1 Introduction

Since the pioneering work of Ott et al.[1], the topics of chaos and chaotic control are growing rapidly in many different fields such as ecological system, chemical systems and biological systems, and so forth[2–10]. We all know that chaotic systems have very complicated dynamical nature, which plays an important role in many fields such as secure communications, information processing, high-performance circuit design for telecommunications[11]. Chaos, which often causes irregular behaviors in practical system, is usually undesirable. In many cases, we wish to avoid and eliminate such behaviors. Recently, many schemes such as Ott et al.[1], feedback and non-feedback control[12–17], observer-based control[12], active control[13], adaptive control[18, 19], inverse optimal control[20], evolutionary algorithm[21], etc., have been presented to implement the chaos control. For more related work, one can see [22–26]. In 2012, Wang and Chen[27] reported the very surprising finding of the following new 3D autonomous chaotic Sprott E system with one stable node or stable focus

$$\left\{{\matrix{{\dot x = y(t)z(t) + a} \hfill \cr{\dot y = {x^2}(t) - y(t)} \hfill \cr{\dot z = 1 - 4x(t).} \hfill \cr}} \right.$$
(1)

When a = 0, it is the Sprott E system[28].When a ≠ 0, the stability of the single equilibrium is fundamentally different[29]. Let yz + a = 0, x2 − y = 0, 1 − 4x = 0, we can obtain that system (1) has only one stable equilibrium \(E({x^{\ast}},\;{y^{\ast}},\;{z^{\ast}}) = ({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;16a)\) if a > 0. Interestingly, Wang and Chen[29] found that system (1) can generate chaotic phenomenon which is shown in Fig. 1 (Fig. 1 shows the waveform portraits and the phase portraits of system (1)).

Fig. 1
figure 1

Chaotic attractor of system (1) with a = 0.006. The initial value is (1.5, 1.5, 1.5)

Full size image

In this paper, we study the stability, and the local Hopf bifurcation nature for system (1). To the best of our knowledge, it is the first try to introduce continuous distributed time-delayed feedback force to control the chaos of the Sprott E system.

The remainder of the paper is organized as follows. In Section 2, we investigate the stability of the equilibrium and the occurrence of local Hopf bifurcations of the Sprott E system with distributed delay feedback. In Section 3, the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to show the validity of chaotic control. Some main conclusions are drawn in Section 5.

2 Stability and bifurcation analysis

In this section, we shall study the stability of the equilibrium and the existence of local Hopf bifurcations.

In order to apply feedback control, we add a continuous distributed time delayed force

$$b\int_{- \infty}^0 {[z(t) - z(t + s)]k(- s){\rm{d}}s}$$

to the third equation of system (1), then system (1) takes the form

$$\left\{{\matrix{{\dot x = y(t)z(t) + a} \hfill \cr{\dot y = {x^2}(t) - y(t)} \hfill \cr{\dot z = 1 - 4x(t) + b\int_{- \infty}^0 {[z(t) - z(t + s)]k(- s){\rm{d}}s}} \hfill \cr}} \right.$$
(2)

where \(a,\;b > 0,\;\smallint _{- \infty}^0k(s)ds = 1,\;\smallint _{- \infty}^0sk(s)ds < + \infty\). Obviously, system (2) has the equilibrium point \(E({x^{\ast}},\;{y^{\ast}},\;{z^{\ast}}) = ({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;16a)\).

Let \(\bar x(t) = x(t) - {x^{\ast}},\;\bar y(t) = y(t) - {y^{\ast}},\;\bar z(t) = z(t) - {z^{\ast}}\) and still denote \(\bar x(t),\;\bar y(t)\) and \(z(t)\) by x(t), y(t) and z(t), respectively, then (2) becomes

$$\left\{{\matrix{{\dot x = {z^{\ast}}y(t) + {y^{\ast}}z(t) + y(t)z(t)} \hfill \cr{\dot y = 2{x^{\ast}}x(t) - y(t) + {x^2}(t)} \hfill \cr{\dot z = - 4x(t) + bz(t) - b\int_{- \infty}^0 {z(t + s)k(- s){\rm{d}}s.}} \hfill \cr}} \right.$$
(3)

The linearization of (3) near E(x*, y*, z*) is given by

$$\left\{{\matrix{{\dot x = {z^{\ast}}y(t) + {y^{\ast}}z(t)} \hfill \cr{\dot y = 2{x^{\ast}}x(t) - y(t)} \hfill \cr{\dot z = - 4x(t) + bz(t) - b\int_{- \infty}^0 {z(t + s)k(- s){\rm{d}}s}} \hfill \cr}} \right.$$
(4)

whose characteristic equation appears as

$$\matrix{{\lambda (\lambda + 1)\left({\lambda - b + b\int_{- \infty}^0 {k(- s){{\rm{e}}^{\lambda s}}{\rm{d}}s}} \right) + 4{y^{\ast}}(\lambda + 1) -} \hfill \cr{2{x^{\ast}}{z^{\ast}}\left({\lambda - b + b\int_{- \infty}^0 {k(- s){{\rm{e}}^{\lambda s}}{\rm{d}}s}} \right) = 0.} \hfill \cr}$$
(5)

In this paper, we consider the weak kernel case, i.e., k(s) = αe−αs, where α > 0. As to the general gamma kernel case, we can make a similar analysis. We give the initial condition of system (4) as

$$\left[ {\matrix{{x(s)} \cr{y(s)} \cr{z(s)} \cr}} \right] = \left[ {\matrix{{{\phi _1}(s)} \cr{{\phi _2}(s)} \cr{{\phi _3}(s)} \cr}} \right],\quad - \infty < s \leq 0.$$

The characteristic equation (5) with the weak kernel case takes the form

$${\lambda ^4} + {\theta _1}(\alpha){\lambda ^3} + {\theta _2}(\alpha){\lambda ^2} + {\theta _3}(\alpha)\lambda + {\theta _4}(\alpha) = 0$$
(6)

where

$$\left\{{\matrix{{{\theta _1}(\alpha) = \alpha - b + 1} \hfill \cr{{\theta _2}(\alpha) = \alpha - b - 2{x^{\ast}}{z^{\ast}} + 4{y^{\ast}}} \hfill \cr{{\theta _3}(\alpha) = 4{y^{\ast}}(1 + \alpha) - 2{x^{\ast}}{z^{\ast}}(\alpha - b)} \hfill \cr{{\theta _4}(\alpha) = 4{y^{\ast}}\alpha .} \hfill \cr}} \right.$$
(7)

In view of the well known Routh-Hurwitz criterion, we can conclude that all the roots of (6) has negative real parts if the following conditions

$$\left\{{\matrix{{{D_1}(\alpha) = {\theta _1}(\alpha) = \alpha - b + 1 > 0} \hfill \cr{{D_2}(\alpha) = {\theta _1}(\alpha){\theta _2}(\alpha) - {\theta _3}(\alpha) =} \hfill \cr{\quad \quad \quad \;\;(\alpha - b + 1)(\alpha - b - 2{x^{\ast}}{z^{\ast}} + 4{y^{\ast}}) -} \hfill \cr{\quad \quad \quad \;\;[4{y^{\ast}}(1 + \alpha) - 2{x^{\ast}}{z^{\ast}}(\alpha - b)] > 0} \hfill \cr{{D_3}(\alpha) = {\theta _3}(\alpha){D_2}(\alpha) - \theta _1^2(\alpha){\theta _4}(\alpha) =} \hfill \cr{\quad \quad \quad \;\;[4{y^{\ast}}(1 + \alpha) - 2{x^{\ast}}{z^{\ast}}(\alpha - b)] \times} \hfill \cr{\quad \quad \quad \;\;\{(\alpha - b + 1)(\alpha - b - 2{x^{\ast}}{z^{\ast}} + 4{y^{\ast}}) -} \hfill \cr{\quad \quad \quad \;\;[4{y^{\ast}}(1 + \alpha) - 2{x^{\ast}}{z^{\ast}}(\alpha - b)]\} -} \hfill \cr{\quad \quad \quad \;\;4{y^{\ast}}\alpha {{(\alpha - b + 1)}^2} > 0} \hfill \cr{{D_4}(\alpha) = {\theta _4}(\alpha){D_3}(\alpha) = 4{y^{\ast}}\alpha {D_3}(\alpha) > 0} \hfill \cr}} \right.$$
(8)

hold true.

Based on the analysis above, we can easily obtain the following Theorem 1.

  • Theorem 1. The equilibrium E(x*, y*, z*) of system (2) with the weak kernel is locally asymptotically stable if the following conditions are fulfilled:

    $$\left\{{\matrix{{\alpha - b + 1 > 0} \hfill \cr{(\alpha - b + 1)(\alpha - b - 2{x^{\ast}}{z^{\ast}} + 4{y^{\ast}}) -} \hfill \cr{\quad [4{y^{\ast}}(1 + \alpha) - 2{x^{\ast}}{z^{\ast}}(\alpha - b)] > 0} \hfill \cr{[4{y^{\ast}}(1 + \alpha) - 2{x^{\ast}}{z^{\ast}}(\alpha - b)] \times} \hfill \cr{\quad \{(\alpha - b + 1)(\alpha - b - 2{x^{\ast}}{z^{\ast}} + 4{y^{\ast}}) -} \hfill \cr{\quad [4{y^{\ast}}(1 + \alpha) - 2{x^{\ast}}{z^{\ast}}(\alpha - b)]\} - 4{y^{\ast}}\alpha {{(\alpha - b + 1)}^2} > 0.} \hfill \cr}} \right.$$

    Let λ i (i = 1, 2, 3, 4) be the roots of (6), then we have

    $$\left\{{\matrix{{{\lambda _1} + {\lambda _2} + {\lambda _3} + {\lambda _4} = - {\theta _1}(\alpha)} \hfill \cr{{\lambda _1}{\lambda _2} + {\lambda _1}{\lambda _3} + {\lambda _1}{\lambda _4} + {\lambda _2}{\lambda _3} + {\lambda _2}{\lambda _4} + {\lambda _3}{\lambda _4} = {\theta _2}(\alpha)} \hfill \cr{{\lambda _1}{\lambda _2}{\lambda _3} + {\lambda _1}{\lambda _2}{\lambda _4} + {\lambda _1}{\lambda _3}{\lambda _4} + {\lambda _2}{\lambda _3}{\lambda _4} = - {\theta _3}(\alpha)} \hfill \cr{{\lambda _1}{\lambda _2}{\lambda _3}{\lambda _4} = {\theta _4}(\alpha).} \hfill \cr}} \right.$$
    (9)

    If there exists an α0 ∈ R+ such that D3 (∈0) = 0 and , then by the Routh-Hurwitz criterion, there exists a pair of purely imaginary roots, say \({\lambda _1} + {{\bar \lambda}_2} = {\rm{i}}{\omega _0}(\omega \neq 0)\), and the other two roots λ3, X4 satisfy: if X3, X4 are real, then λ3 < 0, λ4 < 0; if λ3, λ4 are complex conjugate, then \({\mathop{\rm Re}\nolimits} \{{\lambda _3}\} = {\mathop{\rm Re}\nolimits} \{{\lambda _4}\} = - {{{\theta _1}(\alpha)} \over 2}\). It is easy to calculate that

    $$\matrix{{{{{\rm{d}}({\mathop{\rm Re}\nolimits})({\lambda _1}))} \over {{\rm{d}}\alpha}} = - {{{\theta _1}(\alpha)} \over {2[\theta _1^3(\alpha){\theta _3}(\alpha) + {{({\theta _1}(\alpha){\theta _2}(\alpha) - 2{\theta _3}(\alpha))}^2}]}} \times} \hfill \cr{{{\left. {\quad \quad \quad \quad \quad {{{\rm{d}}{D_3}(\alpha)} \over {{\rm{d}}\alpha}}} \right\vert}_{\alpha = {\alpha _0}}}} \hfill \cr}$$
    (10)

    thus the Hopf bifurcation occurs near E(x*,y*,z*) when α passes through α0.

3 Direction and stability of bifurcating periodic solutions

In this section, by using techniques from normal form and center manifold theory[30], we shall investigate the direction, stability, and period of these periodic solutions bifurcating from the equilibrium E(i*, y*, z*). Let μ = α−α0, then system (3) undergoes the Hopf bifurcation at the equilibrium E(x*,y*,z*) for μ = 0, and then ±iω0 are purely imaginary roots of the characteristic equation at the equilibrium E(x*, y*, z*). System (3) can be written as an functional differential equation (FDE) in C = C([−∞, 0]), R3) as

$$\dot u(t) = A(\mu){u_t} + R(\mu){u_t}$$
(11)

where u(t) = [x(t),y(t),z(t)]T ∈ C and u t (θ) = u(t + θ) = [x(t + θ),y(t + θ), z(t + θ)]T ∈ C, and A and R are given by

((12))

and

((13))

respectively, where φ(θ) = [φ1 (θ), φ2 (θ), φ3 (θ)]T ∈ C and \({f_1} = {\phi _2}(0){\phi _3}(0),\;{f_2} = \phi _1^2(0)\).

For ψ ∈ C ([0, +∞], (R3)*), define

$${A^{\ast}}\psi (s) = \left\{{\matrix{{- {{{\rm{d}}\psi {\rm{(}}s{\rm{)}}} \over {{\rm{d}}s}},\quad s \in (0, + \infty ]} \hfill \cr{{L^{\rm{T}}}\psi (0) + \int_{- \infty}^0 {{F^{\rm{T}}}(t)\psi (- t){\rm{d}}s,\quad s = 0.}} \hfill \cr}} \right.$$

For φ ∈ C([−∞, 0], R3) and ψ ∈ C([0, +∞], (R3)*), define the bilinear form

$$\left\langle {\psi ,\phi} \right\rangle = \bar \psi (0)\phi (0) - \int_{- \infty}^0 {\int_{\xi = 0}^\theta {\bar \psi} (\xi - \theta){\rm{d}}} \eta (\theta)\phi (\xi){\rm{d}}\xi$$

where η(θ) = η(θ, 0), the A = A(0) and A* are adjoint operators. By the discussions in Section 2, we know that ±iω0 are eigenvalues of A(0), and they are also eigenvalues of A* corresponding to iω0 and −iω0, respectively. Assume that \(q(\theta) = {(1,\;{a_1},\;{a_2})^T}{{\rm{e}}^{{\rm{i}}{\omega _{\rm{0}}}\theta}}\) is the eigenvector of A(0) corresponding to iω0, then we have A(0) q(0) = iω0q(0), i.e.,

where

$${\chi ^{(1)}} = \int_{- \infty}^0 {k(- s){{\rm{e}}^{{\rm{i}}{\omega _0}s}}{\rm{d}}s = {\alpha \over {\alpha + {\rm{i}}{\omega _0}}}.}$$

We can obtain

$$q(\theta) = {[1,{a_1},{a_2}]^{\rm{T}}}{{\rm{e}}^{{\rm{i}}{\omega _0}\theta}}$$

where

$${a_1} = {{2{x^{\ast}}} \over {{\rm{i}}{\omega _0} + 1}},\quad {a_2} = {4 \over {b - b{\chi ^{(1)}} - {\rm{i}}{\omega _0}}}.$$

Assume that \({q^{\ast}}(s) = D{[1,\;a_1^{\ast},\;a_2^{\ast}]^T}{{\rm{e}}^{{\rm{i}}{\omega _{\rm{0}}}s}}(0 \leq s < + \infty)\) is the eigenvector of A*(0) corresponding to −iω0, then we have A*(0)q*(0) = iω0q*(0), i.e.,

where

$${\chi ^{(2)}} = \int_{- \infty}^0 {k(- s){{\rm{e}}^{- {\rm{i}}{\omega _0}s}}{\rm{d}}s = {\alpha \over {\alpha - {\rm{i}}{\omega _0}}}.}$$

We can obtain

$${q^{\ast}}(s) = D{[1,a_1^{\ast},a_1^{\ast}]^{\rm{T}}}{{\rm{e}}^{{\rm{i}}{\omega _0}s}}$$

where

$$a_1^{\ast} = {{{z^{\ast}}} \over {1 - {\rm{i}}{\omega _0}}},\quad a_2^{\ast} = {{b{\chi ^{(2)}} - {y^{\ast}}} \over {b + {\rm{i}}{\omega _0}}}.$$

If we choose

$$D = {1 \over {1 + a_1^{\ast}{{\bar a}_1} + a_2^{\ast}{{\bar a}_2} + ba_2^{\ast}{{\bar a}_2}\int_{- \infty}^0 {\theta {{\rm{e}}^{- {\rm{i}}{\omega _0}\theta}}k(- \theta){\rm{d}}\theta}}}$$

then <q*(s),q(θ) ≥ 1 and \(< {q^{\ast}}(s),\;\bar q(\theta) \geq 0\).

Next, we use the same notations as those in Hassard[30] and we first compute the coordinates to describe the center manifold C0 at μ = 0. Let u t be the solution of (11) when μ = 0.

Define

$$z(t) = \left\langle {{q^{\ast}},{u_t}} \right\rangle W(t,\theta) = {u_t}(\theta) - 2{\mathop{\rm Re}\nolimits} \{z(t)q(\theta)\}$$
(14)

on the center manifold C0, and we have

$$W(t,\theta) = W(z(t),\bar z(t),\theta)$$
(15)

where

$$W(z(t),\bar z(t),\theta) = W(z,\bar z) = {W_{20}}{{{z^2}} \over 2} + {W_{11}}z\bar z + {W_{02}}{{{{\bar z}^2}} \over 2} + \cdots$$
(16)

and z and a are local coordinates for center manifold C0 in the direction of q* and \({{\bar q}^{\ast}}\). Noting that W is also real if u t is real, we consider only real solutions. For solutions u t ∈ C0 of (11),

$$\matrix{{\dot z(t) = {\rm{i}}{\omega _0}z + {{\bar q}^{\ast}}(\theta)f[0,W(z,\bar z,\theta)] + 2{\mathop{\rm Re}\nolimits} \{zq(\theta)\} \mathop = \limits^{{\rm{def}}}} \hfill \cr{\quad \quad \;\;{\rm{i}}{\omega _0}z + {{\bar q}^{\ast}}(0){{[{f_1},{f_2},0]}^{\rm{T}}}.} \hfill \cr}$$

That is

$$\dot z(t) = {\rm{i}}{\omega _0}z + g(z,\bar z)$$

where

$$g(z,\bar z) = {g_{20}}{{{z^2}} \over 2} + {g_{11}}z\bar z + {g_{02}}{{{{\bar z}^2}} \over 2} + \cdots .$$

Let ƒ0 = [ƒ1, ƒ2]T. Hence, we have

$$\matrix{{g(z,\bar z) = {{\bar q}^{\ast}}(0){f_0}(z,\bar z) = f(0,{u_t}) =} \hfill \cr{\quad \quad \quad \quad \bar D({a_1}{a_2} + \bar a_1^{\ast}){z^2} + \bar D(2{\mathop{\rm Re}\nolimits} \{{a_1}{{\bar a}_2}\} + 2\bar a_1^{\ast})z\bar z +} \hfill \cr{\quad \quad \quad \quad \bar D({{\bar a}_1}{{\bar a}_2} + \bar a_1^{\ast}){{\bar z}^2} + \bar D\left[ {{1 \over 2}{{\bar a}_1}W_{20}^{(3)}(0) +} \right.} \hfill \cr{\quad \quad \quad \quad {1 \over 2}{{\bar a}_2}W_{20}^{(3)}(0) + {{\bar a}_1}W_{11}^{(3)}(0) + {{\bar a}_2}W_{11}^{(2)}(0) +} \hfill \cr{\quad \quad \quad \quad \bar a_1^{\ast}\left({W_{20}^{(1)}(0) + 2W_{11}^{(1)}(0)} \right){z^2}\bar z + {\rm{high}}\;{\rm{order}}\;{\rm{term}}.} \hfill \cr}$$

Then, we get

$$\matrix{{{g_{20}} = 2\bar D({a_1}{a_2} + \bar a_1^{\ast})} \hfill \cr{{g_{11}} = 2\bar D({\mathop{\rm Re}\nolimits} \{{a_1}{{\bar a}_2}\} + \bar a_1^{\ast})} \hfill \cr{{g_{02}} = 2\bar D({{\bar a}_1}{{\bar a}_2} + \bar a_1^{\ast})} \hfill \cr{{g_{21}} = 2\bar D\left[ {{1 \over 2}{{\bar a}_1}W_{20}^{(3)}(0) + {1 \over 2}{{\bar a}_2}W_{20}^{(3)}(0) +} \right.} \hfill \cr{\quad \quad \;{{\bar a}_1}W_{11}^{(3)}(0) + {{\bar a}_2}W_{11}^{(2)}(0) +} \hfill \cr{\quad \quad \;\bar a_1^{\ast}\left. {\left({W_{20}^{(1)}(0) + 2W_{11}^{(1)}(0)} \right)} \right].} \hfill \cr}$$

Since there exist unknowns \(W_{20}^{(1)}(0),\;W_{20}^{(3)}(0),\;W_{11}^{(1)}(0),\;W_{11}^{(2)}(0),\;W_{11}^{(3)}(0)\), in g21, we still need to compute them.

It follows from (11) and (14) that

((17))

where

$$H(z,\bar z,\theta) = {H_{20}}(\theta){{{z^2}} \over 2} + {H_{11}}(\theta)z\bar z + {H_{02}}(\theta){{{{\bar z}^2}} \over 2} + \cdots .$$
(18)

Comparing the coefficients, we obtain

$$(AW - 2{\rm{i}}{\omega _0}){W_{20}} = - {H_{20}}(\theta)$$
(19)
$$A{W_{11}}(\theta) = - {H_{11}}(\theta).$$
(20)

For θ ∈ [−∞, 0),

$$\matrix{{H(z,\bar z,\theta) = - {{\bar q}^{\ast}}(0){f_0}q(\theta) - {q^{\ast}}(0){{\bar f}_0}\bar q(\theta) =} \hfill \cr{\quad \quad \quad \quad \quad - g(z,\bar z)q(\theta) - \bar g(z,\bar z)\bar q(\theta).} \hfill \cr}$$
(21)

Comparing the coefficients of (21) with (18) gives that

$${H_{20}}(\theta) = - {g_{20}}q(\theta) - {\bar g_{02}}\bar q(\theta)$$
(22)
$${H_{11}}(\theta) = - {g_{11}}q(\theta) - {\bar g_{11}}\bar q(\theta).$$
(23)

From (19), (22) and the definition of A, we get

$${\dot W_{20}}(\theta) = 2{\rm{i}}{\omega _0}{W_{20}}(\theta) + {g_{20}}q(\theta) + \bar {{g_{02}}} \bar q(\theta).$$
(24)

Noting that \(q(\theta) = q(0){{\rm{e}}^{{\rm{i}}{\omega _{\rm{0}}}\theta}}\), we have

$${W_{20}}(\theta) = {{{\rm{i}}{g_{20}}} \over {{\omega _0}}}q(0){{\rm{e}}^{{\rm{i}}{\omega _0}\theta}} + {{{\rm{i}}{{\bar g}_{02}}} \over {3{\omega _0}}}\bar q(0){{\rm{e}}^{{\rm{- i}}{\omega _0}\theta}} + {E_1}{{\rm{e}}^{{\rm{2i}}{\omega _0}\theta}}$$
(25)

where \({E_1} = [{E_1}^{(1)},\;{E_1}^{(2)},\;{E_1}^{(3)}] \in {{\bf{R}}^3}\) is a constant vector. Similarly, from (20), (23) and the definition of A, we have

$${\dot W_{11}}(\theta) = {g_{11}}q(\theta) + \bar {{g_{11}}} \bar q(\theta)$$
(26)
$${W_{11}}(\theta) = - {{{\rm{i}}{g_{11}}} \over {{\omega _0}}}q(0){{\rm{e}}^{{\rm{i}}{\omega _0}\theta}} + {{{\rm{i}}{{\bar g}_{11}}} \over {{\omega _0}}}\bar q(0){{\rm{e}}^{{\rm{- i}}{\omega _0}\theta}} + {E_2}$$
(27)

where \({E_2} = [{E_2}^{(1)},\;{E_2}^{(2)},\;{E_2}^{(3)}] \in {{\bf{R}}^3}\) is a constant vector.

In what follows, we shall seek appropriate E1, E2 in (25), (27), respectively. It follows from the definition of A and (22), (23) that

$$\int_{- 1}^0 {{\rm{d}}\eta {\rm{(}}\theta {\rm{)}}{{\rm{W}}_{20}}(\theta)} = 2{\rm{i}}{\omega _0}{W_{20}}(0) - {H_{20}}(0)$$
(28)

and

$$\int_{- 1}^0 {{\rm{d}}\eta {\rm{(}}\theta {\rm{)}}{{\rm{W}}_{11}}(\theta)} = - {H_{11}}(0)$$
(29)

where η(θ) = η(0, θ).

From (22) and (23), we have

$${H_{20}}(0) = - {g_{20}}q(0) - {\bar g_{02}}\bar q(0) + 2\left[ {\matrix{{{a_1}{a_2}} \cr1 \cr0 \cr}} \right]$$
(30)
$${H_{11}}(0) = - {g_{11}}q(0) - {\bar g_{11}}(0)\bar q(0) + 2\left[ {\matrix{{{\mathop{\rm Re}\nolimits} \{{a_1}{{\bar a}_2}\}} \cr1 \cr0 \cr}} \right].$$
(31)

Noting that

$$\matrix{{\left({{\rm{i}}{\omega _0}I - \int_{- 1}^0 {{{\rm{e}}^{{\rm{i}}{\omega _0}\theta}}{\rm{d}}\eta {\rm{(}}\theta {\rm{)}}}} \right)q(0) = 0} \cr{\left({{\rm{- i}}{\omega _0}I - \int_{- 1}^0 {{{\rm{e}}^{{\rm{- i}}{\omega _0}\theta}}{\rm{d}}\eta {\rm{(}}\theta {\rm{)}}}} \right)\bar q(0) = 0} \cr}$$

and substituting (25) and (30) into (28), we have

$$\left({2{\rm{i}}{\omega _0}I - \int_{- 1}^0 {{{\rm{e}}^{{\rm{2i}}{\omega _0}\theta}}{\rm{d}}\eta {\rm{(}}\theta {\rm{)}}}} \right){E_1} = 2\left[ {\matrix{{{a_1}{a_2}} \cr1 \cr0 \cr}} \right].$$

That is

where

$${\chi ^{(3)}} = \int_{- \infty}^0 {k(- s){{\rm{e}}^{{\rm{2i}}{\omega _0}s}}{\rm{d}}s = {\alpha \over {\alpha + 2{\rm{i}}{\omega _0}}}.}$$

It follows that

$$E_1^{(1)} = {{{\Delta _{11}}} \over {{\Delta _1}}},\;E_1^{(2)} = {{{\Delta _{12}}} \over {{\Delta _1}}},\;E_1^{(3)} = {{{\Delta _{12}}} \over {{\Delta _1}}}$$
(32)

where

Similarly, substituting (26) and (31) into (29), we have

$$\left({\int_{- 1}^0 {{\rm{d}}\eta {\rm{(}}\theta {\rm{)}}}} \right){E_2} = 2\left[ {\matrix{{{\mathop{\rm Re}\nolimits} \{{a_1}{{\bar a}_2}\}} \cr1 \cr0 \cr}} \right].$$

That is

It follows that

$$E_2^{(1)} = {{{\Delta _{21}}} \over {{\Delta _2}}},\;E_2^{(2)} = {{{\Delta _{22}}} \over {{\Delta _2}}},\;E_2^{(3)} = {{{\Delta _{23}}} \over {{\Delta _2}}}$$
(33)

where

From (25), (27), (32) and (33), we can calculate g21 and derive the following values:

$$\left\{{\matrix{{{c_1}(0) = {{\rm{i}} \over {2{\omega _0}}}\left({{g_{20}}{g_{11}} - 2{{\left\vert {{g_{11}}} \right\vert}^2} - {{{{\left\vert {{g_{02}}} \right\vert}^2}} \over 3}} \right) + {{{g_{21}}} \over 2}} \hfill \cr{{\mu _2} = - {{{\mathop{\rm Re}\nolimits} \{{c_1}(0)\}} \over {{\mathop{\rm Re}\nolimits} \{{\lambda ^{\prime}}({\alpha _0})\}}}} \hfill \cr{{\beta _2} = 2{\mathop{\rm Re}\nolimits} ({c_1}(0))} \hfill \cr{{T_2} = - {{{\mathop{\rm Im}\nolimits} \{{c_1}(0)\} + {\mu _2}{\mathop{\rm Im}\nolimits} \{{\lambda ^{\prime}}({\alpha _0})\}} \over {{\omega _0}}}} \hfill \cr}} \right.$$
(34)

which determine the quantities of bifurcating periodic solutions on the center manifold C0 at the critical value α0, namely, we have the following result.

  • Theorem 2. The periodic solution is supercritical (sub-critical) if μ2 > 0 (μ2 < 0); the bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if β2 < 0 (β2 > 0); the periods of the bifurcating periodic solutions increase (decrease) if T2 > 0 (T2 < 0).

4 Computer simulations

In this section, we present some numerical results of system (2) to verify the analytical predictions obtained in the previous section. From Section 3, we may determine the direction of a Hopf bifurcation and the stability of the bifurcation periodic solutions. Let us consider the following system:

$$\left\{{\matrix{{\dot x = y(t)z(t) + 0.006} \hfill \cr{\dot y = {x^2}(t) - y(t)} \hfill \cr{\dot z = 1 - 4x(t) + 0.5\int_{- \infty}^0 {(z(t) - z(t + s))k(- s){\rm{d}}s}} \hfill \cr}} \right.$$
(35)

where k(s)= αe−αs, α > 0. Let yz + 0.006 = 0, x2 − y = 0, 1 − 4x = 0. It is easy to see that system (35) has an equilibrium \({\rm{E}}({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\) and all the conditions indicated in Theorem 2 are satisfied. When τ = 0, the equilibrium \({\rm{E}}({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\) is asymptotically stable. By means of Matlab 7.0, we get α0 ≈ 0.4512. Thus the equilibrium \({\rm{E}}({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\) is stable when α < α0 which is illustrated by the computer simulations (see Fig. 2). When α passes through the critical value a0, the equilibrium \({\rm{E}}({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\) loses its stability and a Hopf bifurcation occurs, i.e., a family of periodic solutions bifurcations from the equilibrium \({\rm{E}}({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\) It follows from the formulae (34) presented in Section 3 that μ2 > 0 and β2 < 0. Then the direction of the Hopf bifurcation is α > α0, and these bifurcating periodic solutions from \({\rm{E}}({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\) around α0 are stable, which are depicted in Fig. 3.

Fig. 2
figure 2

Behavior and phase portraits of system (35) with α = 0.25 < α0 ≈ 0.4512. The equilibrium \({\rm{E}}({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\) is asymptotically stable. The initial value is (1.5, 1.5, 1.5)

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Fig. 3
figure 3

Behavior and phase portraits of system (35) with α = 0.55 > α0 ≈ 0.4512. Hopf bifurcation occurs from the equilibrium \(E({\textstyle{1 \over 4}},\;{\textstyle{1 \over {16}}},\;{\textstyle{1 \over 4}},\; - {\textstyle{{12} \over {125}}})\). The initial value is (1.5, 1.5, 1.5)

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5 Conclusions

In this paper, a feedback control method is applied to suppress chaotic behavior of a Sprott E system within the chaotic attractor. By adding a continuous distributed time delayed force to the third equation of the Sprott E system, we have made a detailed discussion on the local stability of the equilibrium E(x*, y*, z*) and local Hopf bifurcation of the delayed Sprott E system model. We showed that if some conditions are fulfilled, then the Sprott E system is asymptotically stable for α < α0 and when α passes through α0, a sequence of Hopf bifurcations occur around the equilibrium E(x*, y*, z*), namely, a family of periodic orbits bifurcate from the the equilibrium E(x*, y*, z*), which implies the chaos of this system can be suppress. Some numerical simulations are included to visualize the theoretical findings. Moreover, the control method used in this paper can be applicable to other chaotic systems. We will carry our some related work in near future.