E_{\mathrm{1}}}^{\mathrm{*}}\right)}^{\mathrm{3}}-k_{\mathrm{3}}v.\end{array}$(28)

When $k=\mathrm{1}$, then $p=-\frac{\mathrm{\pi }}{\mathrm{10}}$. We get

$\{\begin{array}{l}{p}_{\mathrm{1}}=-\mathrm{25.368}\mathrm{1}-k_{\mathrm{1}}+k_{\mathrm{3}},\\ p_{\mathrm{2}}=-\mathrm{92.638}-\mathrm{4.631}\mathrm{9}{k}_{\mathrm{1}}-\mathrm{25.368}\mathrm{1}{k}_{\mathrm{3}},\\ p_{\mathrm{3}}=-\mathrm{2}\mathrm{470.37}-\mathrm{46.319}{k}_{\mathrm{1}}-\mathrm{93.638}{k}_{\mathrm{3}},\\ p_{\mathrm{4}}=-\mathrm{12}\mathrm{436.2}-\mathrm{214.545}{k}_{\mathrm{1}}-\mathrm{2}\mathrm{470.37}{k}_{\mathrm{3}},\\ p_{\mathrm{5}}=-\mathrm{12}\mathrm{436.2}{k}_{\mathrm{3}}.\end{array}$

If (27) satisfies the conditional constraints, the range of values for $k_{\mathrm{1}}$ and $k_{\mathrm{3}}$ is as follows:

$k_{\mathrm{1}}<-\mathrm{50.297}\mathrm{75}{k}_{\mathrm{3}},k_{\mathrm{3}}<\mathrm{0}.$

When $k_{\mathrm{3}}=-\mathrm{1.5}$, we can obtain $k_{\mathrm{1}}=-\mathrm{58.885}\mathrm{6},-\mathrm{40.370}\mathrm{3,44.249}\mathrm{6}$. Since $k_{\mathrm{1}}<-\mathrm{50.297}\mathrm{75}$, we choose $k_{\mathrm{1}}=-\mathrm{58.885}\mathrm{6}$. With $a=\mathrm{10},b=-\mathrm{3},c=-\mathrm{10},d=\mathrm{20},e=\mathrm{25}$ and the initial condition of $\left(-\mathrm{1,3},-\mathrm{1},-\mathrm{1},-\mathrm{1}\right)$, the waveform plot and phase portrait of the controlled system (25) are presented in Fig. 3.

Fig. 3 Waveform plot and phase portrait of the controlled system (12) at $\mathit{a}=\mathrm{10},\mathit{b}=\mathrm{3},\mathit{c}=\mathrm{10},\mathit{d}=\mathrm{20},\mathit{e}=\mathrm{25},\mathit{k}=\mathrm{10}$

From Fig. 3, it can be seen that the controlled system is gradually stabilizing. Figure 3(a) shows the trend of the system state variable $x$ over time and Fig. 3(b-d) show the phase diagram of the system in space $o-xy$, $o-xyz$ and $o-xyu$, respectively. Figure 3 verifies the effect of anti-control of Hopf bifurcation for the controlled system (18) using the dynamic state feedback control method.

When other parameters remain unchanged, the initial values of the controlled system are changed to $\left(-\mathrm{1,0},\mathrm{0,0},\mathrm{0}\right)$ and $\left(-\mathrm{3,0},\mathrm{0,0},\mathrm{0}\right)$. The phase diagram of the controlled system at different initial values is shown in Fig. 4.

Fig.4 Phase portrait of the controlled system (25) at the different initial values

As can be seen from Fig. 4, when the system adjustment parameters remain unchanged and the initial values change, the system is still in a gradually stable state. The dynamic characteristics of the controlled system cannot be changed by different initial values.

The nonlinear control parameter $k_{\mathrm{2}}$ in the controller cannot change the Hopf bifurcation characteristics of the system. Therefore, we did not analyze the impact of $k_{\mathrm{2}}$ transformation on the system.

Through analysis, it can be seen that the state feedback control method is effective for a chaotic system. The process is simple, avoiding complex calculations.

4 Conclusion

This paper mainly studies a chaotic system with countless equilibrium points, analyzes the coexisting attractors and Hopf bifurcation characteristics of the system, and tests their existence. According to the Hopf bifurcation control theory, a dynamic state feedback control method is proposed, which can be used for the anti-control of Hopf bifurcation. By adjusting the control parameters, Hopf bifurcation can be achieved at any critical value, causing Hopf bifurcation to advance or delay, and making the system transition from unstable to a stable state, thereby achieving the anti-control of Hopf bifurcation in the system.

References

All Figures

	Fig. 1 Waveform plot and phase portrait of the system (1) at $\mathit{a}=\mathrm{10},\mathit{b}=\mathrm{3},\mathit{c}=\mathrm{10},\mathit{d}=\mathrm{20},\mathit{e}=\mathrm{25},\mathit{k}=\mathrm{10}$
In the text

	Fig. 2 Hidden attractors symmetry
In the text

	Fig. 3 Waveform plot and phase portrait of the controlled system (12) at $\mathit{a}=\mathrm{10},\mathit{b}=\mathrm{3},\mathit{c}=\mathrm{10},\mathit{d}=\mathrm{20},\mathit{e}=\mathrm{25},\mathit{k}=\mathrm{10}$
In the text

	Fig.4 Phase portrait of the controlled system (25) at the different initial values
In the text