1 Introduction

As we all know, Chua’s circuit is the first analog system to implement chaos in experiments. Although Chua’s system is comprised of simple ordinary differential equations, it can present some extremely complex dynamical behaviors such as chaos and bifurcation. It is for this reason that Chua’s circuit shows comparative advantage in practical applications and has attracted many researchers [1–15].

People have noticed that the research of Chua’s circuit relies heavily on computer numerical simulations. Many fundamental questions, such as complex chaotic behavior and the existence of a global attractive compact set, are still not solved. Recently, a hidden attractor was proposed by Leonov et al. [16–18], and means basin of attraction does not contain neighborhoods of any equilibria. Based on the idea of a hidden attractor, Leonov et al. discovered a hidden Chua attractor [17] in smooth Chua systems. Chua et al. expressed a conjecture that Andronov–Hopf bifurcation can lead to the birth of hidden attractors in Chua systems [19].

Regarding the original Chua’s system, people use continuous piecewise linear functions with two non-differentiable breakpoints and three segments to characterize the characteristics of Chua’s diode. It can be easily implemented experimentally with simple electronics for piecewise-linear functions. However, we should know that, seriously talking, the characteristics of segmented devices are generally smooth in the actual circuits [20]. Thus, it is also extremely important to consider the complex dynamics in smooth Chua’s systems whose piecewise linear functions are replaced by smooth polynomials [21–38]. Llibre and Valls analyzed the existence of the first integral locally and globally for Chua’s system [34]. By using the Poincaré compactification for a polynomial vector field in \(R^{3}\), Messias researched the dynamics at infinity of the modified smoothly Chua system [36]. The local codimension one, two, and three Hopf bifurcations, which occur in Chua’s differential equations with cubic nonlinearity, was studied in [37]. In addition, Algaba and his coworkers proved the existence of codimension-three Hopf bifurcations for the nontrivial equilibria in Chua’s system[38]. Based on the works mentioned above and the research about the Hopf bifurcations in [38], the modified Chua circuit system is described as the following form:

$$ \textstyle\begin{cases} \dot{x}=\alpha(y-bx-cx^{2}-ax^{3}), \\ \dot{y}=x-y+z, \\ \dot{z}=-\beta y-\gamma z, \end{cases} $$
(1)

where \(a=\pm1\) and \(\alpha,\beta,\gamma,c,b\in R\). Note that for \(\alpha=0\) system (1) is linear and if \(\beta=0\) it is uncoupled. In this paper we extend the result in [38] and consider the case \(a=1\). System (1) will be of the following form:

$$ \textstyle\begin{cases} \dot{x}=\alpha(y- bx- cx^{2}-x^{3}), \\ \dot{y}=x-y+z, \\ \dot{z}=-\beta y-\gamma z. \end{cases} $$
(2)

The rest of this paper is organized as follows. In Sect. 2, distribution of equilibria of system (2) is presented. The problem of zero-Hopf bifurcation of system (2) is addressed in Sect. 3. In Sect. 4, the classical Hopf bifurcation is studied to illustrate the existence of periodic solution. Finally, we conclude this paper in Sect. 5.

2 Distribution of equilibria of Chua’s system

In this section, the distribution of the equilibria in the modified Chua system (2) is studied. The equilibria of system (2) can be obtained from the following equations:

$$ \textstyle\begin{cases} y-bx-cx^{2}-x^{3}=0, \\ x-y+z=0, \\ -\beta y-\gamma z=0. \end{cases} $$
(3)

Hence we can draw the following conclusions.

Theorem 2.1

Denote

$$\begin{aligned}& A_{0}=(\beta+\gamma) \bigl(4\gamma-4b(\beta+\gamma)+c^{2}( \beta+\gamma)\bigr), \\& E_{\pm}=\biggl(-\frac{c}{2} \pm\frac{\sqrt{A_{0}}}{2(\beta+\gamma)},- \frac {\gamma(c\beta+c\gamma\mp\sqrt{A_{0}})}{2(\beta+\gamma)^{2}},\frac{\beta (c\beta+c\gamma\mp\sqrt{A_{0}})}{2(\beta+\gamma)^{2}}\biggr), \\& E^{1}_{\pm}=\biggl(\frac{-c\pm\sqrt{4-4b+c^{2}}}{2},\frac{-c\pm\sqrt{4-4b+c^{2}}}{2},0 \biggr), \\& E^{2}_{\pm}=\biggl(\frac{-c\pm\sqrt{-4b+c^{2}}}{2},0,\frac{-c\mp\sqrt{-4b+c^{2}}}{2} \biggr). \end{aligned}$$

The distribution of equilibria of system (2) are summarized in Table 1 when the parameters α, β, γ, b, and c vary in \(\mathbb{R}^{3}\).

Table 1 The distribution of equilibrium of system (1)
Full size table

3 Zero-Hopf bifurcation of system (2)

It is easy to obtain the characteristic equation associated with the equilibrium \(E_{0}\) in Table 1:

$$ p(\lambda)=\lambda^{3}+(1+b\alpha+\gamma) \lambda^{2}+(-\alpha+b\alpha+\beta +\gamma+b\alpha\gamma)\lambda+b\alpha \beta-\alpha\gamma+b\alpha\gamma=0. $$
(4)

Zero-Hopf bifurcation at the equilibrium \(E_{0}\) in Table 1 could happen in condition that the characteristic equation associated with the equilibrium \(E_{0}\) has a zero real eigenvalue and a pair of pure imaginary eigenvalues. Therefore, we have the following proposition.

Proposition 3.1

The differential system (2) has a zero-Hopf equilibrium localized at equilibrium \(E_{0}\) if the following condition is satisfied:

$$ \begin{aligned} &b=-\frac{-1-\gamma}{\alpha},\qquad \beta=\frac{\gamma(1+\alpha+\gamma )}{1+\gamma}, \\ & \alpha=-\frac{-1-3\gamma-3\gamma^{2}-\gamma^{3}-\omega ^{2}-\gamma\omega^{2}}{1+2\gamma}. \end{aligned} $$
(5)

Moreover, the eigenvalues at the origin are 0, and \(\pm\omega i\).

Now, we give the averaging theory.

Theorem 3.2

(First-order averaging theory [39–43])

We consider the following two initial value problems:

$$ \dot{x}=\varepsilon f(t,x)+\varepsilon^{2}g(t,x, \varepsilon), \qquad x(0)=x_{0}, $$
(6)

and

$$ \dot{y}=\varepsilon f^{0}(y),\qquad y(0)=x_{0}, $$
(7)

where \(x,y,x_{0}\in\Omega\) is an open subset of \(R^{n}\), \(t\in[0,\infty )\), \(\varepsilon\in(0,\varepsilon_{0}]\), f and g are periodic of period T in the variable t, and \(f^{0}(y)\) is the averaged function of \(f(t,x)\) with respect to t, i.e.,

$$ f^{0}(y)=\frac{1}{T} \int_{0} ^{T} f(t,y)\, dt. $$
(8)

Suppose:

  1. (i)

    f, its Jacobian \(\frac{\partial f}{\partial x}\), its Hessian \(\frac{\partial^{2} f}{\partial x^{2}}\), g and its Jacobian \(\frac{\partial g}{\partial x}\) are defined, continuous, and bounded by a constant independent of ε in \([0,\infty)\times\Omega\) and \(\varepsilon\in(0,\varepsilon_{0}]\);

  2. (ii)

    T is a constant independent of ε; and

  3. (iii)

    \(y(t)\) belongs to Ω on the interval of time \([0,1/\varepsilon]\). Then the following statements hold:

    1. (a)

      On the time scale \(1/\varepsilon\), we have that \(x(t)-y(t)=O(\varepsilon)\), as \(\varepsilon\rightarrow0\).

    2. (b)

      If p is a singular point of the averaged system (9) such that the determinant of the Jacobian matrix

      $$ \frac{\partial f^{0}}{\partial y}\bigg|_{y = p} $$
      (9)

      is not zero, then there exists a limit cycle \(\phi(t,\varepsilon) \) of period T for system (8) which is close to p and such that \(\phi (t,\varepsilon)\rightarrow p \) as \(\varepsilon\rightarrow0\).

    3. (c)

      The stability or instability of the limit cycle \(\phi (t,\varepsilon) \) is given by the stability or instability of the singular point p of the averaged system (9). In fact, the singular point p has the stability behavior of the Poincaré map associated to the limit cycle \(\phi(t,\varepsilon) \).

In the rest of this section, we employ the three-dimensional zero-Hopf bifurcation theory and apply symbolic computations to perform the analysis of parametric variations in system (2).

Theorem 3.3

Let

$$ \begin{aligned} &b=-\frac{-1-\gamma}{\alpha}+\varepsilon,\qquad \beta=\frac{\gamma(1+\alpha+\gamma)}{1+\gamma}+ \varepsilon, \\ &\alpha=-\frac{-1-3\gamma-3\gamma^{2}-\gamma^{3}-\omega^{2}-\gamma\omega ^{2}}{1+2\gamma} +\varepsilon. \end{aligned} $$
(10)

If \(\omega\neq0\) and \(\Gamma>0\), where

$$\begin{aligned} \Gamma = & \gamma\bigl(10\gamma^{14}+\gamma^{15}+16 \gamma^{12}\bigl(8+\omega^{2}\bigr)+\gamma ^{13} \bigl(45+2\omega^{2}\bigr) \\ &{}+\gamma^{10}\bigl(422+120 \omega^{2}-10\omega^{4}\bigr) \\ &{} +\gamma^{11}\bigl(266+57\omega^{2}-\omega^{4} \bigr)+2\omega^{4}\bigl(-1+\omega^{2}+\omega ^{4} \bigr) \\ &{}+\gamma^{9}\bigl(518+168\omega^{2}-41 \omega^{4}-4\omega^{6}\bigr) \\ &{}-2\gamma^{8}\bigl(-247-88\omega^{2}+55 \omega^{4}+12\omega^{6}\bigr) \\ &{}-2\gamma ^{6} \bigl(-92-45\omega^{2}+158\omega^{4}+45\omega^{6}+ \omega^{8}\bigr) \\ &{}-\gamma^{7}\bigl(-357-134\omega^{2}+217 \omega^{4}+62\omega^{6}+\omega^{8}\bigr) \\ &{}+2\gamma ^{2}\omega^{2}\bigl(5-53\omega^{2}+14 \omega^{4}+19\omega^{6}+5\omega^{8}+ \omega^{10}\bigr) \\ &{}+\gamma^{5}\bigl(62+86\omega^{2}-383 \omega^{4}-79\omega^{6}+3\omega^{8}+2\omega ^{10}\bigr) \\ &{}+2\gamma^{4}\bigl(6+38\omega^{2}-189 \omega^{4}-16\omega^{6}+11\omega^{8}+4\omega ^{10}\bigr) \\ &{}+\gamma\bigl(\omega^{2}-23\omega^{4}+13 \omega^{6}+15\omega^{8}+3\omega^{10}+\omega ^{12}\bigr) \\ &{}+\gamma^{3}\bigl(1+39\omega^{2}-258 \omega^{4}+16\omega^{6}+43\omega^{8}+13\omega ^{10}+\omega^{12}\bigr)\bigr). \end{aligned}$$

Then the smooth Chua system (2) has a zero-Hopf bifurcation and produces a limit cycle at the equilibrium point \(E_{0}\) located at the origin of coordinates for \(\varepsilon>0\) sufficiently small. The equilibrium point of a planar differential system has the same stability or instability as this limit cycle, and the equilibrium point with eigenvalues

$$ \frac{A_{1}A_{2}\pm\sqrt{-\gamma(1+\gamma)^{4}A_{3}A_{4}}}{2\sqrt{\gamma }(1+2\gamma)\omega^{3}(\gamma+\gamma^{2}-\omega^{2})(\gamma^{2}+\omega^{2})A_{1}}, $$
(11)

where \(A_{i}\) (\(i=1,2,3,4\)) can be found in the Appendix .

Proof of Theorem 3.3

We first write the linear part at the origin of the differential system (2) when condition (5) is satisfied into its real Jordan normal form, i.e., into the following form:

$$ \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0 & -\omega& 0\\ \omega& 0 & 0\\ 0&0&0 \end{array}\displaystyle \right ). $$
(12)

For that we consider the linear change

$$ \begin{aligned} &x =-\biggl(\frac{3\omega+2\gamma\omega}{\gamma^{2}+\omega^{2}}+ \frac{\omega }{\gamma(\gamma^{2}+\omega^{2})}\biggr)u \\ &\hphantom{x ={}}{}+ \biggl(-\frac{-\gamma^{2}+\omega^{2}-1-2\gamma }{\gamma^{2}+\omega^{2}}+ \frac{\omega^{2}}{\gamma(\gamma^{2}+\omega^{2})}\biggr)v \\ &\hphantom{x ={}}{}-\biggl(\frac{\gamma^{2}+\omega ^{2}+1+2\gamma}{\gamma^{2}+\omega^{2}}\biggr)\omega, \\ &y =-\biggl(\frac{2\omega}{\gamma^{2}+\omega^{2}}+\frac{\omega}{\gamma(\gamma ^{2}+\omega^{2})}\biggr)u-\frac{1+2\gamma}{\gamma^{2}+\omega^{2}}v- \frac{1+2\gamma }{\gamma^{2}+\omega^{2}}w , \\ &z = v+w. \end{aligned} $$
(13)

By using the replaced variables \((u,v,w)\), the differential system (2) will be changed to

$$ \textstyle\begin{cases} \dot{u}= \frac{B_{1}}{\gamma^{2}+\omega^{2}}u+\frac{B_{2}}{\omega(\gamma ^{2}+\omega^{2})}v+\frac{\gamma B_{1}}{\omega(\gamma^{2}+\omega^{2})}w, \\ \dot{v}= \frac{1}{\omega^{2}}(u B_{3}+\frac{\alpha(u B_{4}+v B_{5}+w B_{6})^{2}(-c \gamma^{3}+u B_{7}-c \gamma\omega^{2}+v B_{5}+w B_{6})}{B_{8}} \\ \hphantom{\dot{v}={}}{} +w \gamma(-B_{9}+B_{10}+v (-\gamma B_{9}+B_{11}))), \\ \dot{w}= \frac{1}{\omega^{2}}(u B_{12}+\frac{\alpha(u B_{4}+v B_{5}+w B_{6})^{2}(-c \gamma^{3}+u B_{7}-c \gamma\omega^{2}+v B_{5}+w B_{6})}{B_{8}} \\ \hphantom{\dot{w}= {}}{}+v B_{13}+w B_{14}), \end{cases} $$
(14)

where \(B_{i}\) (\(i=1,2,\ldots,10\)) can be found in the Appendix. Then we use the cylindrical coordinates \((u, v)=(r \cos\theta, r \sin \theta)\) and obtain

$$ \textstyle\begin{cases} \dot{r}=-\frac{1}{c_{1}^{2}}(C_{1}\cos\theta(C_{2}\gamma w+C_{2}\omega r\cos \theta+C_{3} r \sin\theta)-\frac{1}{C_{5}\gamma^{2}}\sin\theta(B_{4}^{3} \alpha r^{3} \cos\theta^{3} \\ \hphantom{\dot{r}={}}{}+\gamma^{2}(B_{11}C_{5}C_{6}^{2}+3 B_{5}C_{4}^{2}\alpha w^{2}-C_{5}C_{6}C_{7}\beta\gamma-2c C_{4}C_{6}C_{7}\alpha(1+\gamma) w)r \sin\theta \\ \hphantom{\dot{r}={}}{}-B_{5}^{2}(c C_{6}-3C_{4}w)\alpha\gamma r^{2}\sin\theta^{2}+B_{5}^{3} \alpha r^{3} \sin \theta^{3} \\ \hphantom{\dot{r}={}}{}+B_{4}^{2}\alpha r^{2} \cos\theta^{2}(-cC_{6}\gamma+3C_{4}\gamma w +3 B_{5} r \sin\theta) \\ \hphantom{\dot{r}={}}{}+\omega r \cos\theta(\gamma^{2}(C_{4}w(-2cC_{6}+3C_{4} w)\alpha C_{8}+C_{5}C_{6}(-C_{7}\beta+C_{6}\gamma(1+\gamma) \\ \hphantom{\dot{r}={}}{}+C_{6}\alpha(-1+b+b\gamma )))-2C_{5}C_{7}\alpha\gamma(c C_{6}(1+\gamma)^{2}-3 C_{4}(1+\gamma^{2})w)r \sin \theta \\ \hphantom{\dot{r}={}}{}+3 C_{5} C_{7}^{2}\alpha(1+\gamma)^{3}r^{2} \sin\theta^{2}) \\ \hphantom{\dot{r}={}}{}+w \gamma^{2}(\gamma(b C_{4}C_{6}^{2} \alpha-c C_{4}^{2} C_{6}\alpha w+C_{4}^{3}\alpha w^{2}-C_{5}C_{6}(C_{6}\alpha +C_{7}\beta) \\ \hphantom{\dot{r}={}}{}+C_{6}^{2}C_{7}\gamma)+6C_{4}C_{5}C_{7}\alpha\omega r^{2} \sin{2\theta}))), \\ \dot{\theta}=\frac{1}{c_{1}^{2} r}(C_{1}\sin\theta(C_{2}\gamma w+C_{2}\omega r\cos\theta+C_{3} r \sin\theta)-\frac{1}{C_{5}\gamma^{2}}\cos\theta(B_{4}^{3} \alpha r^{3} \cos\theta^{3} \\ \hphantom{\dot{\theta}={}}{}+\gamma^{2}(B_{11}C_{5}C_{6}^{2}+3 B_{5}C_{4}^{2}\alpha w^{2}-C_{5}C_{6}C_{7}\beta\gamma-2c C_{4}C_{6}C_{7}\alpha(1+\gamma) w)r \sin\theta \\ \hphantom{\dot{\theta}={}}{}-B_{5}^{2}(c C_{6}-3C_{4}w)\alpha\gamma r^{2}\sin\theta^{2}+B_{5}^{3} \alpha r^{3} \sin \theta^{3} \\ \hphantom{\dot{\theta}={}}{}+B_{4}^{2}\alpha r^{2} \cos\theta^{2}(-cC_{6}\gamma+3C_{4}\gamma w +3 B_{5} r \sin\theta) \\ \hphantom{\dot{\theta}={}}{}+\omega r \cos\theta(\gamma^{2}(C_{4}w(-2cC_{6}+3C_{4} w)\alpha C_{8}+C_{5}C_{6}(-C_{7}\beta+C_{6}\gamma(1+\gamma) \\ \hphantom{\dot{\theta}={}}{}+C_{6}\alpha(-1+b+b\gamma )))-2C_{5}C_{7}\alpha\gamma(c C_{6}(1+\gamma)^{2}-3 C_{4}(1+\gamma^{2})w)r \sin \theta \\ \hphantom{\dot{\theta}={}}{}+3 C_{5} C_{7}^{2}\alpha(1+\gamma)^{3}r^{2} \sin\theta^{2}) \\ \hphantom{\dot{\theta}={}}{}+w \gamma^{2}(\gamma(b C_{4}C_{6}^{2} \alpha-c C_{4}^{2} C_{6}\alpha w+C_{4}^{3}\alpha w^{2}-C_{5}C_{6}(C_{6}\alpha +C_{7}\beta) \\ \hphantom{\dot{\theta}={}}{}+C_{6}^{2}C_{7}\gamma)+6C_{4}C_{5}C_{7}\alpha\omega r^{2} \sin{2\theta}))), \\ \dot{\omega}=\frac{w}{C_{1}^{2} C_{5}}(cC_{4}^{2}C_{6}\alpha\gamma w-C_{4}^{3}\alpha \gamma w^{2}-C_{6}^{3}\gamma(1+\gamma) \\ \hphantom{\dot{\omega}={}}{}+C_{6}^{2}(C_{8}\beta+(-bC_{4}+C_{5})\alpha\gamma ))-B_{4}^{3}\alpha r^{3} \cos{\theta^{3}} \\ \hphantom{\dot{\omega}={}}{}-\gamma^{2}(3B_{5}C_{4}^{2}\alpha w^{2}-2cC_{4}C_{6}C_{7}\alpha(1+\gamma )w \\ \hphantom{\dot{\omega}={}}{}-C_{6}^{2}(C_{8}\beta-(1+\gamma)(\gamma^{3}-\omega^{2}-\gamma\omega^{2}) \\ \hphantom{\dot{\omega}={}}{}+\alpha(\gamma-2(-1+b)\gamma^{2}-b\gamma^{3}+b\omega^{2}+b\gamma(-1+\omega ^{2}))))r\sin\theta \\ \hphantom{\dot{\omega}={}}{}+B_{5}^{2}\alpha\gamma(cC_{6}-3C_{4}w)r^{2}\sin{\theta^{2}} \\ \hphantom{\dot{\omega}={}}{}-B_{5}^{3}\alpha r^{3}\sin{\theta^{3}}-B_{4}^{2}\alpha r^{2}\cos{\theta ^{2}}(-cC_{6}\gamma+3C_{4}\gamma w+3B_{5} r \sin\theta) \\ \hphantom{\dot{\omega}={}}{}+\omega r \cos\theta (2cC_{4}C_{6}C_{8}\alpha\gamma^{2} w \\ \hphantom{\dot{\omega}={}}{}-3C_{4}^{2}C_{8}\alpha\gamma^{2} w^{2}+C_{5}C_{6}^{2}\gamma(\beta(1+\gamma)-\gamma (\gamma(1+\gamma)+\alpha(-1+b+b\gamma))) \\ \hphantom{\dot{\omega}={}}{}+2C_{5}C_{7}r(cC_{6}-3C_{4}w)\alpha\gamma(1+\gamma)^{2}\sin\theta -3C_{5}C_{7}^{2}r^{2}\alpha(1+\gamma)^{3}\sin\theta^{2}), \end{cases} $$
(15)

where \(C_{i}\) (\(i=1,2,\ldots,8\)) can be found in the Appendix.

We take condition (6) into system (15), and rescale \((r,w)=(\varepsilon R,\varepsilon W)\) with \(\varepsilon>0\) being a sufficiently small parameter, so system (15) becomes

$$ \textstyle\begin{cases} \frac{dR}{d\theta}=\varepsilon F_{11}(\theta,R,W)+O(\varepsilon^{2}), \\ \frac{dW}{d\theta}=\varepsilon F_{21}(\theta,R,W)+O(\varepsilon^{2}), \end{cases} $$
(16)

where

$$ \begin{aligned} &F_{11}(\theta,R,W)=- \frac{1}{(1+\gamma)D_{1}}\biggl(C_{5}^{3} \omega\cos\theta \bigl(R \omega\cos\theta+\gamma(W+R\sin\theta)\bigr) \\ &\hphantom{F_{11}(\theta,R,W)={}}{}-\frac{1}{\gamma}\sin\theta \bigl(D_{2} W^{2}-W D_{3}+D_{4} R^{2} \cos\theta^{2} \\ &\hphantom{F_{11}(\theta,R,W)={}}{}- \gamma\bigl(\gamma C_{5}^{2}C_{7}-2c(1+\gamma )^{3}C_{7}C_{4}^{2}W+D_{5} \bigr)R\sin\theta \\ &\hphantom{F_{11}(\theta,R,W)={}}{}+c R^{2}(1+\gamma)^{4} C_{7}^{2}C_{4} \sin\theta^{2} \\ &\hphantom{F_{11}(\theta,R,W)={}}{} + C_{5} \omega R \cos\theta(D_{8}W-D_{6}+D_{7} R \sin\theta)\bigr)\biggr), \\ &F_{21}(\theta,R,W)= \frac{1}{\gamma D_{1}}(1+\gamma) \bigl(-c \gamma^{2}C_{4}^{3} W^{2}+\gamma C_{6}D_{9} W-D_{10}R^{2} \cos \theta^{2} \\ &\hphantom{F_{21}(\theta,R,W)={}}{}-\gamma\bigl(2c(1+\gamma )C_{7}C_{4}W \bigr)^{2}R-C_{6}D_{11}\bigr)\sin\theta-D_{12}R^{2} \sin\theta^{2} \\ &\hphantom{F_{21}(\theta,R,W)={}}{}-C_{5}\omega R \cos\theta (D_{15}W-D_{1}3+D_{14}R \sin\theta), \end{aligned} $$
(17)

and \(D_{i}\) (\(i=1,2,\ldots,15\)) can be found in the Appendix.

By using the notation of the averaging theory, we get \(t=\theta\), \(T=2\pi\), \(x=(R,W)\). Then we mark \(g(R,W)=(g_{11}(R,W),g_{21}(R,W))\) and calculate the integrals (17), which are the averaged functions

$$ \textstyle\begin{cases} g_{11}(R,W) =\frac{1}{2\pi} \int_{0}^{2\pi} F_{11}(\theta,R,W)\, d\theta \\ \hphantom{g_{11}(R,W)} =-\frac{R(C_{5}^{3}\omega^{2}-C_{4}(E_{1}+E_{2} W))}{2(1+\gamma)D_{1}}, \\ g_{21}(R,W) =\frac{1}{2\pi} \int_{0}^{2\pi} F_{21}(\theta,R,W)\, d\theta \\ \hphantom{g_{21}(R,W)} =-\frac{(1+\gamma)(-cC_{4}^{2}((1+\gamma)^{2}C_{6} R^{2}+2\gamma^{2}C_{4} W^{2})+2\gamma C_{6}E_{3}W)}{2\gamma D_{1}}, \end{cases} $$
(18)

where

$$ \begin{aligned} &E_{1}= -\gamma^{2} \bigl(1+\gamma(3+\gamma)\bigr) \bigl(1+\gamma\bigl(2+\gamma+\gamma^{2} \bigr)\bigr)-\gamma (1+\gamma)^{3}\omega^{2}+(1+ \gamma)^{3}\omega^{4}, \\ &E_{2}= 2c(1+\gamma)^{3}C_{7}C_{4}, \\ &E_{3}= 1+\gamma\bigl(5+8\gamma+6\gamma^{2}+4 \gamma^{3}+\gamma^{4}+2(1+\gamma)^{2}\omega ^{2}+\omega^{4}\bigr). \end{aligned} $$
(19)

Therefore, from \(g_{11}(R,W)=g_{21}(R,W)=0\), we can obtain a unique positive real solution \((R_{*},W_{*})\) (satisfying \(R_{*}>0\))

$$ R_{*}=\sqrt{\frac{\Gamma}{2c^{2}(1+\gamma)^{4}C_{7}^{2}C_{4}^{4}}},\qquad W_{*}= \frac{E_{4}}{2c(1+\gamma)C_{7}C_{4}^{2}}, $$
(20)

where

$$\begin{aligned} E_{4} = &4\gamma^{6}+\gamma^{7}+ \gamma^{5}\bigl(6+\omega^{2}\bigr)+\gamma^{4} \bigl(8+3\omega^{3}\bigr)+\gamma ^{2}\bigl(1+7 \omega^{2}-2\omega^{4}\bigr)+\gamma^{3}\bigl(5+4 \omega^{2}-\omega^{4}\bigr) \\ &{}-\omega^{2}\bigl(-1+\omega^{2}+\omega^{4} \bigr)-\gamma\omega^{2}\bigl(-5+2\omega^{2}+ \omega^{4}\bigr). \end{aligned}$$
(21)

We recall that Γ is defined in the statement of Theorem 3.2, and get the Jacobian matrix

$$ \frac{\partial(g_{11},g_{21})}{\partial(R,W)}\bigg|_{(R,W)=(R_{*},W_{*})}= \left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} 0 & \frac{(1+\gamma)^{2}C_{7}C_{4}\sqrt{\gamma\Gamma}}{C_{5}C_{6}\omega^{3}\sqrt {2}E_{5}} \\ -\frac{(1+\gamma)^{3}C_{4}\sqrt{\Gamma}}{C_{5}\omega^{3}\sqrt{2\gamma}E_{5}} & -\frac{C_{5}}{\omega C_{7}} \end{array}\displaystyle \right ), $$
(22)

where

$$ E_{5}= (1+\gamma)^{2}(1+2\gamma) \bigl( \gamma+3\gamma^{2}+3\gamma^{3}+\gamma^{4}-\gamma \omega ^{2}-\omega^{2}\bigl(1+\omega^{2}\bigr) \bigr). $$
(23)

Furthermore, the determinant of Jacobian matrix (22) at \((R_{*}, W_{*})\) takes the following value:

$$ \frac{(1+\gamma)^{5}C_{7}C_{4}^{2}\Gamma}{C_{5}^{2}C_{6}\omega^{6} 2E_{5}^{2}} \neq0. $$
(24)

In a nutshell, we prove that when \(\varepsilon>0\) sufficiently small, the periodic solutions corresponding to \((R_{*},W_{*})\) produce a periodic solution that bifurcates from the coordinate origin of the differential system (18). Theorem 3.3 guarantees the existence of a periodic solution corresponding to the point \((R_{*} ,W_{*})\) of the form \((R(\theta, \epsilon ),W(\theta, \epsilon))\) for \(\epsilon>0\) small enough, such that \((R(0, \epsilon),W(0, \epsilon))\rightarrow(R_{*} ,W_{*})\) when \(\epsilon \rightarrow0\). Thus we get the periodic solution of system (14)

$$ u(\theta, \epsilon)=\epsilon R \cos\theta,\qquad v(\theta, \epsilon )=\epsilon R \sin\theta,\qquad w(\theta, \epsilon)=\epsilon W $$
(25)

for \(\epsilon>0\) small enough. From relation (28) to the linear change of variable (13), we can get a periodic solution \((x(\theta),y(\theta ),z(\theta))\) of system (2). Finally, we get the conclusion that the modified Chua system (2) has a periodic solution \((x(\theta),y(\theta ),z(\theta))\) for \(\epsilon>0\) sufficiently small, and when \(\epsilon \rightarrow0\), it tends to the origin of coordinates. The periodic solution starts from the zero-Hopf equilibrium point, and is located at the origin of coordinates when \(\epsilon=0\). So we have completed the proof of Theorem 3.3. □

4 Hopf bifurcation of the modified Chua system

Assume that the characteristic equation of the modified Chua system (2) has a pair of imaginary roots \(\pm i \omega\) (\(\omega\in R^{+}\)). It is not hard to know that when

$$ \beta=\beta_{0}=-\frac{(\alpha b+1) (\alpha (b \gamma+b-1)+\gamma (\gamma+1))}{\gamma+1}, $$
(26)

(4) yields

$$ \lambda_{1} = -b\alpha-\gamma-1< 0,\qquad \lambda_{2,3}=\pm i\sqrt{ \frac {\alpha(\alpha b^{2} (\gamma+1)-\alpha b+\gamma)}{-\gamma-1}}, $$
(27)

where \(b\alpha+\gamma+1>0\), \(\alpha(\alpha b^{2} (\gamma+1)-\alpha b+\gamma)(-\gamma-1)>0\).

Proposition 4.1

Define

$$ \begin{aligned} &T=\bigl\lbrace (\alpha,\beta,\gamma,b) \mid b\alpha+\gamma+1>0, \alpha \bigl(\alpha b^{2} (\gamma+1)-\alpha b+ \gamma\bigr) (-\gamma-1)>0\bigr\rbrace , \\ &\beta=\beta_{0} =-\frac{(\alpha b+1) (\alpha (b \gamma+b-1)+\gamma (\gamma+1))}{\gamma+1}, \end{aligned} $$
(28)

then the Jacobian matrix of system (2) at \(O(0,0,0)\) has a negative real eigenvalue \(-b\alpha-\gamma-1\) and a pair of purely imaginary eigenvalues \(\pm i\sqrt{ \frac{\alpha(\alpha b^{2} (\gamma +1)-\alpha b+\gamma)}{-\gamma-1}}\).

Let β be the Hopf bifurcation parameter, the transverse condition

$$\begin{aligned}& \operatorname{Re} \bigl(\lambda'(\beta_{0})\bigr) \big|_{\lambda=\pm i\sqrt{ \frac{\alpha(\alpha b^{2} (\gamma+1)-\alpha b+\gamma)}{-\gamma-1}}} \\& \quad =\frac{(\gamma+1)^{2}}{2 ((\alpha+3) \gamma+2 \alpha^{2} b^{2} (\gamma+1)+\alpha b (2 (\gamma+1)^{2}-\alpha )+\gamma^{3}+3 \gamma^{2}+1 )}\neq0 \end{aligned}$$
(29)

can also be satisfied. Consequently, we can get the following theorem.

Theorem 4.2

If \((\alpha,\beta,\gamma,b) \in T\) and β varies and passes through the critical value

$$ \beta_{0} =-\frac{(\alpha b+1) (\alpha (b \gamma+b-1)+\gamma (\gamma +1))}{\gamma+1}, $$

system (2) undergoes the Hopf bifurcation at the equilibrium \(O(0,0,0)\).

Firstly, let us retrospect the projection method presented in [44], but after the study of [45–51], used to figure out the first Lyapunov coefficient \(l_{1}\) connected with the stability of a Hopf bifurcation. Consider the following differential equation:

$$ \dot{X}=f(X,\mu), $$
(30)

where \(X\in R^{3}\) and \(\mu\in R^{5} \) are respectively vectors representing phase variables and control parameters. Assume that f is a class of \(C^{\infty}\) in \(R^{3} \times R^{5}\). Suppose that (33) has an equilibrium \(X=X_{0}\) at \(\mu=\mu_{0}\). Denoting the variable \(X=X_{0}\) at \(\mu=\mu_{0}\), and denoting the variable \(X-X_{0}\) also by X, write

$$ F(X)=f(X,\mu_{0}) $$
(31)

as

$$ F(X)=AX+\frac{1}{2}B(X,X)+\frac{1}{6}C(X,X,X)+O \bigl(\|X\|^{4}\bigr), $$
(32)

where \(A=f_{x}(0,\mu_{0})\) and, for \(i=1,2,3\),

$$\begin{aligned}& B(X,Y)=\sum_{j,k=1}^{3}\frac{\partial^{2}F_{i}(\xi)}{\partial\xi_{j}\, \partial\xi_{k}} \bigg\vert _{\xi=0} X_{j}Y_{k}, \\& C(X,Y,Z)=\sum_{j,k,l=1}^{3}\frac{\partial^{3}F_{i}(\xi)}{\partial\xi_{j}\, \partial\xi_{k}\, \partial\xi_{l}} \bigg\vert _{\xi=0} X_{j}Y_{k}Z_{l}. \end{aligned}$$

Assume that A has one pair of complex eigenvalues on the imaginary axis: \(\lambda_{2,3}=\pm w_{0}\) (\(w_{0}>0\)), and these eigenvalues are the only eigenvalues with \(\operatorname{Re} \lambda=0\). Let \(T^{c}\) be the generalized eigenvalues of A corresponding to \(\lambda_{2,3}\). Let \(p, q \in C^{3}\) be vectors such that

$$ Aq=i w_{0} q, \qquad A^{T} p=-i w_{0} p,\qquad \langle p, q \rangle=1, $$
(33)

where \(A^{T}\) is the transpose of the matrix A. Any vector \(y \in T^{c}\) can be expressed as \(y=w q+\overline{w} \bar{q}\), where \(w=\langle p, y \rangle\in C\). The two-dimensional center manifold related to the eigenvalues \(\lambda_{2,3}\) can be parameterized by w and w̅, by way of an immersion of the form \(X=H(w, \overline{w})\), where \(H:C^{2}\rightarrow R^{3}\) has a Taylor expansion of the following form:

$$\begin{aligned} H(w,\overline{w}) = &w q+\overline{w} \bar{q} \\ &{}+\sum_{2\leq j+k \leq3}\frac{1}{j!k!}h_{jk}w^{j} \overline{w}^{k}+O\bigl(|w|^{4}\bigr), \end{aligned}$$
(34)

with \(h_{jk} \in C^{3}\) and \(h_{jk}=\bar{h}_{kj}\). Taking this expression into (34), we get the following differential equation:

$$ H_{w} w'+H_{w} \overline{w}'=F \bigl(H(w,\overline{w})\bigr), $$

where F is given by (32). The complex vectors \(h_{ij}\) are obtained by the coefficients of (34). Considering the coefficients of F, system (30) can be written as the following form on the chart w for a central manifold:

$$ \dot{w}=iw_{0}w+\frac{1}{2}G_{21}w|w|^{2}+O \bigl(|w|^{4}\bigr), $$

with \(G_{21}\in C\). The first Lyapunov coefficient can be presented as

$$ l_{1}=\frac{1}{2} \operatorname{Re} G_{21}, $$

where

$$ G_{21}=\bigl\langle p,C(q,q,\bar{q})+B(\bar{q}, h_{20})+2B(q,h_{11}) \bigr\rangle . $$

If the Jacobian matrix A of an equilibrium point has only a pair of purely imaginary eigenvalues \(\pm i w_{0}\) (\(w_{0}>0\)), and the other eigenvalue with nonzero real part, then the equilibrium point is called a Hopf bifurcation point \((X_{0},\mu_{0})\). It is clear that a two-dimensional center manifold is well defined at a Hopf point. Also, it is invariant and can continue any higher-order derivative to nearby parameter values.

If the parameter-dependent complex eigenvalues cross the imaginary axis with nonzero derivative, the Hopf point is called transversal. In the neighborhood of transverse Hopf points with \(l_{1}\neq0\), the dynamic behavior of system (30) is reduced to a family of parameter-dependent continuations of the center manifold, which is topologically equivalent to the following complex standard form in orbit:

$$ w'=(\eta+iw)w+l_{1} w|w|^{2}, $$

where \(w\in C\), η, w, and \(l_{1}\) are real functions with any higher-order derivative, which are continuations of 0, \(w_{0}\), and the first Lyapunov coefficient at the Hopf point [46]. In this manifold family, we find one family of stable (unstable) periodic orbits as \(l_{1}<0\) (\(l_{1}>0\)), narrowing to a point of equilibrium at the Hopf point.

In the remainder of this part, we apply the three-dimensional Hopf bifurcation method, and symbolic calculations are used to analyze the parameter changes regarding dynamic bifurcations. We consider the bifurcation of system (2) at \(O(0,0,0)\) (system (2) at \(E_{0}\)).

Theorem 4.3

Take the two-parameter family of system (2) into account. The first Lyapunov coefficient related to the equilibrium point O is presented by

$$\begin{aligned} l_{1} =&\frac{1}{2(\alpha b+\gamma+1) ((4 \alpha+3) \gamma+5 \alpha^{2} b^{2} (\gamma+1)+2 \alpha b ((\gamma+1)^{2}-2 \alpha )+\gamma^{3}+3 \gamma^{2}+1 )} \\ &{}\times\bigl(144 \alpha (\gamma+1)^{4} K_{1} (\alpha b+1)^{2} \bigl(\alpha b^{2} (\gamma+1)-\alpha b+\gamma \bigr) \\ &{}\times \bigl(\alpha (b \gamma+b-1)+\gamma (\gamma+1)\bigr)^{4} \bigl(3 K_{3}-2 c^{2} K_{2} \bigr)\bigr), \end{aligned}$$

where \(K_{i}\) (\(i=1,2,3\)) can be found in the Appendix . If \(l_{1}>0\), then the Hopf point at equilibrium O is unstable(weak repelling focus), and for each \(b>b_{0}=\frac{a^{2}}{3-a}\), but near \(b_{0}\), there is an unstable limit cycle around the asymptotically stable equilibrium point O. If \(l_{1}<0\), the Hopf point at the equilibrium point O is asymptotically stable (weak attractor focus), and for each \(b>b_{0}=\frac{a^{2}}{3-a}\), but close to \(b_{0}\), there is a stable limit cycle around the unstable equilibrium point O.

Proof

Under these circumstances (28), the transversality condition (29) is also satisfied. Consequently, the Hopf bifurcation at equilibrium point O happens. The value of the first Lyapunov coefficient \(l_{1}\) determines the stability of the equilibrium point O and it can also exhibit the stability of the equilibrium point and the previous section. The multilinear symmetric functions can be presented as

$$\begin{aligned}& B(x, y)=(-2c x_{1} y_{1},0,0), \\& C(x, y, z)=(-6 x_{1} y_{1} z_{1},0,0). \end{aligned}$$

Furthermore, we can also obtain

$$\begin{aligned}& p=\frac{1}{\Delta} \biggl(i \sqrt{\frac{(-\gamma-1)(\alpha b^{2} (\gamma+1)-\alpha b+\gamma)}{\alpha}}-b (\gamma+1)+1, \\& \hphantom{p={}}{}\gamma+ \frac {1}{i}\sqrt{\frac{\alpha(\alpha b^{2} (\gamma+1)-\alpha b+\gamma )}{-\gamma-1}},1 \biggr), \\& q= \biggl(\frac{\sqrt{\alpha} (\gamma+1) ( (-\sqrt{\alpha } ) (b \gamma+b-1)-i \sqrt{(-\gamma-1)(\alpha b^{2} (\gamma +1)-\alpha b+\gamma)} )}{(\alpha b+1) (\alpha (b \gamma +b-1)+\gamma (\gamma+1))}, \\& \hphantom{q={}}{}\frac{-i \sqrt{\alpha(-\gamma-1) (\alpha b^{2} (\gamma+1)-\alpha b+\gamma)}+\gamma^{2}+\gamma}{(\alpha b+1) (\alpha (b \gamma +b-1)+\gamma (\gamma+1))},1 \biggr) \end{aligned}$$

and

$$\begin{aligned}& h_{11}= \biggl(\frac{2 \alpha c (\gamma+1)^{2} (\alpha b^{2} (\gamma+1)+b ((\gamma+1)^{2}-\alpha )-1 )}{(\alpha b+1)^{2} (\alpha (b \gamma+b-1)+\gamma (\gamma+1)) (\alpha^{2} b^{3} (\gamma+1)+\alpha b^{2} ((\gamma+1)^{2}-\alpha )-\alpha b+\gamma (\gamma+1) )}, \\& \hphantom{h_{11}={}}{}-\frac{2 c \gamma (\gamma+1)^{3}}{(\alpha b+1)^{2} (\alpha (b \gamma +b-1)+\gamma (\gamma+1)) (\alpha^{2} b^{3} (\gamma+1)+\alpha b^{2} ((\gamma+1)^{2}-\alpha )-\alpha b+\gamma (\gamma+1) )}, \\& \hphantom{h_{11}={}}{} -\frac{2 c (\gamma+1)^{2}}{(\alpha b+1) (\alpha^{2} b^{3} (\gamma +1)+\alpha b^{2} ((\gamma+1)^{2}-\alpha )-\alpha b+\gamma (\gamma+1) )} \biggr), \\& h_{20}= (\frac{1}{3 (\alpha b+1)^{2} (\alpha b^{2} (\gamma +1)-\alpha b+\gamma ) (\alpha (b \gamma+b-1)+\gamma (\gamma +1))^{2} K_{4}} \\& \hphantom{h_{20}={}}{}\times 2 \sqrt{\alpha} c (-\gamma-1)^{5/2} \bigl( \sqrt{(-\gamma -1) \bigl(\alpha b^{2} (\gamma+1)-\alpha b+\gamma \bigr)}-i \sqrt{\alpha} (b \gamma+b-1) \bigr)^{2} \\& \hphantom{h_{20}={}}{}\times \bigl(2 i (-\gamma-1)^{3/2} \sqrt{\alpha b^{2} (\gamma+1)-\alpha b+\gamma}+3 \alpha^{3/2} b (b \gamma+b-1) \\& \hphantom{h_{20}={}}{}+\sqrt{\alpha} \bigl(-b (\gamma+1)^{2}+4 \gamma+1 \bigr) \bigr), \\& \hphantom{h_{20}={}}{} -\frac{1}{3 (\alpha b+1)^{2} (\alpha b^{2} (\gamma+1)-\alpha b+\gamma ) (\alpha (b \gamma+b-1)+\gamma (\gamma+1))^{2} K_{4}} \\& \hphantom{h_{20}={}}{}\times 2 c (\gamma+1)^{3} \bigl(\gamma \sqrt{- \gamma-1}+2 i \sqrt{\alpha \bigl(\alpha b^{2} (\gamma+1)-\alpha b+\gamma \bigr)} \bigr) \\& \hphantom{h_{20}={}}{}\times \bigl(\sqrt{\alpha} (b \gamma+b-1)+i \sqrt{(-\gamma-1) \bigl(\alpha b^{2} (\gamma+1)-\alpha b+\gamma\bigr)} \bigr)^{2}, \\& \hphantom{h_{20}={}}{}\times \frac{2 c (-\gamma-1)^{5/2} (\sqrt{(-\gamma-1)(\alpha b^{2} (\gamma+1)-\alpha b+\gamma)}-i \sqrt{\alpha} (b \gamma+b-1) )^{2}}{3(\alpha b+1) (\alpha b^{2} (\gamma+1)-\alpha b+\gamma ) (\alpha (b \gamma+b-1)+\gamma (\gamma+1))K_{4}} ), \\& G_{21} =\frac{2 \sqrt{\alpha} (\gamma+1)(\alpha (b \gamma +b-1)+\gamma (\gamma+1))K_{6}}{K_{5}}, \end{aligned}$$

where Δ and \(K_{i}{i=4,5,6}\) can be found in the Appendix.

Therefore, we obtain the first Lyapunov coefficient

$$\begin{aligned} l_{1} =&\frac{1}{2(\alpha b+\gamma+1) ((4 \alpha+3) \gamma+5 \alpha^{2} b^{2} (\gamma+1)+2 \alpha b ((\gamma+1)^{2}-2 \alpha )+\gamma^{3}+3 \gamma^{2}+1 )} \\ &{}\times\bigl(144 \alpha (\gamma+1)^{4} K_{1} (\alpha b+1)^{2} \bigl(\alpha b^{2} (\gamma+1)-\alpha b+\gamma \bigr) \\ &{}\times \bigl(\alpha (b \gamma+b-1)+\gamma (\gamma+1)\bigr)^{4} \bigl(3 K_{3}-2 c^{2} K_{2} \bigr)\bigr). \end{aligned}$$

Therefore, Theorem 4.3 is proved. □

5 Conclusion

In this paper, Chua’s system with a smooth nonlinearity, described by a cubic polynomial, has been presented. Two kinds of bifurcations of the modified smooth Chua system have been studied theoretically. We explored the distribution of the equilibrium points and researched the limit cycles bifurcating from zero-Hopf equilibrium points of the modified Chua system by using the averaging theory. Moreover, the existence of periodic solutions in the modified Chua system by the classical Hopf bifurcation was derived. In fact, there are many other rich dynamic properties of Chua’s system that are not fully exploited. We hope to propose more new things about Chua’s system in later research.