Zero-Hopf Bifurcations of 3D Quadratic Jerk System

Abstract

This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf bifurcations of canonical system, which constitutes the core contributions of this paper. By averaging theory of first order, we prove that, at most, one limit cycle bifurcates from the zero-Hopf equilibrium. By averaging theory of second order, third order, and fourth order, we show that, at most, two limit cycles bifurcate from the equilibrium. Overall, this paper can help to increase our understanding of local behaviour in the jerk dynamical system with quadratic non-linearity.

1. Introduction

Consider the following system of ordinary differential equations

$$\frac{dx}{dt}=f(x,\mu ),x\in {\mathbb{R}}^n,\mu \in {\mathbb{R}}^s,$$

(1)

where f is sufficiently smooth. Let us assume that $x=0,\mu =0$ is a bifurcation point of the system. The corresponding linear variational equation is

$$\frac{dx}{dt}=Ax,A=Df(0,0),$$

where $Df$ denotes the Jacobian matrix of the vector field f.

When $n=3$, system (1) is of particular interest for bifurcation analysis and chaos, see [1]. Assume that A has a pair of purely imaginary eigenvalues ${\lambda }_{1,2}=\pm \omega \mathrm{i}$. The other eigenvalue ${\lambda }_3$ must be real. Thus, limit cycles may be found from the system under appropriate conditions. Recall that a limit cycle is an isolated closed orbit in the set of all periodic orbits of the system. In the case of ${\lambda }_3=0$, small-amplitude limit cycles may be found in some neighborhood of the origin. This phenomenon is called the zero-Hopf bifurcation.

A jerk equation is a differential equation of the form

$${\displaystyle \frac{d^{3}x}{d{t}^3}}=J\left(x,{\displaystyle \frac{dx}{dt}},{\displaystyle \frac{d^{2}x}{d{t}^2}}\right).$$

Letting $y={\displaystyle \frac{dx}{dt}},z={\displaystyle \frac{d^{2}x}{d{t}^2}}$, the jerk equation can be transformed into

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& J(x,y,z),\hfill \end{array}\right.$$

(2)

which is called the jerk system. In physics, the first three derivatives ${\displaystyle \frac{dx}{dt}},{\displaystyle \frac{d^{2}x}{d{t}^2}},{\displaystyle \frac{d^{3}x}{d{t}^3}}$ are called velocity, acceleration, and jerk, respectively. The system can exhibit both regular and irregular or chaotic dynamical behaviour. It is shown in [2] that both Lorenz system and Rössler system could be written in jerk form.

Consider the general three-dimensional (3D) quadratic jerk system

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& a_0+a_{1}x+a_{2}y+a_{3}z+a_4{x}^2+a_{5}yx+a_{6}zx+a_7{y}^2+a_{8}zy+a_9{z}^2,\hfill \end{array}\right.$$

(3)

which has attracted intense interest. Some examples of Hopf bifurcation analysis can be found in [3,4]. Some examples of zero-Hopf bifurcation analysis can be found in [5,6,7]. Complex dynamics, such as self-excited and hidden chaotic attractors, can be found in [7,8,9]. In these studies, the qualitative features of equilibria play an important role in determining the complex behaviour of the system. For convenience, we call this system the general system from now on.

The rest of the paper is organized, as follows. In Section 2, we study the presence of saddle-node bifurcation in the general system. The next section is about the canonical system, which will play an important role in bifurcation analysis. In Section 4 and Section 5, for the canonical system, transcritical bifurcation and zero-Hopf bifurcations are studied in detail.

2. Saddle-Node Bifurcation

Lemma 1.

Consider the quadratic function $\phi \left(\lambda \right)={\lambda }^2+s_1\lambda +s_2$. It has all roots with non-zero real parts if and only if

$$s_1^2-4s_2>0,s_2\ne 0;$$

or

$$s_1^2-4s_2\le 0,s_1\ne 0.$$

Proof. 

Let $\Delta =s_1^2-4s_2$. There are three possibilities:

(1)

If $\Delta >0$, then the function has two distinct real roots ${\lambda }_1,{\lambda }_2$. Because ${\lambda }_1{\lambda }_2=s_2$, both roots are non-zero if and only if $s_2\ne 0$.

(2)

If $\Delta =0$, then the function has one real double root ${\lambda }_0=-{\displaystyle \frac{s_1}{2}}$. The root is non-zero if and only if $s_1\ne 0$.

(3)

If $\Delta <0$, then the function has two complex conjugated roots ${\lambda }_1,{\lambda }_2$. Because the real parts of these roots are $-{\displaystyle \frac{s_1}{2}}$, $\phi \left(\lambda \right)$ has no roots with zero real parts if and only if $s_1\ne 0$.

Summing up, we have proved this lemma.  □

Lemma 2

(Sotomayor’s theorem ([10], page 338–339)). Consider system (1) with $s=1$. When $\mu ={\mu }_0$, assume that there is an equilibrium $x_0$, for which the following hypotheses are satisfied:

(a) 

The Jacobian matrix $M=Df(x_0,{\mu }_0)$ has a simple eigenvalue $\lambda =0$ with an eigenvector v, and $M^T$ has an eigenvector w corresponding to $\lambda =0$.

(b) 

M has k eigenvalues with negative real parts, and $n-1-k$ eigenvalues with positive real parts, where $1\le k\le n-1$.

(c) 

$\alpha \triangleq w^T{f}_{\mu }(x_0,{\mu }_0)\ne 0$.

(d) 

$\beta \triangleq w^T\left[D^{2}f\left(x_0,{\mu }_0\right)(v,v)\right]\ne 0$.

Subsequently, system (1) exhibits a saddle-node bifurcation at $x_0$ as μ passes through $\mu ={\mu }_0$.

(1) 

There is a smooth curve of equilibria in ${\mathbb{R}}^n\times R$ passing through $(x_0,{\mu }_0)$ and tangent to the hyperplane ${\mathbb{R}}^n\times \left\{{\mu }_0\right\}$.

(2) 

If $\alpha \beta <0$ (resp. $\alpha \beta >0$), there are no equilibria near $x_0$ when $\mu <{\mu }_0$ (resp. $\mu >{\mu }_0$), and two equilibria near $x_0$ when $\mu >{\mu }_0$ (resp. $\mu <{\mu }_0$).

(3) 

The two equilibria near $x_0$ are hyperbolic and they have stable manifolds of dimensions k and $k+1$, respectively.

Let $\mu =a_1^2-4a_0{a}_4$ with $a_4\ne 0$, then system (3) becomes

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& \frac{{a_1}^2-\mu }{4a_4}+a_{1}x+a_{2}y+a_{3}z+a_4{x}^2+a_{5}yx+a_{6}zx+a_7{y}^2+a_{8}zy+a_9{z}^2.\hfill \end{array}\right.$$

(4)

It has two equilibria if $\mu >0$; one equilibrium at $E=(-\frac{a_1}{2a_4},0,0)$ if $\mu =0$; and, no equilibria if $\mu <0$.

Lemma 3.

For system (4), a saddle-node bifurcation occurs at E as the parameter μ passes through $\mu =0$.

Proof. 

When $\mu =0$, the Jacobian matrix of system (4) at the equilibrium E is

$$M:=Df(E,0)=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& a_2-\frac{a_1{a}_5}{2a_4}& a_3-\frac{a_1{a}_6}{2a_4}\end{array}\right).$$

The matrix $M={\left(m_{ij}\right)}_{3\times 3}$ has a simple eigenvalue ${\lambda }_1=0$ with eigenvectors $v={(1,0,0)}^T$, and the matrix $M^T$ has an eigenvector

$$w=\left(\begin{array}{c}-m_{32}\\ -m_{33}\\ 1\end{array}\right).$$

corresponding to ${\lambda }_1=0$.

The other eigenvalues ${\lambda }_{2,3}$ of M are the roots of $\phi \left(\lambda \right):={\lambda }^2+s_1\lambda +s_2$, where

$$s_1=-m_{33},s_2=-m_{32}.$$

Note that $m_{32}$ depends on $a_2$, and $m_{33}$ depends on $a_3$. Thus, according to Lemma 1, we can chose $a_2,a_3$, such that the eigenvalues ${\lambda }_{2,3}$ can not have zero real parts.

Following Lemma 2, we have

$$\begin{array}{ccc}\hfill \alpha & =& w^T{f}_{\mu }(E,0)=-{\displaystyle \frac{1}{4a_4}}\ne 0,\hfill \\ \hfill \beta & =& w^T\left[D^{2}f\left(E,0\right)(v,v)\right]=2a_4\ne 0.\hfill \end{array}$$

Therefore, system (4) experiences a saddle-node bifurcation at the equilibrium E as the parameter $\mu$ passes through $\mu =0$. Because $\alpha \beta <0$, there are no equilibria near E when $\mu <0$, and two equilibria near E when $\mu >0$.  □

Theorem 1.

For the jerk system (3), a saddle-node bifurcation occurs at E as the parameter $a_0$ passes through $a_0={\displaystyle \frac{a_1^2}{4a_4}}$.

Proof. 

Recall that $\mu =a_1^2-4a_0{a}_4$, thus $a_0={\displaystyle \frac{a_1^2}{4a_1}}-{\displaystyle \frac{\mu }{4a_4}}$. This theorem is a direct consequence of Lemma 3.  □

3. Canonical System

Theorem 2.

Suppose that system (3) has an equilibrium, then it can be transformed into the following system

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& \left(c_{1}y-c_{2}z\right)+c_3\left(x+z\right)+c_4\left(y^2+z^2\right)+c_5\left(y^2-z^2\right)-2c_{6}yz\hfill \\ \hfill & +\left(c_{7}y-c_{8}z\right)\left(x+z\right)+c_9{\left(x+z\right)}^2,\hfill \end{array}\right.$$

(5)

where the coefficients $c_i,i=1,2,\cdots ,9$ can be derived from the original system and equilibrium.

Proof. 

Assume that $E:(x_0,0,0)$ is the equilibrium. Applying the translation $x\to x+x_0,y\to y,z\to z$, the system becomes

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& \left(2a_4{x}_0+a_1\right)x+\left(a_5{x}_0+a_2\right)y+\left(a_6{x}_0+a_3\right)z+a_4{x}^2+a_{5}xy+a_{6}xz\hfill \\ \hfill & +a_7{y}^2+a_{8}yz+a_9{z}^2.\hfill \end{array}\right.$$

(6)

After a little algebra, it is easy to see that system (6) can be arranged as system (5) with

$$\begin{array}{ccc}\hfill c_1& =& a_5{x}_0+a_2,\hfill \\ \hfill c_2& =& 2a_4{x}_0-a_6{x}_0+a_1-a_3,\hfill \\ \hfill c_3& =& 2a_4{x}_0+a_1,\hfill \\ \hfill c_4& =& {\displaystyle \frac{a_4-a_6+a_7+a_9}{2}},\hfill \\ \hfill c_5& =& {\displaystyle \frac{a_6+a_7-a_4-a_9}{2}},\hfill \\ \hfill c_6& =& {\displaystyle \frac{a_5-a_8}{2}},\hfill \\ \hfill c_7& =& a_5,\hfill \\ \hfill c_8& =& -a_6+2a_4,\hfill \\ \hfill c_9& =& a_4.\hfill \end{array}$$

Thus we complete the proof.  □

For convenience, from now on, we call system (5) the canonical system.

4. Transcritical Bifurcation

Lemma 4

(Sotomayor’s theorem ([10], page 338–339)). Consider system (1) with $s=1$ and assume that there is a point $x_0\in {\mathbb{R}}^n$, such that $f(x_0,\mu )=0$ for all μ. Furthermore, when $\mu ={\mu }_0$ suppose that the following hypotheses hold:

(a) 

The Jacobian matrix $M=Df(x_0,{\mu }_0)$ has a simple eigenvalue $\lambda =0$ with an eigenvector v, and $M^T$ has an eigenvector w corresponding to $\lambda =0$.

(b) 

M has k eigenvalues with negative real parts, and $n-1-k$ eigenvalues with positive real parts, where $1\le k\le n-1$.

(c) 

$w^T{f}_{\mu }(x_0,{\mu }_0)=0$.

(d) 

$w^T\left[D{f}_{\mu }\left(x_0,{\mu }_0\right)v\right]\ne 0$.

(e) 

$w^T\left[D^{2}f\left(x_0,{\mu }_0\right)(v,v)\right]\ne 0$.

Subsequently, system (1) exhibits a transcritical bifurcation at the equilibrium $x_0$ as μ passes through $\mu =0$.

Theorem 3.

Consider the canonical system (5) with $c_9\ne 0$, a transcritical bifurcation occurs at the origin as $c_3$ passes through $c_3=0$.

Proof. 

When $c_3=0$, the Jacobian matrix of this system at the origin is

$$Df(0,0)=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& c_1& -c_2\end{array}\right).$$

It has a simple eigenvalue ${\lambda }_1=0$ with eigenvector $v={(1,0,0)}^T$, and its transpose has an eigenvector

$$w=\left(\begin{array}{c}-c_1\\ c_2\\ 1\end{array}\right).$$

corresponding to ${\lambda }_1=0$.

The other eigenvalues ${\lambda }_{2,3}$ of the matrix are the roots of $\phi \left(\lambda \right):={\lambda }^2+c_2\lambda -c_1$. Thus, according to Lemma 1, one can choose $c_1,c_2$, such that these eigenvalues cannot have zero real parts.

Following Lemma 4, we have

$$\begin{array}{ccc}\hfill w^T{f}_{c_3}(0,0)& =& 0,\hfill \\ \hfill w^T\left[D{f}_{c_3}\left(0,0\right)v\right]& =& 1,\hfill \\ \hfill w^T\left[D^{2}f\left(0,0\right)(v,v)\right]& =& 2c_9\ne 0.\hfill \end{array}$$

Therefore, the canonical system experiences a transcritical bifurcation at the origin as $c_3$ passes through $c_3=0$.  □

5. Zero-Hopf Bifurcations

For the canonical system, we are interested in the number of small limit cycles bifurcate from the zero-Hopf equilibrium. There is no general theory of this problem. By perturbing an equilibrium inside the canonical system and using averaging theory up to fourth order, we give a partial answer to the problem. For averaging theory of higher order, see Appendix A.

5.1. The Perturbed System in Cartesian Coordinates

The Jacobian matrix of canonical system at the origin is

$$Df\left(0\right)=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ c_3& c_1& -c_2+c_3\end{array}\right),$$

whose characteristic polynomial is

$$\phi (\lambda ;c_1,c_2,c_3):={\lambda }^3+\left(c_2-c_3\right){\lambda }^2-c_1\lambda -c_3$$

(7)

Let $I_1(c_1,c_2,c_3)=c_2-c_3,I_2(c_1,c_2,c_3)=-c_1,I_3(c_1,c_2,c_3)=-c_3$ be some coefficients of the polynomial in $\lambda$.

Recall that a zero-Hopf equilibrium of a 3D system is an isolated equilibrium of the system, whose linear part at the equilibrium has a pair of purely imaginary eigenvalues and a zero eigenvalue. In the next lemma, we characterize when the equilibrium localized at the origin of canonical system is a zero-Hopf equilibrium.

Lemma 5.

For the canonical system, the origin is a zero-Hopf equilibrium if and only if

$$c_2=c_3=0,c_1<0.$$

(8)

Proof. 

Suppose that the origin is a zero-Hopf equilibrium of the canonical system. Thus, the polynomial (7) has a pair of purely imaginary roots and a zero root. According to Proposition 6 in [11], we have

$$I_1(c_1,c_2,c_3)=I_3(c_1,c_2,c_3)=0,I_2(c_1,c_2,c_3)>0,$$

which is equivalent to (8). Hence, we have proved this lemma.  □

Lemma 6.

Suppose that the canonical system has a zero-Hopf equilibrium at the origin. Subsequently, it can be transformed into the following system

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& -y+b_4\left(y^2+z^2\right)+b_5\left(y^2-z^2\right)-2b_{6}yz+\left(b_{7}y-b_{8}z\right)\left(x+z\right)+b_9{\left(x+z\right)}^2,\hfill \end{array}\right.$$

(9)

where $b_i,i=4,5,\cdots ,9$ depend on $c_i,i=1,4,5,\cdots ,9$.

Proof. 

Suppose that the canonical system has a zero-Hopf equilibrium at the origin. According to Lemma 5, we have

$$c_2=c_3=0,c_1<0.$$

Setting $\omega =\sqrt{-c_1}$, the canonical system becomes

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& -{\omega }^{2}y+c_4\left(y^2+z^2\right)+c_5\left(y^2-z^2\right)-2c_{6}yz+\left(c_{7}y-c_{8}z\right)\left(x+z\right)+c_9{\left(x+z\right)}^2.\hfill \end{array}\right.$$

(10)

With the following linear scaling

$$x\to {\displaystyle \frac{1}{\omega }}x,z\to \omega z,t\to {\displaystyle \frac{1}{\omega }}t,$$

(11)

system (10) becomes

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& -y+\mathrm{quadratic} \mathrm{terms}.\hfill \end{array}\right.$$

(12)

Similar to the proof of Theorem 2, it is easy to see that system (12) can be arranged as system (9). Hence, we have completed the proof.  □

Consider the following system

$$\left\{\begin{array}{cc}\hfill \frac{dx}{dt}=& y,\hfill \\ \hfill \frac{dy}{dt}=& z,\hfill \\ \hfill \frac{dz}{dt}=& -y+f(x,y,z)+\sum _{k=1}^3{\epsilon }^k\left(g_{2k-1}(x,y,z)+g_{2k}(x,y,z)\right)+{\epsilon }^4{g}_7(x,y,z)\hfill \\ \hfill & +{\epsilon }^4{\mathcal{O}\left(\right|x,y,z|}^2)+\mathcal{O}\left({\epsilon }^5\right),\hfill \end{array}\right.$$

(13)

where

$$\begin{array}{ccc}\hfill f(x,y,z)& =& b_4(y^2+z^2)+b_5(y^2-z^2)-2b_{6}yz+(b_{7}y-b_{8}z)(x+z)+b_9{(x+z)}^2,\hfill \\ \hfill g_1(x,y,x)& =& (b_{1}y-b_{2}z)+b_3(x+z),\hfill \\ \hfill g_2(x,y,z)& =& b_{10}(y^2+z^2)+b_{11}(y^2-z^2)-2b_{12}yz+(b_{13}y-b_{14}z)(x+z)+b_{15}{(x+z)}^2,\hfill \\ \hfill g_3(x,y,x)& =& (b_{16}y-b_{17}z)+b_{18}(x+z),\hfill \\ \hfill g_4(x,y,z)& =& b_{19}(y^2+z^2)+b_{20}(y^2-z^2)-2b_{21}yz+(b_{22}y-b_{23}z)(x+z)+b_{24}{(x+z)}^2,\hfill \\ \hfill g_5(x,y,z)& =& (b_{25}y-b_{26}z)+b_{27}(x+z),\hfill \\ \hfill g_6(x,y,z)& =& b_{28}(y^2+z^2)+b_{29}(y^2-z^2)-2b_{30}yz+(b_{31}y-b_{32}z)(x+z)+b_{33}{(x+z)}^2,\hfill \\ \hfill g_7(x,y,z)& =& (b_{34}y-b_{35}z)+b_{36}(x+z),\hfill \end{array}$$

and $\epsilon$ is sufficiently small.

5.2. Linear Analysis

When $\epsilon =0$, then the Jacobian matrix of system (13) at the origin has three eigenvalues ${\lambda }_{1,2}=\pm \mathrm{i}$ and ${\lambda }_3=0$. Under appropriate conditions, several limit cycles can bifurcate from the origin of the system for $\left|\epsilon \right|>0$ sufficiently small. This type of bifurcation can be referred to as a zero-Hopf bifurcation.

The Jacobian matrix of system (13) at the origin has characteristic polynomial in $\lambda$:

$$\phi (\lambda ;\epsilon ):={\lambda }^3+\epsilon \left(b_2-b_3\right){\lambda }^2+\left(1-b_1\epsilon \right)\lambda -b_3\epsilon +\mathcal{O}\left({\epsilon }^2\right).$$

(14)

Let ${\lambda }_{1,2,3}\left(\epsilon \right)$ be the roots of this polynomial, ${\lambda }_{1,2}\left(\epsilon \right)$ be the continuous extension of $\pm \mathrm{i}$, and ${\lambda }_3\left(\epsilon \right)$ be the continuous extension of 0.

Lemma 7.

$${\displaystyle \frac{dRe{\lambda }_{1,2}}{d\epsilon }}{|}_{\epsilon =0}=-{\displaystyle \frac{b_2}{2}}.$$

Proof. 

By applying the implicit function theorem to (14), we have

$${\displaystyle \frac{d{\lambda }_{1,2}}{d\epsilon }}{|}_{\epsilon =0}=-\frac{\frac{\partial \phi }{\partial \epsilon }}{\frac{\partial \phi }{\partial \lambda }}{|}_{\epsilon =0,\lambda =\pm \mathrm{i}}=-{\displaystyle \frac{b_2}{2}}\mp \frac{b_1}{2}\mathrm{i}.$$

(15)

Taking the real part of (15), we obtain the conclusion.  □

Lemma 8.

$${\displaystyle \frac{d{\lambda }_3}{d\epsilon }}{|}_{\epsilon =0}=b_3.$$

Proof. 

By using the implicit function theorem, we have

$${\displaystyle \frac{d{\lambda }_3}{d\epsilon }}{|}_{\epsilon =0}=-\frac{\frac{\partial \phi }{\partial \epsilon }}{\frac{\partial \phi }{\partial \lambda }}{|}_{\epsilon =0,\lambda =0}=b_3.$$

(16)

Thus the conclusion follows.  □

The following theorem is about the stability of the origin for $\left|\epsilon \right|>0$ sufficiently small.

Theorem 4.

The origin is asymptotically stable if one of the following conditions holds:

(1) 

$\epsilon <0,b_2<0,b_3>0$;

(2) 

$\epsilon >0,b_2>0,b_3<0$,

where $\left|\epsilon \right|$ is sufficiently small.

Proof. 

Let us consider the first case. According to Lemmas 7 and 8, we have

$${\displaystyle \frac{dRe{\lambda }_{1,2}}{d\epsilon }}{|}_{\epsilon =0}>0,{\displaystyle \frac{d{\lambda }_3}{d\epsilon }}{|}_{\epsilon =0}>0.$$

Thus, for $\epsilon <0$ and near 0, we have

$$Re{\lambda }_{1,2}<0,{\lambda }_3<0.$$

Hence the origin is asymptotically stable.

The proof for the other case is similar. Thus, we complete the proof.  □

5.3. The Perturbed System in Cylindrical Coordinates

By using the linear transformation

$$\left\{\begin{array}{cc}\hfill x=& \epsilon v+\epsilon w,\hfill \\ \hfill y=& \epsilon u,\hfill \\ \hfill z=& -\epsilon v,\hfill \end{array}\right.$$

(17)

system (13) becomes

$$\left\{\begin{array}{cc}\hfill \frac{du}{dt}=& -v,\hfill \\ \hfill \frac{dv}{dt}=& u-G(\epsilon ,u,v,w),\hfill \\ \hfill \frac{dw}{dt}=& G(\epsilon ,u,v,w),\hfill \end{array}\right.$$

(18)

where

$$G(\epsilon ,u,v,w)=\sum _{k=1}^4{\epsilon }^k{G}_k(u,v,w)+\mathcal{O}\left({\epsilon }^5\right),$$

and

$$\begin{array}{ccc}\hfill G_1(u,v,w)& =\hfill & b_{1}u+b_{2}v+b_{3}w+b_4(u^2+v^2)+b_5(u^2-v^2)+2b_{6}uv+(b_{7}u+b_{8}v)w+b_9{w}^2,\hfill \\ \hfill G_2(u,v,w)& =\hfill & b_{16}u+b_{17}v+b_{18}w+b_{10}(u^2+v^2)+b_{11}(u^2-v^2)+2b_{12}uv+(b_{13}u+b_{14}v)w+b_{15}{w}^2,\hfill \\ \hfill G_3(u,v,w)& =\hfill & b_{25}u+b_{26}v+b_{27}w+b_{19}(u^2+v^2)+b_{20}(u^2-v^2)+2b_{21}uv+(b_{22}u+b_{23}v)w+b_{24}{w}^2,\hfill \\ \hfill G_4(u,v,w)& =\hfill & b_{34}u+b_{35}v+b_{36}w+b_{28}(u^2+v^2)+b_{29}(u^2-v^2)+2b_{30}uv+(b_{31}u+b_{32}v)w+b_{33}{w}^2.\hfill \end{array}$$

In cylindrical coordinates

$$u=rcos\theta ,v=rsin\theta ,w=w,$$

(19)

system (18) becomes

$$\left\{\begin{array}{cc}\hfill \frac{dr}{dt}=& -H(\epsilon ,\theta ,r,w)sin\theta ,\hfill \\ \hfill \frac{d\theta }{dt}=& 1-H(\epsilon ,\theta ,r,w)\frac{cos\theta }{r},\hfill \\ \hfill \frac{dw}{dt}=& H(\epsilon ,\theta ,r,w),\hfill \end{array}\right.$$

(20)

where $H(\epsilon ,\theta ,r,w)=G(\epsilon ,rcos\theta ,rsin\theta ,w)$. Thus

$$H(\epsilon ,\theta ,r,w)=\sum _{k=1}^4{\epsilon }^k{h}_k(\theta ,r,w)+\mathcal{O}\left({\epsilon }^5\right),$$

where

$$\begin{array}{ccc}\hfill h_1(\theta ,r,w)& =\hfill & (b_{1}cos\theta +b_{2}sin\theta )r+b_{3}w+(b_4+b_{5}cos2\theta +b_{6}sin2\theta )r^2+(b_{7}cos\theta +b_{8}sin\theta )rw+b_9{w}^2,\hfill \\ \hfill h_2(\theta ,r,w)& =\hfill & (b_{16}cos\theta +b_{17}sin\theta )r+b_{18}w+(b_{10}+b_{11}cos2\theta +b_{12}sin2\theta )r^2+(b_{13}cos\theta +b_{14}sin\theta )rw+b_{15}{w}^2,\hfill \\ \hfill h_3(\theta ,r,w)& =\hfill & (b_{25}cos\theta +b_{26}sin\theta )r+b_{27}w+(b_{19}+b_{20}cos2\theta +b_{21}sin2\theta )r^2+(b_{22}cos\theta +b_{23}sin\theta )rw+b_{24}{w}^2,\hfill \\ \hfill h_4(\theta ,r,w)& =\hfill & (b_{34}cos\theta +b_{35}sin\theta )r+b_{36}w+(b_{28}+b_{29}cos2\theta +b_{30}sin2\theta )r^2+(b_{31}cos\theta +b_{32}sin\theta )rw+b_{33}{w}^2.\hfill \end{array}$$

5.4. Standard Form of Fourth Order

Let $\mathit{x}=(r,w)$. Subsequently, in the region $r>0$, system (20) is equivalent to the following system

$$\frac{d\mathit{x}}{d\theta }=\sum _{k=1}^4{\epsilon }^k{F}_k(\theta ,\mathit{x})+\mathcal{O}\left({\epsilon }^5\right),$$

(21)

where $F_k$ and the remainder $\mathcal{O}\left({\epsilon }^5\right)$ are $2\pi$-periodic in the first variable $\theta$.

Let us write the vector functions $F_k(\theta ,\mathit{x})$ in terms of components, i.e.,

$$F_k(\theta ,\mathit{x})={\left(F_{k,1}(\theta ,r,w),F_{k,2}(\theta ,r,w)\right)}^{\mathrm{T}},k=1,2,3,4,$$

where

$$F_{k,1}(\theta ,r,w)=-sin\theta F_{k,2}(\theta ,r,w),$$

and

$$\begin{array}{ccc}\hfill F_{1,2}(\theta ,r,w)& =& h_1,\hfill \\ \hfill F_{2,2}(\theta ,r,w)& =& h_2+\frac{cos\theta }{r}{h}_1^2,\hfill \\ \hfill F_{3,2}(\theta ,r,w)& =& h_3+\frac{2cos\theta }{r}{h}_1{h}_2+{\left(\frac{cos\theta }{r}\right)}^2{h}_1^3,\hfill \\ \hfill F_{4,2}(\theta ,r,w)& =& h_4+\frac{cos\theta }{r}\left(h_2^2+2h_1{h}_3\right)+3{\left(\frac{cos\theta }{r}\right)}^2{h}_1^2{h}_2+{\left(\frac{cos\theta }{r}\right)}^3{h_1}^4.\hfill \end{array}$$

5.5. First Order Averaging

Let $b=(b_1,b_2,\cdots ,b_{36})$ and

$${\mathcal{C}}^{\left(1\right)}=\left\{b\in {\mathbb{R}}^{36}|b_2{b}_4\left(b_3{b}_8-b_2{b}_9\right)>0,b_8\ne 0\right\}.$$

Theorem 5.

Suppose that $b\in {\mathcal{C}}^{\left(1\right)}$. Subsequently, for any $\epsilon \ne 0$ sufficiently small, system (13) has one limit cycle (denote it by ${\Gamma }_{\epsilon }$) bifurcating from the origin.

Proof. 

Let us recall the averaging theory of higher order in the Appendix A. The first order averaged function of system (21) is

$$\begin{array}{ccc}\hfill f_1\left(\mathit{x}\right)& =& \frac{1}{2\pi }{\int }_0^{2\pi }{F}_1(\theta ,\mathit{x})d\theta \hfill \\ & =& \left(\begin{array}{c}-{\displaystyle \frac{\left(b_{8}w+b_2\right)}{2}}r\\ b_4{r}^2+b_9{w}^2+b_{3}w\end{array}\right).\hfill \end{array}$$

By solving $f_1\left(\mathit{x}\right)=0$, we obtain a unique solution $\mathit{x}=(r^{*},w^{*})$, with

$$r^{*}={\displaystyle \frac{1}{|b_8|}}\sqrt{{\displaystyle \frac{b_2\left(b_3{b}_8-b_2{b}_9\right)}{b_4}}}>0,w^{*}=-{\displaystyle \frac{b_2}{b_8}},$$

(22)

where $b\in {\mathcal{C}}^{\left(1\right)}$. By computation, we find that $det\left[D_{\mathit{x}}{f}_1(r^{*},w^{*})\right]={\displaystyle \frac{b_2(b_3{b}_8-b_2{b}_9)}{b_8}}\ne 0$. Thus, according to Theorem A1 of the Appendix A, for $\epsilon \ne 0$ sufficiently small, there exists a limit cycle $\left(r\right(\theta ,\epsilon ),w(\theta ,\epsilon \left)\right)$ for system (21), which converges to $(r^{*},w^{*})$ as $\epsilon \to 0$.

Going back through the transformations of coordinates (19) and (17), for $\epsilon \ne 0$ sufficiently small, system (13) has a limit cycle

$${\Gamma }_{\epsilon }:(x(\theta ,\epsilon ),y(\theta ,\epsilon ),z(\theta ,\epsilon ))=\left(\epsilon \left(r(\theta ,\epsilon )sin\theta +w(\theta ,\epsilon )\right),\epsilon r(\theta ,\epsilon )cos\theta ,-\epsilon r(\theta ,\epsilon )sin\theta \right)$$

near the origin.  □

Corollary 1.

Let

$$p\left(\epsilon \right)=\frac{b_3{b}_8-2b_2{b}_9}{b_8}\epsilon ,q=\frac{b_2(b_3{b}_8-b_2{b}_9)}{b_8}.$$

Assuming that the condition of Theorem 5 holds. Subsequently, for $\left|\epsilon \right|>0$ sufficiently small, we have the following results about the limit cycle ${\Gamma }_{\epsilon }$ in the Theorem 5.

(a) 

If $p\left(\epsilon \right)>0,q>0$, then the limit cycle is a local repellor.

(b) 

If $p\left(\epsilon \right)<0,q>0$, then the limit cycle is a local attractor.

(c) 

If $q<0$, then the limit cycle has two invariant manifolds, one stable and the other unstable, which are locally formed by two two-dimensional cylinders.

Proof. 

The Jacobian matrix of the first order averaged system

$${\displaystyle \frac{d\mathit{x}}{dt}}=\epsilon f_1\left(\mathit{x}\right)$$

at the equilibrium $(r^{*},w^{*})$ is

$$J:=\left(\begin{array}{cc}0& -{\displaystyle \frac{b_8{r}^{*}}{2}}\epsilon \\ 2b_4{r}^{*}\epsilon & p\left(\epsilon \right)\end{array}\right).$$

In view of the expression for $r^{*}$ in (22), the trace and determinant of J are $p\left(\epsilon \right)$ and $q{\epsilon }^2$, respectively. Therefore, we can determine the stability and instability of the equilibrium $(r^{*},w^{*})$. Thus, according to Llibre et al. [12] (Theorem 1.2.1), the three statements of this Corollary 1 hold. This completes the proof.  □

To end this subsection, let us introduce a notation for later use. Let D is a non-empty subset of ${\mathbb{R}}^m$. Suppose that $g:D\to {\mathbb{R}}^n$ is a function, depending on a parameter $b\in \Lambda \subset {\mathbb{R}}^s$. Let ${\Lambda }_0$ be a non-empty subset of $\Lambda$, we can use $g{|}_{{\Lambda }_0}\left(\mathit{x}\right)$ to denote the function of g restricted to the parameter domain ${\Lambda }_0$. It is acceptable to omit the subscript if there is no risk of confusion.

For an example, let

$$\begin{array}{ccc}\hfill {\mathcal{B}}^{\left(1\right)}& =& \left\{b\in {\mathbb{R}}^{36}|f_1\left(\mathit{x}\right)=f_1(r,w)\equiv 0\right\}\hfill \\ & =& \left\{b\in {\mathbb{R}}^{36}|b_2=b_3=b_4=b_8=b_9=0\right\}.\hfill \end{array}$$

Subsequently, we have $f_1{|}_{{\mathcal{B}}^{\left(1\right)}}\left(\mathit{x}\right)\equiv 0$.

5.6. Second Order Averaging

Let

$$\left\{\begin{array}{cc}\hfill {\alpha }_0& =-\frac{b_{17}\left(b_{14}{b}_{18}-b_{15}{b}_{17}\right)}{{b_{14}}^2},\hfill \\ \hfill {\alpha }_2& =\frac{2b_1{b}_5{b_{14}}^2-2b_5{b}_7{b}_{14}{b}_{17}-b_5{b}_7{b}_{14}{b}_{18}+2b_5{b}_7{b}_{15}{b}_{17}+4b_{10}{b_{14}}^2}{4{b_{14}}^2},\hfill \\ \hfill {\alpha }_4& =\frac{{b_5}^2{b_7}^2\left(b_{15}-2b_{14}\right)}{16{b_{14}}^2}.\hfill \end{array}\right.$$

(23)

Define

$${\mathcal{C}}^{\left(2\right)}=\left\{b\in {\mathcal{B}}^{\left(1\right)}|{\alpha }_2{\alpha }_4<0,{\alpha }_0{\alpha }_4>0,{\alpha }_2^2-4{\alpha }_0{\alpha }_4>0,b_{14}\ne 0\right\}.$$

Lemma 9.

The functions ${\alpha }_0,{\alpha }_2,{\alpha }_4$ are linearly independent.

Proof. 

By direct computation, we have

$$det\left({\displaystyle \frac{\partial \left({\alpha }_0,{\alpha }_2,{\alpha }_4\right)}{\partial \left(b_5,b_{10},b_{15}\right)}}\right)\not\equiv 0.$$

Thus, these functions are linearly independent.  □

Theorem 6.

Assume that $b\in {\mathcal{B}}^{\left(1\right)}$. Subsequently, for $\left|\epsilon \right|>0$ sufficiently small, system (13) can have, at most, two limit cycles by averaging theory of second order. Moreover, if $b\in {\mathcal{C}}^{\left(2\right)}$, then the bound can be reached.

Proof. 

According to Appendix A, we can compute the second order averaged function of system (21), as follows

$$f_2\left(\mathit{x}\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\left[F_2(\theta ,\mathit{x})+D_x{F}_1(\theta ,\mathit{x})y_1(\theta ,\mathit{x})\right]d\theta ={(f_{2,1},f_{2,2})}^{\mathrm{T}},$$

where

$$y_1(\theta ,\mathit{x})={\int }_0^{\theta }{F}_1(s,\mathit{x})ds={(y_{1,1},y_{1,2})}^{\mathrm{T}}.$$

Suppose that $b\in {\mathcal{B}}^{\left(1\right)}$, we have

$$\begin{array}{ccc}\hfill y_{1,1}& =\hfill & {\displaystyle \frac{b_1\left(-1+cos2\theta \right)}{4}}r+\left({\displaystyle \frac{b_5}{3}}-{\displaystyle \frac{b_{5}cos\theta }{2}}+{\displaystyle \frac{b_{5}cos3\theta }{6}}+{\displaystyle \frac{b_{6}sin3\theta }{6}}-{\displaystyle \frac{b_{6}sin\theta }{2}}\right)r^2+{\displaystyle \frac{b_7\left(-1+cos2\theta \right)}{4}}wr,\hfill \\ \hfill y_{1,2}& =\hfill & b_{1}rsin\theta +\left({\displaystyle \frac{b_6}{2}}+{\displaystyle \frac{b_{5}sin2\theta }{2}}-{\displaystyle \frac{b_{6}cos2\theta }{2}}\right)r^2+b_{7}wrsin\theta .\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{ccc}\hfill f_{2,1}& =& -{\displaystyle \frac{b_{17}}{2}}r-{\displaystyle \frac{b_{14}}{2}}wr-{\displaystyle \frac{b_5{b}_7}{8}}{r}^3,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill f_{2,2}& =& b_{18}w+\left({\displaystyle \frac{b_1{b}_5}{2}}+b_{10}\right)r^2+b_{15}{w}^2+{\displaystyle \frac{b_5{b}_7}{2}}w{r}^2.\hfill \end{array}$$

(25)

Note that we are interested in the common solutions of these two polynomials for $(r,w)$ with $r>0$.

Solving $f_{2,1}=0$ for w, we obtain

$$w=-\frac{b_{17}}{b_{14}}-\frac{b_5{b}_7}{4b_{14}}{r}^2.$$

(26)

Substituting (26) into (25), we get

$$P\left(r^2\right):={\alpha }_4{r}^4+{\alpha }_2{r}^2+{\alpha }_0,$$

(27)

where P is a quadratic polynomial in the variable $r^2$. Therefore $f_2\left(\mathit{x}\right)=0$ can have at most two isolated solutions with $r>0$. Hence system (13) can have at most two limit cycles bifurcating from the origin.

By Lemma 9, the functions ${\alpha }_0,{\alpha }_2,{\alpha }_4$ are linearly independent. Therefore ${\mathcal{C}}^{\left(2\right)}\ne \varnothing$. Assume that $b\in {\mathcal{C}}^{\left(2\right)}$, i.e., the discriminant of P with respect to $r^2$ is positive and the number of changes in sign of the coefficients of the terms of $P\left(r^2\right)$ is two. According to Descartes’ rule of sign [13], P has two distinct positive roots (simple roots) for $r^2$, which implies two positive roots $r=r_{1,2}$ for r. Therefore, the original system $f_{2,1}=f_{2,2}=0,r>0$ can have two solutions $(r,w)=(r_i,w_i),i=1,2$. It is easy to check that $det\left({\displaystyle \frac{\partial (f_{2,1},f_{2,2})}{\partial (r,w)}}\right){|}_{(r,w)=(r_i,w_i)}\ne 0$, $i=1,2$. For the general argument, see Appendix B.

Summing up, according to Theorem A1, there exist two limit cycles in the system (21). The correspondingly system (13) has two limit cycles bifurcating from the origin.  □

Before ending the subsection, we present some notations for later use. Let

$$\begin{array}{ccc}\hfill {\mathcal{B}}^{\left(2\right)}& =& \left\{b\in {\mathcal{B}}^{\left(1\right)}|f_2\left(\mathit{x}\right)=f_2(r,w)\equiv 0\right\},\hfill \\ \hfill {\mathcal{B}}_1^{\left(2\right)}& =& \left\{b\in {\mathcal{B}}^{\left(1\right)}|b_5=b_{10}=b_{14}=b_{15}=b_{17}=b_{18}=0\right\},\hfill \\ \hfill {\mathcal{B}}_2^{\left(2\right)}& =& \left\{b\in {\mathcal{B}}^{\left(1\right)}|b_7=b_{14}=b_{15}=b_{17}=b_{18}=0,b_{10}=-\frac{b_1{b}_5}{2}\right\}.\hfill \end{array}$$

It can be checked that

$${\mathcal{B}}^{\left(2\right)}={\mathcal{B}}_1^{\left(2\right)}\cup {\mathcal{B}}_2^{\left(2\right)}.$$

5.7. Third Order Averaging

According to Appendix A, the third order averaged function $f_3\left(\mathit{x}\right)$ of system (21) is defined to be

$$\begin{array}{ccc}\hfill f_3\left(\mathit{x}\right)& =\hfill & {\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\left[F_3(\theta ,\mathit{x})+D_{\mathit{x}}{F}_2(\theta ,\mathit{x})y_1(\theta ,\mathit{x})+{\displaystyle \frac{1}{2}}{D}_{\mathit{x}\mathit{x}}^2{F}_1(\theta ,\mathit{x})\left(y_1(\theta ,\mathit{x}),y_1(\theta ,\mathit{x})\right)+{\displaystyle \frac{1}{2}}{D}_{\mathit{x}}{F}_1(\theta ,\mathit{x})y_2(\theta ,\mathit{x})\right]d\theta \hfill \\ & =\hfill & {(f_{31},f_{32})}^{\mathrm{T}},\hfill \end{array}$$

(28)

where

$$y_2(\theta ,\mathit{x})=2{\int }_0^{\theta }\left[F_2(s,\mathit{x})+D_x{F}_1(s,\mathit{x})y_1(s,\mathit{x})\right]ds.$$

Let

$$\left\{\begin{array}{cc}\hfill {\beta }_0^{\left(1\right)}& =-\frac{b_{26}\left(b_{23}{b}_{27}-b_{24}{b}_{26}\right)}{{b_{23}}^2},\hfill \\ \hfill {\beta }_2^{\left(1\right)}& =\frac{2b_1{b}_{11}{b_{23}}^2-2b_7{b}_{11}{b}_{23}{b}_{26}-b_7{b}_{11}{b}_{23}{b}_{27}+2b_7{b}_{11}{b}_{24}{b}_{26}+4b_{19}{b_{23}}^2}{4{b_{23}}^2},\hfill \\ \hfill {\beta }_4^{\left(1\right)}& =\frac{{b_{11}}^2{b_7}^2\left(b_{24}-2b_{23}\right)}{16{b_{23}}^2}.\hfill \end{array}\right.$$

(29)

Define

$${\mathcal{C}}_1^{\left(3\right)}=\left\{b\in {\mathcal{B}}_1^{\left(2\right)}|{\beta }_2^{\left(1\right)}{\beta }_4^{\left(1\right)}<0,{\beta }_0^{\left(1\right)}{\beta }_4^{\left(1\right)}>0,{\left({\beta }_2^{\left(1\right)}\right)}^2-4{\beta }_0^{\left(1\right)}{\beta }_4^{\left(1\right)}>0,b_{23}\ne 0\right\}.$$

It is easy to check that ${\beta }_0^{\left(1\right)},{\beta }_2^{\left(1\right)},{\beta }_4^{\left(1\right)}$ are linearly independent, thus ${\mathcal{C}}_1^{\left(3\right)}\ne \varnothing$.

Theorem 7.

Assume that $b\in {\mathcal{B}}_1^{\left(2\right)}$. Subsequently, for $\left|\epsilon \right|>0$ sufficiently small, system (13) can have, at most, two limit cycles by averaging theory of third order. Moreover, if $b\in {\mathcal{C}}_1^{\left(3\right)}$, then the bound can be reached.

Proof. 

By direct computation, from (28) we have

$$\begin{array}{ccc}\hfill f_{3,1}{|}_{{\mathcal{B}}_1^{\left(2\right)}}& =& -{\displaystyle \frac{b_{26}}{2}}r-{\displaystyle \frac{b_{23}}{2}}wr-{\displaystyle \frac{b_7{b}_{11}}{8}}{r}^3,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill f_{3,2}{|}_{{\mathcal{B}}_1^{\left(2\right)}}& =& b_{27}w+\left({\displaystyle \frac{b_1{b}_{11}}{2}}+b_{19}\right)r^2+b_{24}{w}^2+{\displaystyle \frac{b_7{b}_{11}}{2}}w{r}^2.\hfill \end{array}$$

(31)

Note that we are interested in the common solutions of these two polynomials for $(r,w)$, with $r>0$.

Solving (30) for w, we obtain

$$w=-{\displaystyle \frac{b_{26}}{b_{23}}}-{\displaystyle \frac{b_7{b}_{11}}{4b_{23}}}{r}^2.$$

(32)

Substituting (32) into (31), we get

$$P\left(r^2\right):={\beta }_4^{\left(1\right)}{r}^4+{\beta }_2^{\left(1\right)}{r}^2+{\beta }_0^{\left(1\right)},$$

(33)

where P is a quadratic polynomial in the variable $r^2$.

The rest of proof is similar to the proof of Theorem 6.  □

Let

$$\left\{\begin{array}{cc}\hfill {\beta }_0^{\left(2\right)}=& -\frac{b_{26}\left(b_{23}{b}_{27}-b_{24}{b}_{26}\right)}{{b_{23}}^2},\hfill \\ \hfill {\beta }_2^{\left(2\right)}=& \frac{{b_1}^2{b}_5{b_{23}}^2+2b_1{b}_{11}{b_{23}}^2-2b_5{b}_{13}{b}_{23}{b}_{26}-b_5{b}_{13}{b}_{23}{b}_{27}+2b_5{b}_{13}{b}_{24}{b}_{26}+2b_5{b}_{16}{b_{23}}^2}{4{b_{23}}^2}\hfill \\ \hfill & +b_{19},\hfill \\ \hfill {\beta }_4^{\left(2\right)}=& \frac{{b_{13}}^2{b_5}^2\left(b_{24}-2b_{23}\right)}{16{b_{23}}^2}.\hfill \end{array}\right.$$

(34)

Define

$${\mathcal{C}}_2^{\left(3\right)}=\left\{b\in {\mathcal{B}}_2^{\left(2\right)}|{\beta }_2^{\left(2\right)}{\beta }_4^{\left(2\right)}<0,{\beta }_0^{\left(2\right)}{\beta }_4^{\left(2\right)}>0,{\left({\beta }_2^{\left(2\right)}\right)}^2-4{\beta }_0^{\left(2\right)}{\beta }_4^{\left(2\right)}>0,b_{23}\ne 0\right\}.$$

It is easy to check that ${\beta }_0^{\left(2\right)},{\beta }_2^{\left(2\right)},{\beta }_4^{\left(2\right)}$ are linearly independent, thus ${\mathcal{C}}_2^{\left(3\right)}\ne \varnothing$.

Theorem 8.

Assume that $b\in {\mathcal{B}}_2^{\left(2\right)}$. Subsequently, for $\left|\epsilon \right|>0$ sufficiently small, system (13) can have, at most, two limit cycles by averaging theory of third order. Moreover, if $b\in {\mathcal{C}}_2^{\left(3\right)}$, then the bound can be reached.

Proof. 

By direct computation, from (28) we have

$$\begin{array}{ccc}\hfill f_{3,1}{|}_{{\mathcal{B}}_2^{\left(2\right)}}& =& -\frac{1}{2}{b}_{26}r-\frac{1}{2}{b}_{23}wr-\frac{1}{8}{b}_{13}{b}_5{r}^3,\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill f_{3,2}{|}_{{\mathcal{B}}_2^{\left(2\right)}}& =& b_{27}w+\left(\frac{1}{4}{b_1}^2{b}_5+\frac{1}{2}{b}_1{b}_{11}+\frac{1}{2}{b}_5{b}_{16}+b_{19}\right)r^2+b_{24}{w}^2+\frac{1}{2}{b}_{13}{b}_{5}w{r}^2.\hfill \end{array}$$

(36)

Note that we are interested in the common solutions of these two polynomials for $(r,w)$ with $r>0$.

Solving (35) for w, we get

$$w=-{\displaystyle \frac{b_{26}}{b_{23}}}-{\displaystyle \frac{b_5{b}_{13}}{4b_{23}}}{r}^2.$$

(37)

Substituting (37) into (36), we obtain

$$P\left(r^2\right):={\beta }_4^{\left(2\right)}{r}^4+{\beta }_2^{\left(2\right)}{r}^2+{\beta }_0^{\left(2\right)},$$

(38)

where P is a quadratic polynomial in the variable $r^2$.

The rest of proof is similar to the proof of Theorem 6.  □

Before ending this subsection, we introduce some notations for later use. Let

$$\begin{array}{ccc}\hfill {\mathcal{B}}^{\left(3\right)}& =& \left\{b\in {\mathcal{B}}^{\left(2\right)}|f_3\left(\mathit{x}\right)=f_3(r,w)\equiv 0\right\},\hfill \\ \hfill {\mathcal{B}}_{1,1}^{\left(3\right)}& =& \left\{b\in {\mathcal{B}}_1^{\left(2\right)}|b_7=b_{23}=b_{24}=b_{26}=b_{27}=0,b_{19}=-{\displaystyle \frac{b_1{b}_{11}}{2}}\right\},\hfill \\ \hfill {\mathcal{B}}_{1,2}^{\left(3\right)}& =& \left\{b\in {\mathcal{B}}_1^{\left(2\right)}|b_{11}=b_{19}=b_{23}=b_{24}=b_{26}=b_{27}=0\right\},\hfill \\ \hfill {\mathcal{B}}_{2,1}^{\left(3\right)}& =& \left\{b\in {\mathcal{B}}_2^{\left(2\right)}|b_5=b_{23}=b_{24}=b_{26}=b_{27}=0,b_{19}=-{\displaystyle \frac{b_1{b}_{11}}{2}}\right\},\hfill \\ \hfill {\mathcal{B}}_{2,2}^{\left(3\right)}& =& \left\{b\in {\mathcal{B}}_2^{\left(2\right)}|b_{13}=b_{23}=b_{24}=b_{26}=b_{27}=0,b_{19}=-\frac{1}{4}{b}_1^2{b}_5-\frac{1}{2}{b}_1{b}_{11}-\frac{1}{2}{b}_5{b}_{16}\right\}.\hfill \end{array}$$

It is easy to check that

$${\mathcal{B}}^{\left(3\right)}={\mathcal{B}}_{1,1}^{\left(3\right)}\cup {\mathcal{B}}_{1,2}^{\left(3\right)}\cup {\mathcal{B}}_{2,1}^{\left(3\right)}\cup {\mathcal{B}}_{2,2}^{\left(3\right)}.$$

(39)

5.8. Fourth Order Averaging

For system (21), the averaged function of fourth order is

$$\begin{array}{ccc}\hfill f_4\left(x\right)& =\hfill & {\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }[F_4(\theta ,x)+D_{\mathit{x}}{F}_3(\theta ,\mathit{x})y_1(\theta ,\mathit{x})+\frac{1}{2}{D}_{\mathit{x}}{F}_2(\theta ,\mathit{x})y_2(\theta ,\mathit{x})\hfill \\ \hfill & \hfill & +\frac{1}{2}{D}_{\mathit{x}\mathit{x}}^2{F}_2(\theta ,\mathit{x})(y_1(\theta ,\mathit{x}),y_1(\theta ,\mathit{x}))+\frac{1}{6}{D}_{\mathit{x}}{F}_1(\theta ,\mathit{x})y_3(\theta ,\mathit{x})\hfill \\ \hfill & \hfill & +\frac{1}{2}{D}_{\mathit{x}\mathit{x}}^2{F}_1(\theta ,\mathit{x})(y_1(\theta ,\mathit{x}),y_2(\theta ,\mathit{x}))+\frac{1}{6}{D}_{\mathit{x}\mathit{x}\mathit{x}}^3{F}_1(\theta ,\mathit{x})(y_1(\theta ,\mathit{x}),y_1(\theta ,\mathit{x}),y_1(\theta ,\mathit{x}))]d\theta \hfill \\ \hfill & =\hfill & {(f_{4,1},f_{4,2})}^{\mathrm{T}},\hfill \end{array}$$

(40)

where

$$\begin{array}{ccc}\hfill y_3(\theta ,\mathit{x})& =& {\int }_0^s(6F_3(s,\mathit{x})+6D_{\mathit{x}}{F}_2(s,\mathit{x})y_1(s,\mathit{x})+3D_{\mathit{x}}{F}_1(s,\mathit{x})y_2(s,\mathit{x})\hfill \\ & & +3D_{\mathit{x}\mathit{x}}^2{F}_1(s,\mathit{x})(y_1(s,\mathit{x}),y_1(s,\mathit{x})))ds.\hfill \end{array}$$

By direct computation, from (40), we have

$$\begin{array}{ccc}\hfill f_{4,1}{|}_{{\mathcal{B}}_{1,1}^{\left(3\right)}}& =& -\frac{1}{8}{b}_{11}{b}_{13}{r}^3-\frac{1}{2}{b}_{35}r-\frac{1}{2}{b}_{32}rw,\hfill \\ \hfill f_{4,2}{|}_{{\mathcal{B}}_{1,1}^{\left(3\right)}}& =& \left(b_{28}+\frac{1}{4}{b_1}^2{b}_{11}+\frac{1}{2}{b}_1{b}_{20}+\frac{1}{2}{b}_{11}{b}_{16}\right)r^2+\frac{1}{2}{b}_{11}{b}_{13}{r}^{2}w+b_{33}{w}^2+b_{36}w,\hfill \\ \hfill f_{4,1}{|}_{{\mathcal{B}}_{1,2}^{\left(3\right)}}& =& -\frac{1}{8}{b}_7{b}_{20}{r}^3-\frac{1}{2}{b}_{35}r-\frac{1}{2}{b}_{32}rw,\hfill \\ \hfill f_{4,2}{|}_{{\mathcal{B}}_{1,2}^{\left(3\right)}}& =& \left(b_{28}+\frac{1}{2}{b}_1{b}_{20}\right)r^2+\frac{1}{2}{b}_7{b}_{20}{r}^{2}w+b_{33}{w}^2+b_{36}w,\hfill \\ \hfill f_{4,1}{|}_{{\mathcal{B}}_{2,1}^{\left(3\right)}}& =& -\frac{1}{8}{b}_{11}{b}_{13}{r}^3-\frac{1}{2}{b}_{35}r-\frac{1}{2}{b}_{32}rw,\hfill \\ \hfill f_{4,2}{|}_{{\mathcal{B}}_{2,1}^{\left(3\right)}}& =& \left(b_{28}+\frac{1}{4}{b}_1^2{b}_{11}+\frac{1}{2}{b}_1{b}_{20}+\frac{1}{2}{b}_{11}{b}_{16}\right)r^2+\frac{1}{2}{b}_{11}{b}_{13}{r}^{2}w+b_{33}{w}^2+b_{36}w,\hfill \\ \hfill f_{4,1}{|}_{{\mathcal{B}}_{2,2}^{\left(3\right)}}& =& -\frac{1}{8}{b}_5{b}_{22}{r}^3-\frac{1}{2}{b}_{35}r-\frac{1}{2}{b}_{32}rw,\hfill \\ \hfill f_{4,2}{|}_{{\mathcal{B}}_{2,2}^{\left(3\right)}}& =& \left(b_{28}+\frac{1}{4}{b_1}^2{b}_{11}+\frac{1}{2}{b}_1{b}_{20}+\frac{1}{2}{b}_{11}{b}_{16}+\frac{1}{2}{b}_1{b}_5{b}_{16}+\frac{1}{2}{b}_5{b}_{25}+\frac{1}{8}{b_1}^3{b}_5\right)r^2\hfill \\ & & +\frac{1}{2}{b}_5{b}_{22}{r}^{2}w+b_{33}{w}^2+b_{36}w.\hfill \end{array}$$

For $b\in {\mathcal{B}}_{1,1}^{\left(3\right)}$, we introduce

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\gamma }_0^{\left(1\right)}& =\frac{b_{33}{b_{35}}^2}{{b_{32}}^2}-\frac{b_{36}{b}_{35}}{b_{32}},\hfill \\ \hfill {\gamma }_2^{\left(1\right)}& =b_{28}+\frac{1}{4}{b_1}^2{b}_{11}+\frac{1}{2}{b}_1{b}_{20}+\frac{1}{2}{b}_{11}{b}_{16}-\frac{1}{4}\frac{b_{11}{b}_{13}\left(2b_{35}+b_{36}\right)}{b_{32}}+\frac{1}{2}\frac{b_{33}{b}_{11}{b}_{13}{b}_{35}}{{b_{32}}^2},\hfill \\ \hfill {\gamma }_4^{\left(1\right)}& =-\frac{{b_{11}}^2{b_{13}}^2}{8b_{32}}+\frac{b_{33}{b_{11}}^2{b_{13}}^2}{16{b_{32}}^2}.\hfill \end{array}\right.\end{array}$$

For $b\in {\mathcal{B}}_{1,2}^{\left(3\right)}$, we introduce

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\gamma }_0^{\left(2\right)}& =\frac{b_{33}{b_{35}}^2}{{b_{32}}^2}-\frac{b_{36}{b}_{35}}{b_{32}},\hfill \\ \hfill {\gamma }_2^{\left(2\right)}& =b_{28}+\frac{1}{2}{b}_1{b}_{20}-\frac{b_7{b}_{20}\left(2b_{35}+b_{36}\right)}{4b_{32}}+\frac{b_{33}{b}_7{b}_{20}{b}_{35}}{2{b_{32}}^2},\hfill \\ \hfill {\gamma }_4^{\left(2\right)}& =-\frac{{b_7}^2{b_{20}}^2}{8b_{32}}+\frac{b_{33}{b_7}^2{b_{20}}^2}{16{b_{32}}^2}.\hfill \end{array}\right.\end{array}$$

For $b\in {\mathcal{B}}_{2,1}^{\left(3\right)}$, we introduce

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\gamma }_0^{\left(3\right)}& =\frac{b_{33}{b_{35}}^2}{{b_{32}}^2}-\frac{b_{36}{b}_{35}}{b_{32}},\hfill \\ \hfill {\gamma }_2^{\left(3\right)}& =b_{28}+\frac{1}{4}{b_1}^2{b}_{11}+\frac{1}{2}{b}_1{b}_{20}+\frac{1}{2}{b}_{11}{b}_{16}-\frac{b_{11}{b}_{13}\left(2b_{35}+b_{36}\right)}{4b_{32}}+\frac{b_{33}{b}_{11}{b}_{13}{b}_{35}}{{2b_{32}}^2},\hfill \\ \hfill {\gamma }_4^{\left(3\right)}& =-\frac{{b_{11}}^2{b_{13}}^2}{8b_{32}}+\frac{b_{33}{b_{11}}^2{b_{13}}^2}{16{b_{32}}^2}.\hfill \end{array}\right.\end{array}$$

Finally, for $b\in {\mathcal{B}}_{2,2}^{\left(3\right)}$, we introduce

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\gamma }_0^{\left(4\right)}& =\frac{b_{33}{b_{35}}^2}{{b_{32}}^2}-\frac{b_{36}{b}_{35}}{b_{32}},\hfill \\ \hfill {\gamma }_2^{\left(4\right)}& =b_{28}+\frac{1}{4}{b_1}^2{b}_{11}+\frac{1}{2}{b}_1{b}_{20}+\frac{1}{2}{b}_{11}{b}_{16}+\frac{1}{2}{b}_1{b}_5{b}_{16}+\frac{1}{2}{b}_5{b}_{25}+\frac{1}{8}{b_1}^3{b}_5\hfill \\ & -\frac{b_5{b}_{22}\left(2b_{35}+b_{36}\right)}{4b_{32}}+\frac{b_{33}{b}_5{b}_{22}{b}_{35}}{2{b_{32}}^2},\hfill \\ \hfill {\gamma }_4^{\left(4\right)}& =-\frac{{b_5}^2{b_{22}}^2}{8b_{32}}+\frac{b_{33}{b_5}^2{b_{22}}^2}{16{b_{32}}^2}.\hfill \end{array}\right.\end{array}$$

Proceed as we did in previous subsections, we have the following result.

Theorem 9.

Assume that $b\in {\mathcal{B}}^{\left(3\right)}$. Subsequently, for $\left|\epsilon \right|>0$ sufficiently small, system (13) has, at most, two limit cycles by averaging theory of fourth order. Furthermore, the bound can be reached if

$$b\in \bigcup _{s=1}^4{\mathcal{C}}_s^{\left(4\right)}$$

(41)

where

$$\begin{array}{c}\hfill {\mathcal{C}}_1^{\left(4\right)}=\left\{b\in {\mathcal{B}}_{1,1}^{\left(3\right)}|{\gamma }_2^{\left(1\right)}{\gamma }_4^{\left(1\right)}<0,{\gamma }_0^{\left(1\right)}{\gamma }_4^{\left(1\right)}>0,{\left({\gamma }_2^{\left(1\right)}\right)}^2-4{\gamma }_0^{\left(1\right)}{\gamma }_4^{\left(1\right)}>0,b_{32}\ne 0\right\},\\ \hfill {\mathcal{C}}_2^{\left(4\right)}=\left\{b\in {\mathcal{B}}_{1,2}^{\left(3\right)}|{\gamma }_2^{\left(2\right)}{\gamma }_4^{\left(2\right)}<0,{\gamma }_0^{\left(2\right)}{\gamma }_4^{\left(2\right)}>0,{\left({\gamma }_2^{\left(2\right)}\right)}^2-4{\gamma }_0^{\left(2\right)}{\gamma }_4^{\left(2\right)}>0,b_{32}\ne 0\right\},\\ \hfill {\mathcal{C}}_3^{\left(4\right)}=\left\{b\in {\mathcal{B}}_{2,1}^{\left(3\right)}|{\gamma }_2^{\left(3\right)}{\gamma }_4^{\left(3\right)}<0,{\gamma }_0^{\left(3\right)}{\gamma }_4^{\left(3\right)}>0,{\left({\gamma }_2^{\left(3\right)}\right)}^2-4{\gamma }_0^{\left(3\right)}{\gamma }_4^{\left(3\right)}>0,b_{32}\ne 0\right\},\\ \hfill {\mathcal{C}}_4^{\left(4\right)}=\left\{b\in {\mathcal{B}}_{2,2}^{\left(3\right)}|{\gamma }_2^{\left(4\right)}{\gamma }_4^{\left(4\right)}<0,{\gamma }_0^{\left(4\right)}{\gamma }_4^{\left(4\right)}>0,{\left({\gamma }_2^{\left(4\right)}\right)}^2-4{\gamma }_0^{\left(4\right)}{\gamma }_4^{\left(4\right)}>0,b_{32}\ne 0\right\}.\end{array}$$

6. Conclusions

In this work, some types of bifurcations of equilibria are studied for quadratic jerk system. First, we study the saddle-node bifurcation for the general jerk system. Subsequently, for convenience, we introduce the concept of canonical system. Finally, the other bifurcations: transcritical bifurcation and zero-Hopf bifurcation are also studied for canonical system. By using the averaging theory up to fourth order, we prove that at most two limit cycles bifurcate from the zero-Hopf equilibrium, and this bound is sharp.