Simple algebraic necessary and sufficient conditions for Lyapunov stability of a Chen system and their applications

Abstract

In this paper, we study the Lyapunov stability problem of a Chen chaotic system. Because of the positive elements of the main diagonal of a linearized Chen system, compared to the coefficient of a linearized Lorenz system which are all negative, it is more difficult to deal with the stability analysis. Since it has the properties of invariance and symmetry, different Lyapunov functions in different regions are constructed to solve stability problems with geometric and algebraic methods. Then, simple algebraic necessary and sufficient conditions of global exponential stability, global asymptotic stability and global instability of equilibrium $S_{0} (0, 0, 0)$ are proposed. We obtain the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability and local instability of equilibria $S_{\pm} (\pm \sqrt{b (2 c - a)}, \pm \sqrt{b (2 c - a)}, (2 c - a))$ . Furthermore, the smallest conservative linear feedback controllers are used to globally exponentially stabilize equilibria.

Keywords

Chen system Lyapunov stability global exponential stability global asymptotic stability instability

Introduction

Since Lorenz’s chaotic system was found in 1963, the phenomena of chaos has attracted many researchers’ interests, and many chaotic models are found in various aspects of the natural world and society. In the most recent half a century, by using computer simulation, electronic circuits and other methods, many mathematicians, physicists and other researchers in different areas focus on studying the properties of chaotic systems. Some new systems which are not topologically equivalent to the Lorenz system are found, i.e. the Chen system (Chen, 1999) and the Yang–Chen system (Yang and Chen, 2008). Many valuable results have been proposed in recent years (Algaba et al., 2015; Leonov, 2001; Liao, 2014; Liao and Luo, 2010; Liao et al., 2007; Liao et al., 2015; Liu and Pang, 2012; Luo et al., 2006; Ma and Miao, 2015b; Ma et al., 2015a; Ma et al., 2016; Sparrow, 1976; Stwart, 2002; Wang et al., 2012; Yuan et al., 2016a; Yuan et al., 2016b).

It is difficult to thoroughly understand the global dynamical behaviours since it is impossible to analytically solve these differential equations of nonlinear dynamical systems. It is particularly desirable to find a strict mathematical proof of asymptotic behaviours for these systems. Therefore, Stwart (2012) declared that a strict mathematical proof of existence of singular attractor for Lorenz system is one of the public mathematical problems in the 21st century.

It is generally believed that the conditions for producing chaos of a continuous dynamical system include:

the system has at least one positive Lyapunov exponent in a small area;

the system is ultimately bounded (or Lagrange stable).

That is, the trajectory of the system far away from the equilibrium point converges to a specific bounded set. Furthermore, computing the Lyapunov exponent is significant to make (b) hold. Therefore, proof of the ultimate boundedness for dynamical systems is a critical problem. Soviet academic Leonov (2001) is the first person who studied this problem. He obtained a cylindrical estimator and one oval estimator for the Lorenz system, respectively. Based on the above results, Liao (2004) and Liao et al. (2007) not only gave a simplified proof for the Lorenz system, but also improved and extended the well-known results. Particularly, Liao (2004) first proposed the new concept of global exponential stability in the sense of Lagrange. Recently, by using geometric and algebraic methods, Liao et al. (2015) proposed a constructive proof of globally exponentially attractive set for Chen system.

It is well known that stability is a prerequisite for the other applications. Therefore, the stability study of a dynamical system is of much importance and has received considerable attention. Ma et al. (2013) presented a novel L-K functional, and the less conservative delay-dependent stability criteria was obtained based on a partial differential equation technique and a linear operator approach. A relaxed Lyapunov–Krasovskii functional approach was proposed by Zhang et al. (2013) for a class of distributed delay neural networks, and the approach was shown to be very efficient for reducing the conservatism of delay-dependent stability conditions. The extended dissipativity concept was first proposed by Zhang et al. (2015), which gives a unified framework for the analysis and design of multiple performance constrained systems. However, it is more difficult to deal with stability analysis for a nonlinear dynamical system. In the work by Liao and Luo (2010), we completed Lyapunov stability analysis of the equilibrium of the Lorenz system. Some very simple algebraically necessary and sufficient conditions for global exponential stability, global asymptotic stability and instability were given, and the results were applied to chaotic control very well. But for the Chen system, does it has the same results? There has been no answer for a long time because it is a very difficult problem.

In this paper, as a sequential and extensional work of our recent results (Liao et al., 2015), we analyse the Lyapunov stability of these equilibria $S_{0}$ and $S_{\pm}$ . Since $(x, y, z) \to (- x, - y, z)$ , by using the properties of invariance and symmetry of the Chen system, we construct different Lyapunov functions in different regions to solve the problems of geometric and algebraic methods. Furthermore, by using a lemma, we conclude that all states of the Chen system are Lyapunov stable if states y and z are Lyapunov stable. Then, simple algebraic necessary and sufficient conditions of global exponential stability, global asymptotic stability and global instability of equilibrium $S_{0} (0, 0, 0)$ are proposed. We also obtain the relevant expressions for the corresponding parameters for the local exponential stability, local asymptotic stability and the local instability of equilibria $S_{\pm}$ . As an application of our results, the smallest conservative linear feedback controllers are used to globally exponentially stabilize equilibria $S_{\pm}$ or any given non-equilibrium point. Furthermore, the controller can make the system asymptotically track any given bounded trajectory. The necessary and sufficient conditions presented here are very useful for chaos control, chaos synchronization and other applications. It also enriches the integrity and systematicness for the Lorenz system family and provides inspiration and examples for the research of chaotic systems.

The rest of this paper is organized as follows. In the ‘Sufficient and necessary conditions for the Lyapunov stability of the equilibrium $S_{0}$ ‘ section, three necessary and sufficient conditions for the Lyapunov stability of the equilibrium $S_{0}$ are proposed. The relevant expression for the corresponding parameters for the local exponential stability, local asymptotic stability and local instability of equilibria $S_{\pm}$ is given in the ‘Sufficient and necessary condition for the local stability of the equilibria $S_{\pm}$ ‘ section, and the smallest conservative linear feedback controllers are used to globally exponentially stabilize equilibria $S_{\pm}$ or any given non-equilibrium point. Numerical simulations are presented to demonstrate the correctness of our theoretic results in the ‘Simulation’ section. The final section is the conclusion.

Sufficient and necessary conditions for the Lyapunov stability of the equilibrium $S_{0}$

Consider the Chen chaotic system as follows (Chen, 1999)

\begin{matrix} {\begin{matrix} \frac{d x}{d t} = a (y - x) \\ \frac{d y}{d t} = (c - a) x - xz + cy \\ \frac{d z}{d t} = xy - bz \end{matrix} \end{matrix}

(1)

where $a, b, c \in R$ are positive constants and $X (t) = (x (t),$ $y (t), z (t))$ is the state with $X_{0} = X (t_{0})$ . It is easy to see that $S_{0} = (0, 0, 0)$ is an equilibrium of the system given by equation (1). There exists two other equilibria $S_{\pm} = (\pm \sqrt{b (2 c - a)}, \pm \sqrt{b (2 c - a)}, (2 c - a))$ if $2 c \geq a$ .

In the work by Liao et al. (2015), the Lagrange stability and chaotic synchronization of the Chen system were studied, but the results of the Lyapunov stability analysis of the equilibrium points $S_{0}$ and $S_{\pm}$ were not seen before. So, Lyapunov stability properties for the above three equilibria will be discussed here. First, necessary and sufficient conditions for the Lyapunov stability of the system given by equation (1) on $S_{0}$ will be studied in this section. For this purpose, a lemma is given.

Lemma 1. If the trajectory $y (t)$ of the system given by equation (1) is globally exponentially stable, that is, there exist two constants $k > 0$ and $α > 0$ , it holds that

| y (t) | \leq k e^{- α (t - t_{0})}, \forall t \geq t_{0}

(2)

where the initial time $t_{0} \geq 0$ . Then, the trajectory $x (t)$ of the system given by equation (1) is also globally exponentially stable.

Proof. From the first equation of the system given by equation (1), one obtains that

\begin{matrix} x (t) = x_{0} e^{- a (t - t_{0})} + \int_{t_{0}}^{t} e^{- a (t - τ)} ay (τ) d τ \end{matrix}

(2)

Consider the inequality given by equation (2), one has

\begin{matrix} | x (t) | \leq | x_{0} | e^{- a (t - t_{0})} + a \int_{t_{0}}^{t} e^{- \bar{α} (t - τ)} k e^{- \bar{α} (τ - t_{0})} d τ \\ = | x_{0} | e^{- a (t - t_{0})} + ak \int_{t_{0}}^{t} e^{- \bar{α} (t - t_{0})} d τ \\ = | x_{0} | e^{- a (t - t_{0})} + ak e^{- \frac{\bar{α}}{2} (t - t_{0})} e^{- \frac{\bar{α}}{2} (t - t_{0})} (t - t_{0}) \end{matrix}

(2)

where $\bar{α} = \min {a, α}$ . Since $li m_{t \to + \infty} e^{- \frac{\bar{α}}{2} (t - t_{0})} (t - t_{0}) = 0$ , there exists a constant $T_{0} \geq t_{0}$ , for all $t \geq T_{0}$ , such that $e^{- \frac{\bar{α}}{2} (t - t_{0})} (t - t_{0}) \leq 1$ . Then, it holds that

\begin{matrix} | x (t) | \leq | x_{0} | e^{- a (t - t_{0})} + ak e^{- \frac{\bar{α}}{2} (t - t_{0})} \\ \leq (| x_{0} | + ak) e^{- \frac{\bar{α}}{2} (t - t_{0})}, t \geq T_{0} \end{matrix}

(3)

Then, $x (t)$ converges to zero exponentially. The proof is complete.□

Theorem 1. The necessary and sufficient condition for the global exponential stability of the equilibrium $S_{0}$ of the system given by equation (1) is $c < \frac{a}{2}$ .

Proof. Necessity. If $S_{0}$ is globally exponentially stable, then $S_{0}$ should be the only equilibrium of the system given by equation (1). Thus, it holds that $c \leq \frac{a}{2}$ .

In addition, the coefficient matrix of the local linearized system of the system given by equation (1) is a Hurwitz matrix, that is

\begin{matrix} (\begin{matrix} - a & a & 0 \\ c - a & c & 0 \\ 0 & 0 & - b \end{matrix}) < 0 \Leftrightarrow (\begin{matrix} - a & a \\ c - a & c \end{matrix}) < 0, since b > 0 \\ \Leftrightarrow {\begin{matrix} - a < 0 \\ - 2 ac + a^{2} > 0 \end{matrix} \\ \Leftrightarrow a^{2} > 2 ac, a > 0 \\ \Leftrightarrow c < \frac{a}{2}, a > 0 \end{matrix}

The necessity has been proved.

Sufficiency. The proof includes three steps.

Step 1. Since $c < \frac{a}{2}$ , consider the following Lypunov function

\begin{matrix} V_{1} = λ x^{2} + \frac{1}{2} y^{2} - μ x y + \frac{1}{2} z^{2} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} λ & - \frac{μ}{2} & 0 \\ - \frac{μ}{2} & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{2} \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{1} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

(4)

where $μ and λ$ are positive constants which will be determined later.

When $(y, z) \in {\bar{Ω}}_{1} = {(y, z) | y \in R, z \leq ξ, 0 < ξ << 1}$ , take the derivative of $V_{1}$ along the system given by equation (1), it holds that

\begin{matrix} \frac{d V_{1}}{d t} = 2 λ x \overset{\cdot}{x} + y \overset{\cdot}{y} - μ \overset{\cdot}{x} y - μ x \overset{\cdot}{y} + z \overset{\cdot}{z} \\ = [- 2 λ a + μ (a - c)] x^{2} + [2 λ a + (μ - 1) (a - c)] xy \\ - (μ a - c) y^{2} + μ x^{2} (z - ξ) + μ ξ x^{2} - b z^{2} \\ \leq {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} Q_{1} (\begin{matrix} x \\ y \\ z \end{matrix}) \leq \frac{λ_{M} (Q_{1})}{λ_{M} (P_{1})} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{1} (\begin{matrix} x \\ y \\ z \end{matrix}) \\ = \frac{λ_{M} (Q_{1})}{λ_{M} (P_{1})} V_{1} \end{matrix}

where

Q_{1} = (\begin{matrix} - 2 λ a + μ (a - c) + μ ξ & λ a + \frac{(a - c) (μ - 1)}{2} & 0 \\ λ a + \frac{(a - c) (μ - 1)}{2} & - μ a + c & 0 \\ 0 & 0 & - b \end{matrix})

Then

V_{1} (x (t), y (t), z (t)) \leq V_{1} (x (t_{0}), y (t_{0}), z (t_{0})) e^{\frac{λ_{M} (Q_{1})}{λ_{M} (P_{1})} (t - t_{0})}

Thus, one has

x^{2} (t) + y^{2} (t) + z^{2} (t) \leq \frac{V_{1} (x (t_{0}), y (t_{0}), z (t_{0}))}{λ_{m} (P_{1})} e^{\frac{λ_{M} (Q_{1})}{λ_{M} (P_{1})} (t - t_{0})}

(5)

If $P_{1} > 0$ and $Q_{1} < 0$ , the trajectory $(x, y, z)$ of the system given by equation (1) begins from $z \leq ξ$ will enter into $z \geq ξ$ or tends to $S_{0}$ exponentially.

In the following, we try to find positive parameters $λ$ and $μ$ such that $P_{1}$ is positive definite and $Q_{1}$ is negative definite, respectively.

It is easy to verify that $P_{1}$ is positive definite if $λ > \frac{μ^{2}}{2}$ .

For the purpose to find the condition that $Q_{1}$ is negative definite, two matrices Q and $Q_{ξ}$ are introduced here

Q = (\begin{matrix} - 2 λ a + μ (a - c) & λ a + \frac{(a - c) (μ - 1)}{2} \\ λ a + \frac{(a - c) (μ - 1)}{2} & - μ a + c \end{matrix})

(6)

and

\begin{matrix} Q_{ξ} = (\begin{matrix} - 2 λ a + μ (a - c) + μ ξ & λ a + \frac{(a - c) (μ - 1)}{2} \\ λ a + \frac{(a - c) (μ - 1)}{2} & - μ a + c \end{matrix}) \end{matrix}

(7)

Since $b > 0$ , one can see that $Q_{1}$ is negative definite if and only if $Q_{ξ}$ is negative definite. According to the principle of continuous function dependence, if Q is negative definite and we choose $0 < ξ << 1$ properly, $Q_{ξ}$ is also negative definite. Thus, we only focus on the property of matrix Q.

In order to reduce a parameter, let $a = kc$ with $k > 2$ , Q becomes

\begin{matrix} Q = (\begin{matrix} - 2 λ kc + μ (k - 1) c & λ kc + \frac{(k - 1) (μ - 1) c}{2} \\ λ kc + \frac{(k - 1) (μ - 1) c}{2} & - μ kc + c \end{matrix}) \end{matrix}

(8)

Then, Q is negative definite if and only if

{\begin{matrix} - 2 λ k + μ (k - 1) < 0 \\ (2 λ k - μ (k - 1)) (μ k - 1) - (λ k + \frac{(k - 1) (μ - 1)}{2})^{2} > 0 \end{matrix}

(9)

The first inequality holds if $λ \geq \frac{μ}{2}$ . Then, we need to find a condition to make the second inequality hold. Let

f (λ) = (2 λ k - μ (k - 1)) (μ k - 1) - (λ k + \frac{(k - 1) (μ - 1)}{2})^{2}

(10)

For obtaining the proper parameter $λ$ , for any given $μ > 1, k > 2$ , let

\begin{matrix} f' (λ) = 2 k (μ k - 1) - 2 λ k^{2} - (k - 1) (μ - 1) k = 0 \end{matrix}

(11)

one has $λ^{*} = \frac{2 (μ k - 1) - (k - 1) (μ - 1)}{2 k} \geq \frac{μ}{2} > 0$ and $f'' (λ^{*}) = - 2 k^{2} < 0$ . Furthermore, $f (λ^{*}) = (k + 2 μ - 4) (μ k - 1) > 0$ since $k > 2 and μ > 1$ . Hence, if we select $λ = λ^{*}$ , then, $Q < 0$ , and the corresponding matrix $Q_{1} < 0$ if $0 < ξ << 1$ .

Now, we focus on finding the proper parameters $μ$ and k such that $P_{1} > 0$ . Then, $μ$ and k should be chosen such that the following conditions hold

{\begin{matrix} λ = \frac{2 (μ k - 1) - (k - 1) (μ - 1)}{2 k} = \frac{μ + 1}{2} + \frac{μ - 3}{2 k} > 0 \\ λ > \frac{μ^{2}}{2} \\ μ > 1, k > 2 \end{matrix}

(12)

Consequently, if $k > 2, 1 < μ < \frac{k + 1 + \sqrt{5 k^{2} - 10 k + 1}}{2 k},$ $λ = λ^{*}$ , then, $P_{1} > 0, Q_{1} < 0$ .

Step 2. Construct a radial unbounded and positive definite Lyapunov function on $y \geq 0, z \geq ly, x \geq 0, 0 < l << 1$ (correspondingly, $(y, z) \in {\bar{Ω}}_{2} = {(y, z) | y \geq 0, z \geq ly, 0 < l << 1}$ )

\begin{matrix} V_{2} = \frac{(1 + ε) (a - c)}{2 a} x^{2} + \frac{1 + ε}{2} y^{2} + \frac{1}{2} z^{2} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} \frac{(1 + ε) (a - c)}{2 a} & 0 & 0 \\ 0 & \frac{1 + ε}{2} & 0 \\ 0 & 0 & \frac{1}{2} \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{2} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

(13)

When $x ε l \geq (1 + ε) (c + 1) (\Rightarrow x \geq \frac{(1 + ε) (c + 1)}{ε l} \geq \frac{(1 + ε) c + 1}{ε l} \overset{Δ}{=} x^{*}$ ), take the derivative of $V_{2}$ along the system given by (1), it yields

\begin{matrix} \frac{d V_{2}}{d t} = \frac{(1 + ε) (a - c) x}{a} \frac{d x}{d t} + (1 + ε) y \frac{d y}{d t} + z \frac{d z}{d t} \\ = - (1 + ε) (a - c) x^{2} + (1 + ε) (a - c) xy \\ + (1 + ε) (c - a) xy + (1 + ε) c y^{2} - ε xyz - b z^{2} \\ \leq - (1 + ε) (a - c) x^{2} + (1 + ε) c y^{2} - ε lx y^{2} - b z^{2} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} - (1 + ε) (a - c) & 0 & 0 \\ 0 & - ε lx + (1 + ε) c & 0 \\ 0 & 0 & - b \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \leq {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} - (1 + ε) (a - c) & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - b \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} Q_{2} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

Hence

V_{2}^{.} (X (t)) \leq \frac{λ_{M} (Q_{2})}{λ_{M} (P_{2})} V_{2} (X (t))

where $λ_{M} (P_{2}) and λ_{M} (Q_{2})$ are the maximum eigenvalue of $P_{2}$ and $Q_{2}$ , respectively. It is easy to verify that $P_{2} > 0$ and $Q_{2} < 0$ , then

V_{2} (X (t)) \leq V_{2} (X (t_{0})) e^{\frac{λ_{M} (Q_{2})}{λ_{M} (P_{2})} (t - t_{0})}

(14)

Consequently, $(y (t), z (t))$ converges to zero exponentially on the condition that $x (t) > x^{*}$ .

Step 3. Construct a radial unbounded and positive definite Lyapunov function on $y \leq 0, z \geq - hy, x \geq 0, 0 < h << 1$ (correspondingly, $(y, z) \in {\bar{Ω}}_{3} = {(y, z) | y \leq 0, z \geq - hy, 0 < h << 1}$ )

\begin{matrix} V_{3} = \frac{(1 - \tilde{ε}) (a - c)}{2 a} x^{2} + \frac{1 - \tilde{ε}}{2} y^{2} + \frac{z^{2}}{2} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} \frac{(1 - \tilde{ε}) (a - c)}{2 a} & 0 & 0 \\ 0 & \frac{1 - \tilde{ε}}{2} & 0 \\ 0 & 0 & \frac{1}{2} \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{3} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

(15)

where $0 < \tilde{ε} < 1$ .

When $x \geq \frac{(1 - \tilde{ε}) c + 1}{\tilde{ε} h}$ , take the derivative of $V_{3}$ along the system given by equation (1), it yields

\begin{matrix} \frac{d V_{3}}{d t} = \frac{(1 - \tilde{ε}) (a - c)}{a} x \frac{d x}{d t} + (1 - \tilde{ε}) y \frac{d y}{d t} + z \frac{d z}{d t} \\ = - (1 - \tilde{ε}) (a - c) x^{2} + (1 - \tilde{ε}) (a - c) xy \\ + (1 - \tilde{ε}) (c - a) xy + (1 - \tilde{ε}) c y^{2} + \tilde{ε} xyz - b z^{2} \\ \leq - (1 - \tilde{ε}) (a - c) x^{2} + (1 - \tilde{ε}) c y^{2} - x \tilde{ε} h y^{2} - b z^{2} \\ \leq {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} - (1 - \tilde{ε}) (a - c) & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - b \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} Q_{3} (\begin{matrix} x \\ y \\ z \end{matrix}) \leq \frac{λ_{M} (Q_{3})}{λ_{M} (P_{3})} V_{3} \end{matrix}

where $λ_{M} (P_{3}), λ_{M} (Q_{3})$ are the maximum Eigenvalue of $P_{3}$ and $Q_{3}$ , respectively. It is easy to verify that $P_{3} > 0$ and $Q_{3} < 0$ . Therefore, $(y (t), z (t))$ exponentially converges to zero on the condition that $x \geq \frac{(1 - \tilde{ε}) c + 1}{\tilde{ε} h}$ .

According to Liao et al. (2015), the system given by equation (1) is Lagrange stable, the state $(y (t), z (t))$ of the system given by equation (1) will converge to a bounded area $Ω$ ( $Ω$ is presented in the work by Liao et al, 2015), it also can be seen in Figure 1) in finite time, which is bounded by the area $\bar{Ω} = {\bar{Ω}}_{1} \cup {\bar{Ω}}_{2} \cup {\bar{Ω}}_{3}$ constructed by steps 1–3, if $x \geq max {\frac{(1 + ε) c + 1}{ε l}, \frac{(1 - \tilde{ε}) c + 1}{\tilde{ε} h}}$ and h and l are selected appropriately. $L_{i}, i = 1, 2, 3$ , are positive constants. Thus, by using Lemma 1, $x (t)$ converges to zero exponentially since $(y (t), z (t))$ converges to zero exponentially.

Consequently, $S_{0}$ is globally exponentially stable. The proof is complete.□

Theorem 2. The necessary and sufficient condition for the global asymptotic stability of equilibrium $S_{0}$ of the system given by equation (1) is $c = \frac{a}{2}$ .

Proof. Necessity. Since the necessary condition for the global asymptotic stability of $S_{0}$ is that the equilibrium is unique, it has $2 c - a \leq 0$ , that is, $c \leq \frac{a}{2}$ . As it is proved in Theorem 1 that $S_{0}$ is global exponentially stable if $c < \frac{a}{2}$ , the necessary condition for the global asymptotic stability rather than the global exponential stability of $S_{0}$ is that $c = \frac{a}{2}$ .

Sufficiency. Construct a positive semidefinite Lyapunov function as follows

\begin{matrix} V_{4} = \frac{1}{2} x^{2} + \frac{1}{2} y^{2} + \frac{1}{2} z^{2} - xy \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} \frac{1}{2} & - \frac{1}{2} & 0 \\ - \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{2} \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

(16)

Take the derivative of $V_{4}$ along the system given by equation (1), it yields

\begin{matrix} \frac{d V_{4}}{dt} = x \overset{\cdot}{x} + y \overset{\cdot}{y} + z \overset{\cdot}{z} - \overset{\cdot}{x} y - x \overset{\cdot}{y} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} - c & \frac{a}{2} & 0 \\ \frac{a}{2} & - c & 0 \\ 0 & 0 & - b \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \leq - b z^{2} \end{matrix}

Then, z of the system given by equation (1) globally asymptotically converges to zero.

According to the LaSalle invariant principle (LaSalle, 1976):

plugging $z = 0$ into the third equation, if $y = 0, x \neq 0$ , it has $xy = 0$ ;

plugging $y = 0$ into the second equation, and one gets $x = 0$ ;

plugging $x = 0$ into the first equation, and we have $y = 0$ .

Therefore, it holds that ${lim}_{t \to \infty} (x, y, z) \to S_{0} (0, 0, 0)$ .

Now, consider $c = \frac{a}{2}$ as the limit case for $c_{n} < c$ , namely, ${lim}_{n \to \infty} c_{n} = c = \frac{a}{2}$ . Similar to the sufficiency proof of Theorem 1, for every $c_{n} \leq c = \frac{a}{2}$ , the corresponding parameters satisfy: $0 < ξ_{n} << 1, 0 < η_{n} << 1, 0 < h_{n} << 1, 0 < l_{n} << 1, λ_{n} \geq μ_{n} > 1$ (shown in Figure 2).

$L_{i}^{(n)}, i = 1, 2, 3, n = 1, 2, 3, \dots$ , are positive constants. For $i = 1, 2, 3$ , $L_{i}^{(n)} > L_{i}^{(n + 1)}$ , we have the following results.

(1) Similar to the proof of first part of sufficiency in Theorem 1, there is a radial unbounded and positive definite function

\begin{matrix} V_{1}^{(n)} = λ_{n} x^{2} + \frac{1}{2} y^{2} - μ_{n} xy + \frac{1}{2} z^{2} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{1}^{(n)} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

(17)

Take the derivative of $V_{1}^{(n)}$ along the system given by equation (1) on condition that $z \leq ξ_{n}$ , it has

\begin{matrix} \frac{d V_{1}^{(n)}}{d t} = 2 λ_{n} x \overset{\cdot}{x} + y \overset{\cdot}{y} - μ_{n} \overset{\cdot}{x} y - μ_{n} x \overset{\cdot}{y} + z \overset{\cdot}{z} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} Q_{1}^{(n)} (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \leq \frac{λ_{M} (Q_{1}^{(n)})}{λ_{M} (P_{1}^{(n)})} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{1}^{(n)} (\begin{matrix} x \\ y \\ z \end{matrix}) \\ = \frac{λ_{M} (Q_{1}^{(n)})}{λ_{M} (P_{1}^{(n)})} V_{1}^{(n)} \end{matrix}

where

Q_{1}^{(n)} = (\begin{matrix} A_{n} & B_{n} & 0 \\ B_{n} & - μ_{n} a + c_{n} & 0 \\ 0 & 0 & - b \end{matrix})

with $A_{n} = - 2 λ_{n} a + μ_{n} (a - c_{n}) + μ_{n} ξ_{n}$ , $B_{n} = λ_{n} a + \frac{(a - c_{n}) (μ_{n} - 1)}{2} .$

Thus, one has

V_{1}^{(n)} (t) \leq V_{1}^{(n)} (t_{0}) e^{\frac{λ_{M} (Q_{1}^{(n)})}{λ_{M} (P_{1}^{(n)})} (t - t_{0})}

(18)

(2) Construct a Lyapunov function $V_{2}^{(n)}$ under the condition that $y \geq 0, z \geq l_{n} y, x \geq 0, 0 < l_{n} << 1$

\begin{matrix} V_{2}^{(n)} = \frac{(1 + ε) (a - c_{n})}{2 a} x^{2} + \frac{1 + ε}{2} y^{2} + \frac{1}{2} z^{2} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} \frac{(1 + ε) (a - c_{n})}{2 a} & 0 & 0 \\ 0 & \frac{1 + ε}{2} & 0 \\ 0 & 0 & \frac{1}{2} \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{2}^{(n)} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

(19)

Follow the proof of the second step of sufficiency in Theorem 1, when $x \geq \frac{(1 + ε) (c_{n} + 1)}{ε l_{n}} \overset{Δ}{=} x_{n}^{*}$ , take the derivative of $V_{2}^{(n)}$ along the system given by equation (1), it has

\begin{matrix} \frac{{d V}_{2}^{(n)}}{dt} = \frac{(1 + ε) (a - c_{n}) x}{a} \overset{\cdot}{x} + (1 + ε) y \overset{\cdot}{y} + z \overset{\cdot}{z} \\ \leq {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} - (1 + ε) (a - c_{n}) & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - b \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} Q_{2}^{(n)} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

Then, we have

V_{2}^{(n)} (t) \leq V_{2}^{(n)} (t_{0}) e^{\frac{λ_{M} (Q_{2}^{(n)})}{λ_{M} (P_{2}^{(n)})} (t - t_{0})}

(20)

So, $(y (t), z (t)) \to 0$ if $t \to + \infty$ or enter into the area of $y \leq 0$ .

(3) Construct a Lyapunov function $V_{3}^{(n)}$ under the condition of $y \leq 0, z \geq - h_{n} y, x \geq 0, 0 < h_{n} << 1$

\begin{matrix} V_{3}^{(n)} = \frac{(1 - \tilde{ε}) (a - c_{n})}{2 a} x^{2} + \frac{1 - \tilde{ε}}{2} y^{2} + \frac{z^{2}}{2} \\ = {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} \frac{(1 - \tilde{ε}) (a - c_{n})}{2 a} & 0 & 0 \\ 0 & \frac{1 - \tilde{ε}}{2} & 0 \\ 0 & 0 & \frac{1}{2} \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} P_{3}^{(n)} (\begin{matrix} x \\ y \\ z \end{matrix}) \end{matrix}

(21)

where $0 < \tilde{ε} < 1$ . When $x \geq \frac{(1 - \tilde{ε}) c_{n} + 1}{\tilde{ε} h_{n}}$ , take the derivative of $V_{3}$ along the system given by equation (1), it yields

\begin{matrix} \frac{d V_{3}^{(n)}}{d t} & = \frac{(1 - \tilde{ε}) (a - c_{n})}{a} x \overset{\cdot}{x} + (1 - \tilde{ε}) y \overset{\cdot}{y} + z \overset{\cdot}{z} \\ \leq {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} (\begin{matrix} - (1 - \tilde{ε}) (a - c_{n}) & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - b \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ \overset{Δ}{=} {(\begin{matrix} x \\ y \\ z \end{matrix})}^{T} Q_{3}^{(n)} (\begin{matrix} x \\ y \\ z \end{matrix}) \leq \frac{λ_{M} (Q_{3}^{(n)})}{λ_{M} (P_{3}^{(n)})} V_{3}^{(n)} \end{matrix}

Then, $(y (t), z (t)) \to 0$ when $t \to + \infty$ , according to Lemma 1, it holds that $(x, y, z) \to 0$ .

It has been proven that $(y, z) \to 0$ and $x \to 0$ under the condition of:

$z \leq ξ_{n}$ ;

$z \geq l_{n} y, y \geq 0$ ;

$z \geq - h_{n} y, y \leq 0$ .

Assume ${lim}_{n \to + \infty} c_{n} = c = \frac{a}{2}$ , ${lim}_{n \to + \infty} ξ_{n} = 0$ and ${lim}_{n \to + \infty} η_{n} = 0$ , then, ${lim}_{n \to + \infty} h_{n} = 0$ and ${lim}_{n \to + \infty} l_{n} = 0$ . Thus, we have $(y, z) \to 0$ and $x \to 0$ . As a result, $S_{0}$ is globally asymptotically stable. The proof is complete. □

Theorem 3. $S_{0}$ is unstable if and only if $c > \frac{a}{2}$ .

Proof. Necessity. If $S_{0}$ of the system given by equation (1) is unstable, due to Theorem 1 and Theorem 2, it only holds that $c > \frac{a}{2}$ .

Sufficiency. The coefficient matrix of the linearized system of the system given by equation (1) is as follows

A = (\begin{matrix} - a & a & 0 \\ c - a & c & 0 \\ 0 & 0 & - b \end{matrix})

(22)

Since

det (\begin{matrix} λ + a & - a \\ a - c & λ - c \end{matrix}) = λ^{2} + (a - c) λ - 2 ac + a^{2}

one characteristic value is given by

\begin{matrix} λ_{1} = \frac{- (a - c) + \sqrt{{(a - c)}^{2} - 4 (a^{2} - 2 ac)}}{2} \\ > \frac{- (a - c) + \sqrt{{(a - c)}^{2}}}{2} = 0 \end{matrix}

Then, matrix A has at least one positive characteristic value. Hence, for the system given by equation (1), the corresponding linearized system on $S_{0}$ is unstable. Furthermore, from the Lyapunov first-order approximation theory (Liao, 2011), it is easy to obtain that $S_{0}$ of the nonlinear system given by equation (1) is also unstable. The proof is complete.□

Sometimes, we want to remove the chaotic phenomena with a simple control input in a system so that it can work steadily. No matter what kind of feedback controller, the minimum feedback and less conservative controllers are desirable to be added to the system. We have added less conservative linear feedback controllers to the Lorenz chaotic system, Lü chaotic system, Yang–Chen chaotic system and Li Yuxia chaotic system such that they are globally exponentially stable (Liao and Luo, 2010). As a corollary, a linear feedback controller will be added to the Chen chaotic system when it is unstable.

Corollary 1.When $c > \frac{a}{2}$ , add a negative feedback controller $- k (x + y)$ to the second equation of the Chen system as follows

{\begin{matrix} \frac{d x}{d t} = a (y - x) \\ \frac{d y}{d t} = (c - a) x - xz + cy - k (x + y) \\ \frac{d z}{d t} = xy - bz \end{matrix}

(23)

Then, $S_{0}$ is globally exponentially stable if $k > c - \frac{a}{2}$ ; $S_{0}$ is globally asymptotically (non-exponentially) stable if $k = c - \frac{a}{2}$ .

Proof. Let $\bar{c} = c - k$ , the system given by equation (23) becomes

{\begin{matrix} \frac{d x}{d t} = a (y - x) \\ \frac{d y}{d t} = (\bar{c} - a) x - xz + \bar{c} y \\ \frac{d z}{d t} = xy - bz \end{matrix}

(24)

Referring to Theorem 1 and 2, if $\bar{c} < \frac{a}{2} \Rightarrow k > c - \frac{a}{2}$ , $S_{0}$ is global exponentially stable; if $\bar{c} = \frac{a}{2} \Rightarrow k = c - \frac{a}{2}$ , $S_{0}$ is global asymptotically (non-exponentially) stable. The proof is complete.□

Sufficient and necessary condition for the local stability of the equilibria $S_{\pm}$

In this section, the Lyapunov stability condition for the equilibria $S_{\pm}$ will be studied. Since there is only one global stable equilibrium point for a system, the stability of $S_{\pm}$ will be considered in a local sense. For the purpose of avoiding duplication, we only propose the equivalent conditions for the local exponential stability, the proof for local asymptotic stability and instability of $S_{\pm}$ is omitted. On the other hand, a linear feedback controller will be used to make equilibria $S_{\pm}$ globally exponentially stable and globally asymptotically stable.

Assume that $(x^{*}, y^{*}, z^{*})$ is one of the two equilibria $S_{\pm}$ . Let $\bar{x} = x - x^{*}, \bar{y} = y - y^{*} and \bar{z} = z - z^{*}$ , it has

{\begin{matrix} \frac{d \bar{x}}{d t} = a (\bar{y} - \bar{x}) \\ \frac{d \bar{y}}{d t} = (c - a) \bar{x} - \bar{x} \bar{z} - x^{*} \bar{z} - z^{*} \bar{x} + c \bar{y} \\ \frac{d \bar{z}}{d t} = \bar{x} \bar{y} + x^{*} \bar{y} + y^{*} \bar{x} - b \bar{z} \end{matrix}

(25)

The corresponding linearized system is

{\begin{matrix} \frac{d \bar{x}}{d t} = a (\bar{y} - \bar{x}) \\ \frac{d \bar{y}}{d t} = (c - a - z^{*}) \bar{x} + c \bar{y} - x^{*} \bar{z} \\ \frac{d \bar{z}}{d t} = y^{*} \bar{x} + x^{*} \bar{y} - b \bar{z} \end{matrix}

(26)

Let $A_{+} (A_{-})$ be the coefficient matrix of the system given by equation (26) with $(x^{*}, y^{*}, z^{*}) = S_{+} (S_{-})$ , we have the following results.

Theorem 4. The characteristic polynomial of $A_{+} (A_{-})$ is given by

\begin{matrix} f (λ) = \det (λ - A_{+}) \\ = λ^{3} + (a + b - c) λ^{2} + bc λ + 4 abc - 2 a^{2} b \\ = \det (λ - A_{-}) \end{matrix}

(27)

Then, the following conditions are equivalent.

Case 1. $f (λ)$ is a Hurwitz polynomial.

$\Leftrightarrow (a + b - c) bc > 4 abc - 2 a^{2} b$

$\Leftrightarrow t h e$ linearized system given by equation (26) of the system given by equation (25) is exponentially stable.

$\Leftrightarrow the$ system given by equation (1) is locally exponentially stable on $S_{+} (S_{-}$ ).

Case 2. The coefficient matrix $A_{+} (A_{-})$ has an eigenvalue with a positive real part.

$\Leftrightarrow the$ linearized system of the system Case 2. (1) on $S_{+} (S_{-})$ is unstable.

$\Leftrightarrow (a + b - c) bc < 4 abc - 2 a^{2} b .$

Case 3. The linearized system of the system given by equation (1) on $S_{+} (S_{-})$ is stable while non-asymptotically stable.

$\Leftrightarrow (a + b - c) bc = 4 abc - 2 a^{2} b$

For the purpose of avoiding duplication, the proof process is omitted. One can refer to the work of Liao and Luo (2010) if necessary. In the following, we focus on finding linear control inputs such that $S_{+} (S_{-})$ is globally exponentially stable, globally asymptotically stable and unstable.

Theorem 5. Add three feedback control inputs $- k_{1} \bar{x}$ , $- k_{2} \bar{y}$ and $- k_{3} \bar{z}$ to three equations of the system given by equation (25), respectively, it becomes

{\begin{matrix} \frac{d \bar{x}}{d t} = a (\bar{y} - \bar{x}) - k_{1} \bar{x} \\ \frac{d \bar{y}}{d t} = (c - a) \bar{x} - \bar{x} \bar{z} - x^{*} \bar{z} - z^{*} \bar{x} + c \bar{y} - k_{2} \bar{y} \\ \frac{d \bar{z}}{d t} = \bar{x} \bar{y} + x^{*} \bar{y} + y^{*} \bar{x} - b \bar{z} - k_{3} \bar{z} \end{matrix}

(28)

If the following matrix G is negative definite, then, $(\bar{x}, \bar{y}, \bar{z}) = (0, 0, 0)$ is globally exponentially stable. And thus, the corresponding equilibrium points $S_{+} (S_{-})$ are globally exponentially stable

G = (\begin{matrix} - a - k_{1} & \frac{c - z^{*}}{2} & \frac{y^{*}}{2} \\ \frac{c - z^{*}}{2} & c - k_{2} & 0 \\ \frac{y^{*}}{2} & 0 & - b - k_{3} \end{matrix})

(29)

Proof. Construct a Lyapunov function for the system given by equation (28) as follows

V_{5} = \frac{1}{2} {\bar{x}}^{2} + \frac{1}{2} {\bar{y}}^{2} + \frac{1}{2} {\bar{z}}^{2}

(30)

Take the derivative of $V_{5}$ along the system given by equation (28), it has

\begin{matrix} \frac{d V_{5}}{d t} = - (a + k_{1}) {\bar{x}}^{2} + (c - k_{2}) {\bar{y}}^{2} - (b + k_{3}) {\bar{z}}^{2} \\ + (c - z^{*}) \bar{x} \bar{y} + y^{*} \bar{x} \bar{z} \\ = {(\begin{matrix} \bar{x} \\ \bar{y} \\ \bar{z} \end{matrix})}^{T} (\begin{matrix} - a - k_{1} & \frac{c - z^{*}}{2} & \frac{y^{*}}{2} \\ \frac{c - z^{*}}{2} & c - k_{2} & 0 \\ \frac{y^{*}}{2} & 0 & - b - k_{3} \end{matrix}) (\begin{matrix} \bar{x} \\ \bar{y} \\ \bar{z} \end{matrix}) \\ = {(\begin{matrix} \bar{x} \\ \bar{y} \\ \bar{z} \end{matrix})}^{T} G (\begin{matrix} \bar{x} \\ \bar{y} \\ \bar{z} \end{matrix}) \\ \leq λ_{M} (G) ({\bar{x}}^{2} + {\bar{y}}^{2} + {\bar{z}}^{2}) = 2 λ_{M} (G) V_{5} \end{matrix}

Then, if G is negative definite, $λ_{M} (G) < 0$ , we have

V (\bar{X} (t)) \leq V (\bar{X} (t_{0})) e^{2 λ_{M} (G) (t - t_{0})}

(31)

Therefore, ${\bar{x}}^{2} (t) + {\bar{y}}^{2} (t) + {\bar{z}}^{2} (t) \leq ({\bar{x}}^{2} (t_{0}) + {\bar{y}}^{2} (t_{0}) + {\bar{z}}^{2} (t_{0})) e^{2 λ_{M} (G) (t - t_{0})}$ . Consequently, the equilibrium $(\bar{x}, \bar{y}, \bar{z}) = (0, 0, 0)$ of the system given by equation (28) is globally exponentially stable. And thus, the corresponding equilibrium point $S_{+} (S_{-})$ of the system given by equation (1) is globally exponentially stable. The proof is complete. □

Remark 1. If the parameters $k_{1}, k_{2} and k_{3}$ are selected to satisfy the following inequalities

{\begin{matrix} k_{1} > \frac{c - z^{*}}{2} + \frac{y^{*}}{2} - a \\ k_{2} > \frac{c - z^{*}}{2} + c \\ k_{3} > \frac{y^{*}}{2} - b \end{matrix}

(32)

the coefficient matrix G is negative definite, therefore, the results of Theorem 5 hold.

As a corollary of Theorem 5, with negative feedback control inputs, a globally exponentially state tracking result is proposed in the following.

Assuming $(x_{1}, y_{1}, z_{1})$ is any bounded trajectory of system (1) with $| x_{1} | \leq M, | y_{1} | \leq M, | z_{1} | \leq M$ . Let $\tilde{x} = x - x_{1}, \tilde{y} = y - y_{1} and \tilde{z} = z - z_{1}$ , one has

{\begin{matrix} \frac{d \tilde{x}}{d t} = a (\tilde{y} - \tilde{x}) \\ \frac{d \tilde{y}}{d t} = (c - a) \tilde{x} - \tilde{x} \tilde{z} - x_{1} \tilde{z} - z_{1} \tilde{x} + c \tilde{y} \\ \frac{d \tilde{z}}{d t} = \tilde{x} \tilde{y} + x_{1} \tilde{y} + y_{1} \tilde{x} - b \tilde{z} \end{matrix}

(33)

Add three negative feedback control terms $- k_{1} \tilde{x},$ $- k_{2} \tilde{y} and - k_{3} \tilde{z}$ on three equations of the system given by equation (33), respectively. It has

{\begin{matrix} \frac{d \tilde{x}}{d t} = a (\tilde{y} - \tilde{x}) - k_{1} \tilde{x} \\ \frac{d \tilde{y}}{d t} = (c - a) \tilde{x} - \tilde{x} \tilde{z} - x \tilde{z} - z \tilde{x} + c \tilde{y} - k_{2} \tilde{y} \\ \frac{d \tilde{z}}{d t} = \tilde{x} \tilde{y} + x \tilde{y} + y \tilde{x} - b \tilde{z} - k_{3} \tilde{z} \end{matrix}

(34)

Corollary 2. If the following matrix Q is negative definite, then, the state of the system given by equation (1) can globally exponentially synchronize the trajectory $(x_{1}, y_{1}, z_{1})$

Q = (\begin{matrix} - a - k_{1} & \frac{| c | + M}{2} & \frac{M}{2} \\ \frac{| c | + M}{2} & c - k_{2} & 0 \\ \frac{M}{2} & 0 & - b - k_{3} \end{matrix})

(34)

Simulation

In this section, several cases of the Chen system are proposed to illustrate the theoretical results obtained in the previous sections. A fourth order Runge–Kutta method is used to obtain the simulation results with MATLAB software.

Case 1. If $a = 35, b = 3 and c = 10$ , it is easy to verify that $c = 10 < \frac{a}{2} = 17.5$ . From Theorem 1, $S_{0}$ is globally exponentially stable. With the initial state $(x_{0}, y_{0}, z_{0}) = (2, - 2, 3)$ , one can see the convergence of the state trajectories in Figure 3.

Case 2. If $a = 35, b = 3 and c = 17.5$ , it is easy to verify that $c = 17.5 = \frac{a}{2} = 17.5$ . From Theorem 2, $S_{0}$ is globally asymptotically stable. With the initial state $(x_{0}, y_{0}, z_{0}) = (2, - 2, 3)$ , one can see the convergence of the state trajectories in Figure 4.

Case 3. If $a = 35, b = 3 and c = 25$ , it is easy to verify that $c = 25 > \frac{a}{2} = 17.5$ . By Theorem 3, $S_{0}$ is unstable. With the initial state $(x_{0}, y_{0}, z_{0}) = (2, - 2, 3)$ , as it is shown in Figure 5, one can see that the state trajectories cannot converge to $S_{0}$ any more.

According to Corollary 1, choose feedback control input $- k (x + y)$ for the second equation of systems (1). Let $k = c - a / 2 + 1$ , then, the controlled system given by equation (23) is globally exponentially stable, and the trajectories of the corresponding states are shown in Figure 6. Let $k = c - a / 2$ , then, the controlled system given by equation (23) is globally asymptotically stable, and the trajectories of the corresponding states are shown in Figure 7.

Case 4. If $a = 35, b = 3 and c = 18$ , it is easy to verify that $c = 18 > \frac{a}{2} = 17.5$ . According to Theorem 3, $S_{0}$ is unstable. Since $(a + b - c) bc = 1080 > 4 abc - 2 a^{2} b = 210$ , from Theorem 4, the equilibrium points $S_{\pm}$ are locally exponentially stable. With the initial states $(x_{0}, y_{0}, z_{0}) = (2, 2, 3)$ or $(x_{0}, y_{0}, z_{0}) = (- 2, - 2, 3)$ , one can see that the state trajectories converge to $S_{+}$ or $S_{-}$ in Figures 8 and 9, respectively.

Case 5. If $a = 35, b = 3, c = \frac{- (3 a - b) + \sqrt{{(3 a - b)}^{2} + 8 a^{2}}}{2}$ , then, $(a + b - c) bc = 4 abc - 2 a^{2} b$ . According to Theorem 4, the equilibrium points $S_{\pm}$ are locally stable (non-asymptotically stable). With the initial states $(x_{0}, y_{0}, z_{0}) = (3.9, 3.9, 5)$ or $(x_{0}, y_{0}, z_{0}) = (- 3.9, - 3.9, 5)$ , one can see the state trajectories in Figures 10 and 11, respectively.

Conclusions

In this paper, the necessary and sufficient conditions for global exponential stability, global asymptotic stability and instability of the equilibrium $S_{0}$ for the Chen system are proposed at first. Then, the conditions of local exponential stability and stability for $S_{+}$ and $S_{-}$ are presented. When any of the three equilibrium points are not stable, a linear feedback controller is introduced. Simulations are shown to illustrate the effectiveness of our proposed results. The necessary and sufficient conditions for the Lyapunov stability of dynamical systems is very important and worthy of study. The proof methods and techniques used here can be used to study other systems, such as the Lü system, Yang–Chen system, etc.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Joint Key Grant of National Natural Science Foundation of China and Zhejiang Province (grant number U1509217), the National Natural Science Foundation of China (grant number 71473073), the Hubei province science and technology support program (grant number 2015BAA001) and the Hubei SMEs Innovation Fund Project (grant number 2015DAL069).

References

Algaba

Domínguez-Moreno

Merino

(2015) Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems. Nonlinear Dynamics 79: 885–902.

Chen

(1999) Another chaotic attractor. International Journal of Bifurcation and Chaos 9: 1465–1466.

Chen

Lü

(2003) Dynamics Analysis, Control and Synchronization of Lorenz System. Beijing: Science Press.

LaSalle

(1976) The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics. Philadelphia, PA: SIAM.

Leonov

(2001) Bound for attractors and the existence of homoclinic orbits in the Lorenz system. Journal of Applied Mathematics and Mechanics 65: 19–32.

(2005) Estimating the bounded for the Lorenz family of chaotic systems. Chaos, Solitons & Fractals 23: 529–534.

(2005) Research on Anticontrol of Hyperchaos for Continuous-time Systems. PhD Thesis, Guangdong University of Technology, Guangzhou, China.

Liao

(2001) Mathematical Theory and Application of Stability. Wuhan: Huazhong Normal University Press.

Liao

(2004) On new results of global attractive set and positively sets for the Lorenz chaotic system and the application in chaos control and synchronization. Science China (Technology Science) 34: 1404–1419.

10.

Liao

(2010) Theory, Method and Application of Stability. 2nd ed. Wuhan: Huazhong University of Science and Technology Press.

11.

Liao

Luo

(2010) Lyapunov stability of simple algebraic necessary and sufficient conditions of Lorenz chaotic systems and their application. Science China (Technology Science) 40(8): 1086–1095.

12.

Liao

(2006) Study on the global property of the smooth Chua’s syatem. International Journal of Bifurcation and Chaos 16: 2815–2841.

13.

Liao

Xie

(2005) On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization. Science China (Information Science) 48: 304–321.

14.

Liao

Luo

(2007) Analysis on the global exponential set and positive invariant set of the Lorenz family. Science China (Technology Science) 37: 757–769.

15.

Liao

et al . (2015) Constructive proof of globally exponentially attractive and positively invariant set of the chaotic Chen’s system. Science China Information Sciences 45(1): 129–144.

16.

Liu

Pang

(2012) Dynamics of the general Lorenz family. Nonlinear Dynamics 67: 1595–1611.

17.

Luo

Mao

Shen

(2006) New criteria on exponential stability of neutral stochastic differential delay equations. Systems and Control Letters 55(10): 826–834.

18.

Miao

(2015b) Output consensus for heterogeneous multi-agent systems with linear dynamics. Applied Mathematics and Computation 271: 548–555.

19.

Feng

(2013) Delay-dependent stability criteria for reaction-diffusion neural networks with time-varying delays. IEEE Transactions on Cybernetics 43(6): 1913–1920.

20.

Lewis

(2015a) Cooperative containment of discrete-time linear multi-agent systems. International Journal of Roubust and Nonlinear Control 25: 1007–1018.

21.

Lewis

et al . (2016) Cooperative output regulation of singular heterogeneous multiagent systems. IEEE Transactions on Cybernetics 46(6): 1471–1475.

22.

Sparrow

(1976) The Lorenz Equations: Bifurcation, Chaos and Strange Attractors. New York, Heidelberg, Berlin: Springer-Verlag.

23.

Stwart

(2002) The Lorenz attractor exists. Nature 406: 948–949.

24.

Wang

Huang

Zhao

(2012) Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives. Nonliear Dynamics 69(3): 999–1007.

25.

Yang

Chen

(2008) A chaotic system with one saddle and two stable node-foci. International Journal of Bifurcation and Chaos 18: 1393–1414.

26.

Yang

Wang

(2007) Theory and Application of Nonlinear Dynamics. Beijing: National Defense Industry Press.

27.

Liao

(2006) New estimations for globally attractive and positive invariant set of the family of the Lorenz systems. International Journal of Bifurcation and Chaos 16: 3383–3390.

28.

Liao

Xie

et al . (2009) A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family. Communications in Nonlinear Science and Numerical Simulation 14: 2886–2896.

29.

Yuan

DWC

(2016a) Regularized primal-dual subgradient method for distributed constrained optimization. IEEE Transactions on Cybernetics 46(9): 2109–2118.

30.

Yuan

DWC

Hong

(2016b) On convergence rate of distributed stochastic gradient algorithm for convex optimization with inequality constraints. SIAM Journal on Control and Optimization 54(5): 2872–2892.

31.

Zhang

Lam

(2015) Stability analysis of distributed delay neural networks based on relaxed Lyapunov-Krasovskii functionals. IEEE Transactions on Neural Networks and Learning Systems 26(7): 1480–1492.

32.

Zhang

Zheng

(2013) Filtering of Markovian jump delay systems based on a new performance index. IEEE Transactions on Circuits and Systems-I: Regular Papers 60(5): 1250–1263.

33.

Zhou

Huang

Liao

et al . (2013) Stability analysis and control of a new smooth Chua’s system. Abstract and Applied Analysis 2013: 620286 (10pages).

34.

Zhou

Liu

Huang

et al . (2015) Global synchronization of a new Chua’s system. International Journal of Bifurcation and Chaos 25(4): 1550053 (13pages).

Simple algebraic necessary and sufficient conditions for Lyapunov stability of a Chen system and their applications

Abstract

Keywords

Introduction

Sufficient and necessary conditions for the Lyapunov stability of the equilibrium S 0

Sufficient and necessary condition for the local stability of the equilibria S ±

Simulation

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References

Sufficient and necessary conditions for the Lyapunov stability of the equilibrium $S_{0}$

Sufficient and necessary condition for the local stability of the equilibria $S_{\pm}$