1 Introduction

Scientists have found that unexpected evolution patterns arise frequently in numerous nonlinear systems in physics, chemistry, biology, engineering, economics, and so forth. The most peculiar aspect of these patterns is their random-like behavior although the systems are deterministic, ie., the deterministic nature of these systems does not make them predictable. This behavior is known as chaos which is due to sensitive dependence on the initial conditions. It has been known that chaotic dynamics [1,2,3,4,5,6,7,8,9,10] generally consists of a motion where the nonlinear system state moves for a while in the neighborhood of one of the unstable periodic orbits(UPOs) and then falls close to a different unstable periodic orbit. Based on this fact, Ott et al. [11] proposed the first strategy to stabilize UPOs utilizing the sensitivity to initial conditions. From then on, chaos control has received a great deal of interests among the researchers, and a variety of chaotic systems have been proven to be able to be stabilized by several different techniques such as delayed feedback control(DFC) [12,13,14,15,16,17,18,19], prediction-based control [20,21,22], robust control [23, 24], slide mode control [25,26,27], adaptive control [28,29,30], energy-based feedback control [31].

Among them, delayed feedback control [12] has gained wide acceptance due to that the control input vanishes after the stabilization is achieved. Although the DFC is successful in various chaotic systems, the stability analysis of the closed-loop system often does not have an analytical form. One of the reasons is that the dynamics is described by a delay differential equation(DDE). Furthermore, some researchers have given several analytical results [32, 33], showing that the DFC has an odd number limitation. Since then, several improved DFC methods including extended DFC [8, 34,35,36,37,38], periodic DFC [39,40,41,42], and double delayed feedback control (DDFC) [43] have been developed to make the DFC more applicable. Recently, the DFC has been extended to solve practical application problems in engineering. For example, Wu et al. [44] investigated the effects of a delayed feedback scheme in the parameter space and demonstrated that wave segments can be stabilized. Costa and Savi [45] applied the extended DFC on a smart system composed of a pendulum coupled with shape-memory alloy elements. Paul and Banerjee [46] proposed a nonlocal time-delayed feedback control technique to control the spatiotemporal patterns in coupled map lattice(CML) systems and demonstrated its efficacy in a network of coupled digital phase-locked loops which is real-world CML system.

Although the delay differential equations usually have infinite-dimensional phase spaces, its stability can be described by an infinite-dimensional transition matrix [47]. The system is considered to be asymptotically stable if all the characteristic multipliers of the corresponding transition matrix are in modulus less than one. Insperger [48] proposed an act-and-wait control concept for continuous-time systems with feedback delay associated with infinite poles. The control strategy is that the feedback input is periodically switched on and off. It has been shown that if the waiting duration is larger than the feedback delay time, then the system is represented by a finite-dimensional monodromy matrix and the stability can be described by a finite number of eigenvalues. Thus, the infinite-dimensional pole placement problem is reduced to a finite-dimensional one.

The act-and-wait control method is an effective technique to reduce the number of poles for systems with large feedback delay, which makes the pole placement problem easier. In recent years, act-and-wait concept has been further tested through experiments [49] and theoretically extended to discrete-time systems [50], autonomous systems [51], non-autonomous dynamical systems [52], linear periodic time-varying systems [53, 54], etc. [55,56,57,58]. Act-and-wait scheme requires to obtain the monodromy matrix associated with the closed-loop system. However, it is usually hard to find a homogeneous expression of the corresponding monodromy matrix, due to the existence of the delay term. Based on a high-precision direct integration algorithm [59, 60], this paper explores the stabilization of periodic solutions to chaotic systems with an act-and-wait-fashioned delayed feedback control framework.

The paper aims to overcome the difficulties in obtaining stability conditions of the chaotic systems under the act-and-wait DFC and presents a high-precision direct integration method for calculating the monodromy matrix corresponding to the closed-loop system. The rest of the paper is organized as follows: Sect. 2 introduces the act-and-wait modification of DFC and formulates the condition where the controlled system becomes to be finite dimensional. Section 3 discusses the linear stability analysis of UPOs controlled by act-and-wait delayed feedback control and presents a periodically time-varying precise integration method for calculating the monodromy matrix corresponding to the closed-loop system. In Sec. 4, it is given two illustrative numerical examples to show the effectiveness of the proposed method. Finally, it is concluded in Sect. 5.

2 Problem formulation

Consider an nth-order nonlinear system which shows a chaotic behavior described by an n-dimensional first-order vector differential equation as follows:

$$\begin{aligned} \dot{{\mathbf {x}}}(t)=f({\mathbf {x}}(t),u(t)), \end{aligned}$$
(1)

where \({\mathbf {x}} \in \mathfrak {R}^n\) is the state vector, \(\dot{{\mathbf {x}}}\) denotes the derivative of \({\mathbf {x}}\) with respect to the time variable t, \(u\in \mathfrak {R}^m\) is the control input, \(f(\cdot )\) is an n-dimensional nonlinear vector function and assumed to be continuously differentiable. Suppose that the uncontrolled system has a T-periodic solution \({\mathbf {x}}(t)=\xi (t)=\xi (t+T)\) that satisfies the equation \({\dot{\xi }}(t)=f(\xi (t),0)\) and in a chaotic state. The task is to design a control law to suppress the chaotic behavior and make a more regular motion which may be a periodic motion. Many nonlinear dynamical systems in engineering science can be described by the above general set of differential equation, such as pendulum, mass-spring mechanical system, oscillator, microbeams. [1, 40, 61, 62].

The delayed feedback control (DFC) [34] is given by

$$\begin{aligned} u(t)=K\left( {\mathbf {x}}(t)-{\mathbf {x}}(t-T)\right) , \end{aligned}$$
(2)

where K is the gain matrix of the controller. In the delayed feedback control method, since the control input u(t) is computed based on the difference between the current state and the delayed state, u(t) vanishes after the UPO is stabilized. Although DFC can be relatively simply implemented in experiments, its theory is rather difficult, since the time-delay dynamics takes place in infinite-dimensional phase spaces. Generally, it is difficult to determine the feedback gain with which the desired UPO is stabilized.

As shown in Fig. 1, the act-and-wait controller is introduced as follows:

$$\begin{aligned} u(t)={ s(t)}K\left( {\mathbf {x}}(t)-{\mathbf {x}}(t-T)\right) , \end{aligned}$$
(3)

where s(t) is a T-periodic function that switches the controller on and off alternately at every integer multiples of the period T defined as

$$\begin{aligned} { s(t)}& = \left\{ \begin{array}{ll} 0 \,\,& \text{ if } \; 0 \le (t\, \text{mod}\, 2T)< T, \\ 1 \,\, & \text{if} \; T \le (t\, \text{mod}\, 2T) < 2T . \end{array} \right. \end{aligned}$$
(4)
Fig. 1
figure 1

Block diagram of a chaotic system under the act-and-wait DFC

Full size image

The act-and-wait control method is originally used for n-dimensional continuous-time control systems with feedback delay. It has been shown that the infinite-dimensional pole placement problem of the delayed system can be reduced to an n-dimensional one, if the feedback is periodically switched off and on. The point is that the switched system can be described by a monodromy matrix, consequently, stability properties are described by n eigenvalues.

3 Stability analysis of act-and-wait DFC

The closed-loop system is linearized around the UPO,\(\xi (t)\):

$$\begin{aligned} \left\{ \begin{array}{l} \delta \dot{{\mathbf {x}}}(t) = A(t) \delta {\mathbf {x}}(t) + B(t) u(t), \\ u(t)={s(t)}K\left( \delta {\mathbf {x}}(t)- \delta {\mathbf {x}}(t-T)\right) , \end{array} \right. \end{aligned}$$
(5)

where \(\delta {\mathbf {x}}(t):={\mathbf {x}}(t)-\xi (t)\) is the deviation of the solution from the target UPO, and \(\delta {\mathbf {x}}(t-T)\) is its T-delayed version. The T-periodic matrixes A(t), B(t) are the Jacobian matrixes of the system evaluated on the UPO such that,

$$\begin{aligned} A(t) \triangleq \frac{\partial f({\mathbf {x}}(t),u(t))}{\partial {\mathbf {x}}}|_{{\mathbf {x}}=\xi (t),u(t)=0}, \nonumber \\ B(t) \triangleq \frac{\partial f({\mathbf {x}}(t),u(t))}{\partial u}|_{{\mathbf {x}}=\xi (t),u(t)=0}. \end{aligned}$$
(6)

Here, note that the linearized version of the closed-loop system (5) is locally asymptotically stable at the UPO. Moveover, system (5) is similar to a linear periodic time-varying system such that [54]

$$\begin{aligned} {\dot{x}}(t) = \left( A(t)+B(t){ s(t)}K\right) x(t)- B(t){ s(t)}Kx(t-T), \end{aligned}$$
(7)

where \(x(t)\in \mathfrak {R}^n\) is the state vector. From Eqs. (7) and (4), if \(t \in [2kT, 2kT+T),k=0,1,2,\ldots\), then

$$\begin{aligned} {\dot{x}}(t) = A(t)x(t), \end{aligned}$$
(8)

while if \(t \in [2kT+T, 2kT+2T), k=0,1,2,\ldots\), then

$$\begin{aligned} {\dot{x}}(t) = \left( A(t)+B(t)K\right) x(t)- B(t)Kx(t-T). \end{aligned}$$
(9)

3.1 Monodromy matrix of the closed-loop system

If we define the state-transition matrix of system (8) to be \(\textit{M}(\cdot )\), the monodromy matrix associated with the T-periodic system can be given by \({\textit{M}}(T,0) \in \mathfrak {R}^{n\times n}\). Moreover, define \(\Gamma (\cdot )\) as the state-transition matrix for the periodic system

$$\begin{aligned} {\dot{x}}(t) = \left( A(t)+B(t)K\right) x(t). \end{aligned}$$
(10)

It is obvious that

$$\begin{aligned} {\dot{x}}(t-T)& = {\textit{M}} (t-T, 2kT) x(2kT),\nonumber \\&t \in [2kT+T, 2kT+2T). \end{aligned}$$
(11)

Therefore, substituting Eq. (11) into Eq. (9), it can be obtained

$$\begin{aligned} {\dot{x}}(t)& = \left( A(t)+B(t)K\right) x(t) \nonumber \\&\quad - B(t)K{\textit{M}} (t-T, 2kT) x(2kT). \end{aligned}$$
(12)

By left multiplying matrices \(\Gamma ^{-1}(t,2kT+T)\) on both sides of Eq. (12) and formulation derivation, it is obtained

$$\begin{aligned} x(t)& = \left\{ \Gamma (t-2kT-T, 0) {\textit{M}} (T,0) \right. \nonumber \\&\left. -\int _{2KT+T}^{t} \Gamma (t, {\bar{t}}) B( {\bar{t}} ) K {\textit{M}} ({\bar{t}}-T,2kT) d{\bar{t}} \right\} \cdot x(2kT),\nonumber \\&\, t \in [2kT+T, 2kT+2T), \,{\bar{t}} \in [2kT+T, t]. \end{aligned}$$
(13)

Setting \({\hat{t}}={\bar{t}}-2kT\) yields

$$\begin{aligned}&\int _{2KT+T}^{t} \Gamma (t, {\bar{t}}) B( {\bar{t}} ) K {\textit{M}} ({\bar{t}}-T,2kT) d{\bar{t}} \nonumber \\&\quad =\int _{T}^{t-2KT} \Gamma (t, {\hat{t}}+2kT) B( {\hat{t}}+2kT ) K {\textit{M}} ({\hat{t}}\nonumber \\&\qquad +2kT-T,2kT) d{\hat{t}}, \nonumber \\&\qquad t \in [2kT+T, 2kT+2T). \end{aligned}$$
(14)

Since \(\Gamma (t, {\hat{t}}+2kT)=\Gamma (t-2KT, {\hat{t}}), \textit{M} ({\hat{t}}+2kT-T,2kT)=\textit{M} ({\hat{t}}-T,0)\) and \(B( {\hat{t}}+2kT )=B( {\hat{t}} )\), it is obtained from Eq. (14)

$$\begin{aligned}&\int _{2KT+T}^{t} \Gamma (t, {\bar{t}}) B( {\bar{t}} ) K {\textit{M}} ({\bar{t}}-T,2kT) d{\bar{t}} \nonumber \\&\quad =\int _{T}^{t-2KT} \Gamma (t-2kT, {\hat{t}}) B( {\hat{t}} ) K {\textit{M}} ({\hat{t}}-T,0) d{\hat{t}}. \end{aligned}$$
(15)

Then, it follows from Eqs. (13) and (15) that

$$\begin{aligned} x(t) = \varPsi (t) \cdot x(2kT), \, t \in [2kT+T, 2kT+2T), \end{aligned}$$
(16)

where

$$\begin{aligned} \varPsi (t)& = \Gamma (t-2kT-T, 0) {\textit{M}} (T,0) \nonumber \\&\quad -\int _{T}^{t-2kT} \Gamma (t-2kT, {\hat{t}}) B( {\hat{t}} ) K {\textit{M}} ({\hat{t}}-T,0) d{\hat{t}}. \end{aligned}$$
(17)

By setting \(t = 2kT+2T\), Eq. (17) becomes

$$\begin{aligned} \varPsi (2T)& = \Gamma (T, 0) {\textit{M}} (T,0) \nonumber \\&\quad -\int _{T}^{2T} \Gamma (2T, {\hat{t}}) B( {\hat{t}} ) K {\textit{M}} ({\hat{t}}-T,0) d{\hat{t}}. \end{aligned}$$
(18)

Equation (18) constructs a monodromy matrix for the closed-loop system (7) with the doubled period 2T such that

$$\begin{aligned} x(2kT+2T) = \varPsi (2T) \cdot x(2kT),\, k=0,1,2,\ldots . \end{aligned}$$
(19)

Therefore, the stability of the closed-loop system is considered to be decided by the eigenvalues of the matrix \(\varPsi (2T)\).

3.2 The high-precision direct integration method

Let initial state of system (8) be \(x(t_0) = x_0\). If we define a sequence of vectors \(x_i(t) \in \mathfrak {R}^n (i=[0, \infty ))\) on the closed interval \([t_0, t]\) as

$$\begin{aligned} \left\{ \begin{array}{ll} x_0(t) = x_0, \\ x_1(t) = x_0 + \int _{t_0}^t A(s_1)x_0(s_1)\mathrm{d}s_1, \\ x_2(t) = x_0 + \int _{t_0}^t A(s_1)x_1(s_1)\mathrm{d}s_1, \\ \vdots \\ x_i(t) = x_0 + \int _{t_0}^t A(s_1)x_{i-1}(s_1)\mathrm{d}s_1, \end{array} \right. \end{aligned}$$
(20)

then, \(x_i(t)\) can be rewritten as a sum of terms involving integrals of the matrix A(t).

$$\begin{aligned} x_i(t)& = x_0 + \int _{t_0}^t A(s_1)x_0\mathrm{d}s_1 \nonumber \\&\quad + \int _{t_0}^t A(s_1) \int _{t_0}^{s_1} A(s_2)x_0 \mathrm{d}s_2 \mathrm{d}s_1 + \ldots \nonumber \\&\quad + \int _{t_0}^t A(s_1) \int _{t_0}^{t} A(s_2) \cdots \int _{t_0}^{s_{i-1}} A(s_i) x_0 \mathrm{d}s_i\cdots \mathrm{d}s_1. \end{aligned}$$
(21)

Assume the isometric division of time domain \([t_0, t_f]\) to be

$$\begin{aligned} t_{i+1} - t_i = \tau \equiv T/N \in [t_0, t_f],\, i=0,1,2,\ldots . \end{aligned}$$
(22)

Define \({\textit{M}}(t,t_0)\) which is a function of two variables to be the transition matrix of system (8). It is obvious that there exists

$$\begin{aligned} x(t_{i+1)} = {\textit{M}}(t_{i+1},t_i)x(t_i). \end{aligned}$$
(23)

The transition matrix \({\textit{M}}(t,t_0)\) can be expressed in the generalized Peano–Baker series as [63]

$$\begin{aligned} {\textit{M}}(t,t_0)& = I + \int _{t_0}^t A(s_1)\mathrm{d}s_1 \nonumber \\&\quad + \int _{t_0}^t A(s_1) \int _{t_0}^{s_1} A(s_2) \mathrm{d}s_2 \mathrm{d}s_1 + \ldots \nonumber \\&\quad + \int _{t_0}^t A(s_1) \int _{t_0}^{t} A(s_2) \cdots \int _{t_0}^{s_{i-1}} A(s_i) \mathrm{d}s_i\cdots \mathrm{d}s_1. \end{aligned}$$
(24)

Then, a high-precision direct integration algorithm, named as HPDI algorithm, for calculating the transition matrix \({\textit{M}}(t_{k+1},t_k)\) is established as follows.

Step 1::

Finely divide the interval of \([t_k, t_{k+1}]\) into

$$\begin{aligned} t_k = t_0^{k}< t_1^{k}< t_2^{k}< \cdots < t_m^{k} = t_{k+1}, \end{aligned}$$

where \(m=2^Q\). Here, Q is a fine parameter, \(t_{j+1}^{k}-t_{j}^{k} = \tau /m, j=0,1,\ldots ,m-1\).

Step 2::

Compute, respectively, \(A(t_j^{k})\), \(A( (t_{j}^{k} + t_{j+1}^{k})/2 )\) and \(A(t_{j+1}^{k})\) based on Eq. (6). Here, the periodic solutions \(\xi (t)\) are calculated numerically by fourth-order Runge–kutta method.

Step 3::

From the properties of transfer matrix, it is obvious that

$$\begin{aligned} {\textit{M}} (t_{k+1}, t_k) = {\textit{M}} (t_{m}^{k}, t_{m-1}^{k}) {\textit{M}} (t_{m-1}^{k}, t_{m-2}^{k}) \cdots {\textit{M}} (t_{1}^{k}, t_{0}^{k}). \end{aligned}$$

Each transition matrix needs to be taken its third-order approximation of the following Peano–Baker series.

$$\begin{aligned} \tilde{{\textit{M}}} (t_{j+1}^{k}, t_{j}^{k})& = I + [A(t_j^{k}) + 4A( (t_{j}^{k} + t_{j+1}^{k})/2 )\nonumber \\&\quad + A(t_{j+1}^{k})] d\tau /(3!) \nonumber \\&\quad + [2A(t_{j+1}^{k})A(t_{j}^{k}) + A(t_{j}^{k})A(t_{j+1}^{k})] (d\tau )^2/(3!) \nonumber \\&\quad + [A(t_{j}^{k})]^3 (d\tau )^3 /(3!). \end{aligned}$$
(25)

If we define

$$\begin{aligned} \tilde{{\textit{M}}} (t_{k+1}, t_k) = \prod _{j=0}^{m-1} \tilde{{\textit{M}}} (t_{j+1}^{k}, t_{j}^{k}), \end{aligned}$$
(26)

then \(\tilde{{\textit{M}}} (t_{k+1}, t_k)\) is the approximation of the transition matrix \({\textit{M}} (t_{k+1}, t_k)\).

Step 4::

Calculate \(\tilde{{\textit{M}}} (t_{k+1}, t_k)\).

$$\begin{aligned}&\Lambda _j ^0 \Leftarrow [A(t_j^{k}) + 4A( (t_{j}^{k} + t_{j+1}^{k})/2 ) + A(t_{j+1}^{k})] d\tau /(3!) \\&\quad + [2A(t_{j+1}^{k})A(t_{j}^{k}) + A(t_{j}^{k})A(t_{j+1}^{k})] (d\tau )^2/(3!) \\&\quad + [A(t_{j}^{k})]^3 (d\tau )^3 /(3!),\, j=0,1,2,\ldots ,m-1; \end{aligned}$$

For \(i=1\) to Q

$$\begin{aligned} \Lambda _j ^i& = \Lambda _{2j+1} ^{i-1} + \Lambda _{2j} ^{i-1} + \Lambda _{2j+1} ^{i-1} \Lambda _{2j} ^{i-1}, \\ j& = 0,1,2,\ldots ,2^{Q-i} -1; \end{aligned}$$

End

$$\begin{aligned} \tilde{{\textit{M}}} (t_{k+1}, t_{k}) \Leftarrow I + \Lambda _0 ^Q. \end{aligned}$$
(27)
Step 5::

Repeat executions of Step 2-4 for N times. Then, it can be obtained \(\tilde{{\textit{M}}} (t_{1}, t_{0}), \tilde{{\textit{M}}} (t_{2}, t_{1}), \cdots\), \(\tilde{{\textit{M}}} (t_{N}, t_{N-1}),\), respectively.

Step 6::

Calculate \({\textit{M}} (t, t_{0})\). If \(t=T\), then

$$\begin{aligned} {\textit{M}} (t, t_{0}) \simeq \tilde{{\textit{M}}} (t_{N}, t_{N-1}) \cdots \tilde{{\textit{M}}} (t_{2}, t_{1}) \tilde{{\textit{M}}} (t_{1}, t_{0}), \end{aligned}$$
(28)

else if \(t<T\), then

$$\begin{aligned} {\textit{M}} (t, t_{0}) \simeq \tilde{{\textit{M}}} (t, t-\tau ) \cdots \tilde{{\textit{M}}} (t_{2}, t_{1}) \tilde{{\textit{M}}} (t_{1}, t_{0}). \end{aligned}$$
(29)

Based on the above HPDI algorithm, the transition matrices \({\textit{M}} (T, 0)\) and \({\textit{M}} ({\hat{t}}-T, 0)\) in Eq. (18) can be calculated by Eqs. (28) and (29), respectively.

In order to calculate the transition matrices \(\Gamma (T, 0)\) and \(\Gamma (2T, {\hat{t}})\) in Eq. (18), it is considered that \({\hat{A}}(t) = A(t)+B(t)K.\) Then, similar to that of the transition matrices \({\textit{M}} (T, 0)\) and \({\textit{M}} ({\hat{t}}-T, 0)\) , \(\Gamma (T, 0)\) and \(\Gamma (2T, {\hat{t}})\) can be calculated based on the same precise integration algorithm. Therefore, the monodromy matrix \(\varPsi (2T)\) in Eq. (18) which is associated with the closed-loop system (7) can be computed. By selecting an appropriate gain matrix K, the state trajectory of the system (7) converges toward a periodic solution.

4 Illustrative examples

In this section, we give two examples to examine the effectiveness of the above method.

4.1 Example 1

Consider a nonlinear system defined by a two-dimensional state vector \({\mathbf {x}} = [{\mathbf {x}}_1, {\mathbf {x}}_2]^T\), a control input vector \(u=(u_1,u_2)\) and the vector field

$$\begin{aligned} \dot{{\mathbf {x}}}& = f({\mathbf {x}},u) \nonumber \\\triangleq & \left[ \begin{array}{l} -{\mathbf {x}}_1\left( {\mathbf {x}}_1^2+{\mathbf {x}}_2^2-({\mathbf {x}}_1^2 +{\mathbf {x}}_2^2)^2\right) + 2\pi {\mathbf {x}}_2 + u_1 \\ -{\mathbf {x}}_2\left( {\mathbf {x}}_1^2+{\mathbf {x}}_2^2-({\mathbf {x}}_1^2 +{\mathbf {x}}_2^2)^2\right) - 2\pi {\mathbf {x}}_1 + u_2 \end{array} \right] . \end{aligned}$$
(30)

The above system (30) is a modified version of the exemplified system of the so-called Poincar\(\acute{e}\)–Andronov–Hopf bifurcation [61]. The linear variational equation associated with the closed-loop system under the act-and-wait controller (with \(T=1\) s) is given by Eq. (5), where

$$\begin{aligned} A(t)& = \left[ \begin{array}{cc} 2 \cos ^2(2\pi t) & -\sin (4\pi t) + 2\pi \\ -\sin (4\pi t) - 2\pi & 2\sin ^2(2\pi t) \end{array} \right] , \end{aligned}$$
(31)
$$\begin{aligned} B(t)& = \left[ \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right] . \end{aligned}$$
(32)

The system (30) shows unstable behavior if there is no control input. To evaluate the asymptotic behavior of solutions under the control law (2), it needs to examine the eigenvalues of the monodromy matrix \(\varPsi (2T)\) associated with the closed-loop system (7). Based on the proposed precise integration algorithm, by calculating numerically the value of \(\varPsi (2T)\) for a certain range of feedback gain parameters and search for an appropriate feedback gain matrix \(K \in \mathfrak {R}^{2\times 2}\).

Fig. 2
figure 2

The phase portrait of \({\mathbf {x}}_1-{\mathbf {x}}_2\) of the system (30) without control input corresponding to the time of \(t=[0,1.8]\)

Full size image

Note that for the feedback gain matrix \(K = \left[ 2.2, 4.0;\;4.5 ,-3.2\right] ,\) the corresponding monodromy matrix is calculated from Eq. (18) based on the HPDI algorithm discussed in Sect. 3.2 as

$$\begin{aligned} \varPsi (2T) = \left[ \begin{array}{cc} 0.7052 & 0.0493 \\ 3.5138 & 0.3244 \end{array} \right] . \end{aligned}$$
(33)

which has the eigenvalues \(\lambda _1 = 0.9726\) and \(\lambda _2 = 0.057\).

Figure 2 plots a phase portrait of \(({\mathbf {x}}_1,{\mathbf {x}}_2)\) of the uncontrolled system (30) (with the control input \(u=0\)) under the initial condition \({\mathbf {x}}_0=(1,-\,0.1)\). From the time \(t=1.5\) s, the state trajectory begins to diverge from the periodic orbit. The periodic system without control input, hence, shows unstable behavior.

Fig. 3
figure 3

The time history of \({\mathbf {x}}\) and u of the system (30) stabilized by act-and-wait DFC

Full size image
Fig. 4
figure 4

The phase portrait of \({\mathbf {x}}_1-{\mathbf {x}}_2\) of the system (30) stabilized by act-and-wait delayed feedback control; it is a colored 3D contour of time mapped on the bottom plain in which color-bar represents the time t (s)

Full size image

Figures 3 and 4, respectively, show a behavior and the phase portrait of the controlled system starting from the initial state \({\mathbf {x}}_0=(1,-\,0.1)\). Figure 3 shows a behavior of the controlled system, while Figure 4 plots a phase portrait of \(({\mathbf {x}}_1,{\mathbf {x}}_2)\). It is known from Figs. 3 and 4 that the state trajectory converges to the periodic orbit, i.e., system (30) reaches a stabilization state via the proposed method.

4.2 Example 2

As the next example, it is considered the well-known Lorenz system. The system is defined by a three-dimensional state vector \({\mathbf {x}} = [{\mathbf {x}}_1, {\mathbf {x}}_2, {\mathbf {x}}_3]^T\), a control input vector \(u=(u_1,u_2)\) and the vector field

$$\begin{aligned} \dot{{\mathbf {x}}} = f({\mathbf {x}},u) \triangleq \left[ \begin{array}{c} -\sigma ({\mathbf {x}}_1-{\mathbf {x}}_2) \\ \gamma {\mathbf {x}}_1 - {\mathbf {x}}_2 - {\mathbf {x}}_1{\mathbf {x}}_3 + u \\ {\mathbf {x}}_1 {\mathbf {x}}_2 - b {\mathbf {x}}_3 \end{array} \right] , \end{aligned}$$
(34)

where \(\sigma = 10\), \(\gamma = 28\) and \(b = 8/3\).

Fig. 5
figure 5

The phase portrait of \({\mathbf {x}}_1-{\mathbf {x}}_2-{\mathbf {x}}_3\) of the system (34) without control input corresponding to the time of \(t=[0,50s]\)

Full size image

If choosing the period-one UPO with the period \(T=1.5586\)s and the initial conditions \({\mathbf {x}}_0=[-15.467,-15.411, 36.598]\), the linear variational equation associated with the closed-loop system under the act-and-wait controller is given by Eq. (5), where

$$\begin{aligned} A(t) = \left[ \begin{array}{ccc} -\sigma &\quad \sigma &\quad 0 \\ \gamma - {\mathbf {x}}_3 &\quad -1 &\quad -{\mathbf {x}}_1 \\ {\mathbf {x}}_2 &\quad {\mathbf {x}}_1 &\quad -b \end{array} \right] , \qquad B(t) = \left[ \begin{array}{ccc} 0&1&0 \end{array} \right] ^T. \end{aligned}$$
(35)
Fig. 6
figure 6

The stabilization control input of the system (34) stabilized by act-and-wait delayed feedback control

Full size image
Fig. 7
figure 7

The phase portrait of \({\mathbf {x}}_i-{\mathbf {x}}_j(i=1,2; j=2,3)\) of the system (34) stabilized by act-and-wait delayed feedback control; it is a colored 3D contour of time mapped on the bottom plain in which color-bar represents the time t (s)

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The monodromy matrix associated with the uncontrolled system is given by

$$\begin{aligned} \textit{M} (T,0) = \left[ \begin{array}{ccc} 1.2257 & \quad 1.3786 &\quad -\,1.4541\\ 7.3979 & \quad 9.9107 &\quad -\,9.4232\\ 4.0485 & \quad 6.0767 &\quad -\,5.4226 \end{array} \right] , \end{aligned}$$
(36)

which has the eigenvalues 4.7148, 0, and 0.9991. As shown in Fig. 5, the uncontrolled system (34) shows chaotic behavior.

Then, performing the procedures similar to those described in the previous example, it can be achieved the successful control of the UPO.

For the feedback gain matrix \(K = \left[ -4.1, -5.0 ,-3.8\right]\), the corresponding monodromy matrix \(\varPsi (2T)\) is calculated from Eq.(18) as

$$\begin{aligned} \varPsi (2T) = \left[ \begin{array}{ccc} 0.0080 & \quad 0.0543 & \quad -\,0.0281 \\ 1.1896 & \quad 3.4808 & \quad -\,2.2845 \\ 1.6234 & \quad 4.5691 & \quad -\,3.0435 \end{array} \right] , \end{aligned}$$
(37)

which has the eigenvalues \(\lambda _1=0.6927\), \(\lambda _2=-\,0.0001\) and \(\lambda _3=-\,0.2473\).

Figure 6 plots the time history of the control input of the system (34) stabilized by the proposed method. The trajectory of the orbits in the phase plane \({\mathbf {x}}_i-{\mathbf {x}}_j(i=1,2; j=2,3)\) from t=0 till t=120 s via the stabilization control input u is shown in Fig. 7. The initial condition \({\mathbf {x}}_0=(-15.46,-15.41, 36.59)\). The phase portraits of \({\mathbf {x}}_1-{\mathbf {x}}_2, {\mathbf {x}}_1-{\mathbf {x}}_3,{\mathbf {x}}_2-{\mathbf {x}}_3\) are shown in (i), (ii), (iii), respectively. Fig. 7a depicts the trajectory of \(0\sim 80\) s and Fig. 7b plots that of \(80\sim 120\) s. From Fig. 7b and Fig. 6, it is obvious that the trajectory of the orbits from about 70 s converged closer to a closed curve which means that the system attained a stable orbit via the act-and-wait delayed feedback control.

5 Conclusion

This paper developed a stabilization method for the periodic orbits embedded in nonlinear chaotic systems via an act-and-wait delayed feedback control. The proposed method derived a finite-sized corresponding monodromy matrix for the closed-loop system under a switching mechanism that turns the delayed feedback controller on and off alternately at every integer multiples of the period of the desired UPO. By analyzing the eigenvalues of the corresponding monodromy matrix, it was obtained conditions under which the state trajectory converges toward a periodic solution. Due to the fact that it is hard to find a homogeneous expression of the monodromy matrix corresponding to the act-and-wait DFC, we present a periodically time-varying precise integration method by which the transition matrices can be calculated numerically with a high precision. Furthermore, two numerical examples showed that the dynamics of nonlinear systems can be stabilized to periodic orbits by means of the proposed strategy.