Analysis, Adaptive Control and Synchronization of a Novel 3-D Chaotic System with a Quartic Nonlinearity and Two Quadratic Nonlinearities

Abstract

In this work, we announce a seven-term novel 3-D chaotic system with a quartic nonlinearity and two quadratic nonlinearities. The proposed chaotic system is highly chaotic and it has interesting qualitative properties. The phase portraits of the novel chaotic system are illustrated and the dynamic properties of the highly chaotic system are discussed. The novel 3-D chaotic system has three unstable equilibrium points. We show that the equilibrium point at the origin is a saddle point, while the other two equilibrium points are saddle foci. The novel 3-D chaotic system has rotation symmetry about the $x_3$ axis, which shows that every non-trivial trajectory of the system must have a twin trajectory. The Lyapunov exponents of the novel 3-D chaotic system are obtained as $L_1 = 8.6606$, $L_2 = 0$ and $L_3 = -26.6523$, while the Kaplan-Yorke dimension of the novel chaotic system is obtained as $D_{KY} = 2.3249$. Since the Maximal Lyapunov Exponent (MLE) of the novel chaotic system has a large value, viz. $L_1 = 8.6606$, the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we apply adaptive control method to derive new results for the global chaos control of the novel chaotic system with unknown parameters. We also apply adaptive control method to derive new results for the global chaos synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results derived in this work.

1 Introduction

Chaotic systems are defined as nonlinear dynamical systems which are sensitive to initial conditions, topologically mixing and with dense periodic orbits. Sensitivity to initial conditions of chaotic systems is popularly known as the butterfly effect.

Chaotic systems are either conservative or dissipative. The conservative chaotic systems are characterized by the property that they are volume conserving. The dissipative chaotic systems are characterized by the property that any asymptotic motion of the chaotic system settles onto a set of measure zero, i.e. a strange attractor. In this research work, we shall announce and discuss a novel 3-D dissipative highly chaotic circulant chaotic system with six sinusoidal nonlinearities.

The Lyapunov exponent of a chaotic system is a measure of the divergence of points which are initially very close and this can be used to quantify chaotic systems. Each nonlinear dynamical system has a spectrum of Lyapunov exponents, which are equal in number to the dimension of the state space. The largest Lyapunov exponent of a nonlinear dynamical system is called the maximal Lyapunov exponent (MLE).

In the last few decades, Chaos theory has become a very important and active research field, employing many applications in different disciplines like physics, chemistry, biology, ecology, engineering and economics, among others.

Some classical paradigms of 3-D chaotic systems in the literature are Lorenz system [31], Rössler system [45], ACT system [1], Sprott systems [52], Chen system [14], Lü system [32], Cai system [13], Tigan system [64], etc.

Many new chaotic systems have been discovered in the recent years such as Zhou system [128], Zhu system [129], Li system [27], Sundarapandian systems [57, 61], Vaidyanathan systems [72, 74, 76–79, 83, 90, 100, 101, 103, 109, 111, 114, 117, 118, 120], Pehlivan system [37], Sampath system [47], Pham system [39], etc.

Chaos theory and control systems have many important applications in science and engineering [2, 9–12, 130]. Some commonly known applications are oscillators [23, 51], lasers [28, 125], chemical reactions [17, 38, 87, 88, 91, 93, 94, 98], biology [15, 25, 82, 84–86, 89, 92, 96, 97], ecology [18, 54], encryption [26, 127], cryptosystems [44, 65], mechanical systems [4–8], secure communications [16, 34, 126], robotics [33, 35, 122], cardiology [41, 124], intelligent control [3, 29], neural networks [20, 22, 30], finance [19, 53], memristors [40, 123],etc.

The control of a chaotic system aims to stabilize or regulate the system with the help of a feedback control. There are many methods available for controlling a chaotic system such as active control [55, 66, 67], adaptive control [56, 68, 73, 75, 81, 99, 110, 116, 119], sliding mode control [70, 71], backstepping control [36, 113, 121], etc.

Major works on synchronization of chaotic systems deal with the complete synchronization (CS) which has the design goal of using the output of the master system to control the slave system so that the output of the slave system tracks the output of the master system asymptotically with time.

There are many methods available for chaos synchronization such as active control [21, 48, 49, 104, 106, 112], adaptive control [46, 50, 58–60, 69, 95, 102, 105], sliding mode control [62, 80, 108, 115], backstepping control [42, 43, 63, 107], etc.

In this research work, we announce a seven-term novel 3-D chaotic system with a quartic nonlinearity and two quadratic nonlinearities. Section 2 describes the 3-D dynamic equations and phase portraits of the seven-term novel 3-D chaotic system.

Section 3 details the qualitative analysis and properties of the novel 3-D chaotic system. The Lyapunov exponents of the novel chaotic system are obtained as $L_1 = 8.6606$, $L_2 = 0$ and $L_3 = -26.6523$, while the Kaplan-Yorke dimension of the novel chaotic system is obtained as $D_{KY} = 2.3249$. Since the maximal Lyapunov exponent of the novel chaotic system has a large value, viz. $L_1 = 8.6606$, the novel chaotic system is highly chaotic.

In Sect. 4, we derive new results for the global chaos control of the novel highly chaotic system with unknown parameters. In Sect. 5, we derive new results for the global chaos synchronization of the identical novel highly chaotic systems with unknown parameters. Section 6 contains a summary of the main results derived in this work.

2 A Novel 3-D Chaotic System

In this section, we describe a seven-term novel chaotic system, which is given by the 3-D dynamics

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{x}_1 &{} = &{} a (x_2 - x_1) + q x_3 + x_2 x_3 \\ \dot{x}_2 &{} = &{} b x_1 - x_1 x_3 \\ \dot{x}_3 &{} = &{} - c x_3 + p x_1^4 \\ \end{array} \right. \end{aligned}$$

(1)

where $x_1, x_2, x_3$ are the states and a, b, c, p, q are constant, positive parameters.

The novel 3-D system (1) is a seven-term polynomial system with a quartic nonlinearity and three quadratic nonlinearities.

The system (1) exhibits a highly chaotic attractor for the parameter values

$$\begin{aligned} a = 12, \ \ b = 55, \ \ c = 6, \ \ p = 25, \ \ q = 0.2 \end{aligned}$$

(2)

For numerical simulations, we take the initial conditions as

$$\begin{aligned} x_1(0) = 2.6, \ \ \ x_2(0) = 1.8, \ \ \ x_3(0) = 2.5 \end{aligned}$$

(3)

Figure 1 depicts the 3-D phase portrait of the novel 3-D chaotic system (1), while Figs. 2, 3 and 4 depict the 2-D projection of the novel 3-D chaotic system (1) on the $(x_1, x_2)$, $(x_2, x_3)$ and $(x_1, x_3)$ planes, respectively. Figures 1, 2, 3 and 4 show that the novel 3-D chaotic system (1) exhibits a highly chaotic attractor.

Fig. 1

3-D phase portrait of the novel 3-D chaotic system

Fig. 2

2-D projection of the novel 3-D chaotic system on the $(x_1, x_2)$ plane

Fig. 3

2-D projection of the novel 3-D chaotic system on the $(x_2, x_3)$ plane

Fig. 4

2-D projection of the novel 3-D chaotic system on the $(x_1, x_3)$ plane

3 Analysis of the Novel 3-D Chaotic System

In this section, we give a dynamic analysis of the 3-D novel highly chaotic system (1). We take the parameter values as in the chaotic case (2), i.e. $a = 12$, $b = 55$, $c = 6$, $p = 25$ and $q = 0.2$.

3.1 Dissipativity

In vector notation, the novel chaotic system (1) can be expressed as

$$\begin{aligned} \dot{\mathbf {x}} = f(\mathbf {x}) = \left[ \begin{array}{c} f_1(x_1, x_2, x_3) \\ f_2(x_1, x_2, x_3) \\ f_3(x_1, x_2, x_3) \\ \end{array} \right] , \end{aligned}$$

(4)

where

$$\begin{aligned} \left\{ \begin{array}{ccl} f_1(x_1, x_2, x_3) &{} = &{} a (x_2 - x_1) + q x_3 + x_2 x_3 \\ f_2(x_1, x_2, x_3) &{} = &{} b x_1 - x_1 x_3 \\ f_3(x_1, x_2, x_3) &{} = &{} - c x_3 + p x_1^4 \\ \end{array} \right. \end{aligned}$$

(5)

Let $\varOmega$ be any region in ${\mathbf R}^3$ with a smooth boundary and also, $\varOmega (t) = \varPhi _t(\varOmega ),$ where $\varPhi _t$ is the flow of f. Furthermore, let V(t) denote the volume of $\varOmega (t)$.

By Liouville’s theorem, we know that

$$\begin{aligned} \dot{V}(t) = \int \limits _{\varOmega (t)} \ (\nabla \cdot f) \, dx_1 \, dx_2 \, dx_3 \end{aligned}$$

(6)

The divergence of the novel chaotic system (4) is found as

$$\begin{aligned} \nabla \cdot f = {\partial f_1 \over \partial x_1} + {\partial f_2 \over \partial x_2} + {\partial f_3 \over \partial x_3} = - (a + c) = - \mu < 0 \end{aligned}$$

(7)

since $\mu = a + c = 18 > 0$.

Inserting the value of $\nabla \cdot f$ from (7) into (6), we get

$$\begin{aligned} \dot{V}(t) = \int \limits _{\varOmega (t)} \ (- \mu ) \, dx_1 \, dx_2 \, dx_3 = - \mu V(t) \end{aligned}$$

(8)

Integrating the first order linear differential equation (8), we get

$$\begin{aligned} V(t) = \exp (- \mu t) V(0) \end{aligned}$$

(9)

Since $\mu > 0$, it follows from Eq. (9) that $V(t) \rightarrow 0$ exponentially as $t \rightarrow \infty$. This shows that the novel chaotic system (1) is dissipative.

Hence, the system limit sets are ultimately confined into a specific limit set of zero volume, and the asymptotic motion of the novel chaotic system (1) settles onto a strange attractor of the system.

3.2 Equilibrium Points

We take the parameter values as in the chaotic case (2), i.e. $a = 12$, $b = 55$, $c = 6$, $p = 25$ and $q = 0.2$.

It is easy to see that the system (1) has three equilibrium points, viz.

$$\begin{aligned} E_0 = \left[ \begin{array}{r} 0 \\ 0 \\ 0 \\ \end{array} \right] , \ \ E_1 = \left[ \begin{array}{r} 1.9061 \\ 0.1772 \\ 55.0000 \\ \end{array} \right] , \ \ E_2 = \left[ \begin{array}{r} -1.9061 \\ -0.5056 \\ 55.0000 \\ \end{array} \right] \end{aligned}$$

(10)

The Jacobian of the system (1) at any point $\mathbf {x} \in {\mathbf R}^3$ is calculated as

$$\begin{aligned} J(\mathbf {x}) = \left[ \begin{array}{ccccccc} - a &{}&{}&{} a + x_3 &{}&{}&{} q + x_2 \\ b - x_3 &{}&{}&{} 0 &{}&{}&{} - x_1 \\ 4 p x_1^3 &{}&{}&{} 0 &{}&{}&{} - c\\ \end{array} \right] = \left[ \begin{array}{ccccccc} - 12 &{}&{}&{} 12 + x_3 &{}&{}&{} 0.2 + x_2 \\ 55 - x_3 &{}&{}&{} 0 &{}&{}&{} - x_1 \\ 100 x_1^3 &{}&{}&{} 0 &{}&{}&{} - 6\\ \end{array} \right] = \end{aligned}$$

(11)

The Jacobian of the system (1) at the equilibrium $E_0$ is obtained as

$$\begin{aligned} J_0 = J(E_0) = \left[ \begin{array}{ccccccc} - 12 &{}&{}&{} 12 &{}&{}&{} 0.2 \\ 55 &{}&{}&{} 0 &{}&{}&{} 0 \\ 0 &{}&{}&{} 0 &{}&{}&{} - 6\\ \end{array} \right] \end{aligned}$$

(12)

We find that the matrix $J_0 = J(E_0)$ has the eigenvalues

$$\begin{aligned} \lambda _1 = -6, \ \ \lambda _2 = -32.3818, \ \ \lambda _3 = 20.3818 \end{aligned}$$

(13)

This shows that the equilibrium point $E_0$ is a saddle-point, which is unstable.

The Jacobian of the system (1) at the equilibrium $E_1$ is obtained as

$$\begin{aligned} J_1 = J(E_1) = \left[ \begin{array}{ccccccc} - 12 &{}&{}&{} 67 &{}&{}&{} 0.3772 \\ 0 &{}&{}&{} 0 &{}&{}&{} -1.9061 \\ 692.5275 &{}&{}&{} 0 &{}&{}&{} - 6\\ \end{array} \right] \end{aligned}$$

(14)

We find that the matrix $J_1 = J(E_1)$ has the eigenvalues

$$\begin{aligned} \lambda _1 = -53.0246, \ \ \lambda _{2, 3} = 17.5123 \pm 36.8953 i \end{aligned}$$

(15)

This shows that the equilibrium point $E_1$ is a saddle-focus, which is unstable.

The Jacobian of the system (1) at the equilibrium $E_2$ is obtained as

$$\begin{aligned} J_2 = J(E_2) = \left[ \begin{array}{ccccccc} - 12 &{}&{}&{} 67 &{}&{}&{} -0.3056 \\ 0 &{}&{}&{} 0 &{}&{}&{} 1.9061 \\ -692.5275 &{}&{}&{} 0 &{}&{}&{} - 6\\ \end{array} \right] \end{aligned}$$

(16)

We find that the matrix $J_2 = J(E_2)$ has the eigenvalues

$$\begin{aligned} \lambda _1 = -52.6091, \ \ \lambda _{2, 3} = 17.3045 \pm 37.1708 i \end{aligned}$$

(17)

This shows that the equilibrium point $E_2$ is a saddle-focus, which is unstable.

3.3 Symmetry

It is easy to see that the system (1) is invariant under the change of coordinates

$$\begin{aligned} (x_1, x_2, x_3) \mapsto (- x_1, - x_2, x_3) \end{aligned}$$

(18)

Thus, it follows that the 3-D novel chaotic system (1) has rotation symmetry about the $x_3$-axis.

As a consequence, we conclude that any non-trivial trajectory $\left[ \begin{array}{c} x_1(t) \\ x_2(t) \\ x_3(t) \\ \end{array} \right]$ of the system (1) must have a twin trajectory $\left[ \begin{array}{r} -x_1(t) \\ -x_2(t) \\ x_3(t) \\ \end{array} \right]$ of the system (1).

3.4 Lyapunov Exponents and Kaplan-Yorke Dimension

We take the parameter values of the novel system (1) as in the chaotic case (2), i.e.

$$\begin{aligned} a = 12, \ \ b = 55, \ \ c = 6, \ \ p = 25, \ \ q = 0.2 \end{aligned}$$

(19)

We take the initial state of the novel system (1) as given in (3), i.e.

$$\begin{aligned} x_1(0) = 2.6, \ \ x_2(0) = 1.8, \ \ x_3(0) = 2.5 \end{aligned}$$

(20)

Then the Lyapunov exponents of the system (1) are numerically obtained as

$$\begin{aligned} L_1 = 8.6606, \ \ L_2 = 0, \ \ L_3 = -26.6523 \end{aligned}$$

(21)

Figure 5 shows the dynamics of the Lyapunov exponents of the novel system (1). From Fig. 5, we note that the Maximal Lyapunov Exponent (MLE) of the novel system (1) is given by $L_1 = 8.6606$, which is a large value. This shows that the novel system (1) is highly chaotic.

We also note that the sum of the Lyapunov exponents in (21) is negative, i.e.

$$\begin{aligned} L_1 + L_2 + L_3 = - 17.9917 < 0 \end{aligned}$$

(22)

This shows that the novel chaotic system (1) is dissipative.

Also, the Kaplan-Yorke dimension of the novel chaotic system (1) is found as

$$\begin{aligned} D_{KY} = 2 + {L_1 + L_2 \over |L_3|} = 2.3249, \end{aligned}$$

(23)

which is fractional.

Also, the relatively large value of the Kaplan-Yorke dimension of the novel 3-D chaotic system (1), i.e. $D_{KY} = 2.3249$, indicates that the novel 3-D chaotic system exhibits highly complex behaviour. Hence, the novel chaotic system (1) has applications in cryptosystems, secure communication devices, etc.

Fig. 5

Dynamics of the Lyapunov exponents of the novel chaotic system

4 Adaptive Control of the Novel 3-D Chaotic System

In this section, we use adaptive control method to derive an adaptive feedback control law for globally stabilizing the novel 3-D chaotic system with unknown parameters.

Thus, we consider the novel highly chaotic system given by

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{x}_1 &{} = &{} a (x_2 - x_1) + q x_3 + x_2 x_3 + u_1 \\ \dot{x}_2 &{} = &{} b x_1 - x_1 x_3 + u_2 \\ \dot{x}_3 &{} = &{} - c x_3 + p x_1^4 + u_3 \\ \end{array} \right. \end{aligned}$$

(24)

In (24), $x_1,$ $x_2$, $x_3$ are the states and $u_1,$ $u_2$, $u_3$ are the adaptive controls to be found using estimates for the unknown system parameters.

We consider the adaptive feedback control law

$$\begin{aligned} \left\{ \begin{array}{ccl} u_1 &{} = &{} - \hat{a}(t) (x_2 - x_1) - \hat{q}(t) x_3 - x_2 x_3 - k_1 x_1 \\ u_2 &{} = &{} - \hat{b}(t) x_1 + x_1 x_3 - k_2 x_2 \\ u_3 &{} = &{} \hat{c}(t) x_3 - \hat{p}(t) x_1^4 - k_3 x_3 \\ \end{array} \right. \end{aligned}$$

(25)

where $k_1, k_2, k_3$ are positive gain constants.

Substituting (25) into (24), we get the closed-loop plant dynamics as

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{x}_1 &{} = &{} \left[ a - \hat{a}(t) \right] (x_2 - x_1) + \left[ q - \hat{q}(t) \right] x_3 - k_1 x_1 \\ \dot{x}_2 &{} = &{} \left[ b - \hat{b}(t) \right] x_1 - k_2 x_2 \\ \dot{x}_3 &{} = &{} - \left[ c - \hat{c}(t) \right] x_3 + \left[ p - \hat{p}(t) \right] x_1^4 - k_3 x_3\\ \end{array} \right. \end{aligned}$$

(26)

The parameter estimation errors are defined as

$$\begin{aligned} \left\{ \begin{array}{ccl} e_a(t) &{} = &{} a - \hat{a}(t) \\ e_b(t) &{} = &{} b - \hat{b}(t) \\ e_c(t) &{} = &{} c - \hat{c}(t) \\ e_p(t) &{} = &{} p - \hat{p}(t) \\ e_q(t) &{} = &{} q - \hat{q}(t) \\ \end{array} \right. \end{aligned}$$

(27)

Using (27), we can simplify the plant dynamics (26) as

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{x}_1 &{} = &{} e_a (x_2 - x_1) + e_q x_3 - k_1 x_1 \\ \dot{x}_2 &{} = &{} e_b x_1 - k_2 x_2 \\ \dot{x}_3 &{} = &{} - e_c x_3 + e_p x_1^4 - k_3 x_3\\ \end{array} \right. \end{aligned}$$

(28)

Differentiating (27) with respect to t, we obtain

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{e}_a(t) &{} = &{} - \dot{\hat{a}}(t) \\ \dot{e}_b(t) &{} = &{} - \dot{\hat{b}}(t) \\ \dot{e}_c(t) &{} = &{} - \dot{\hat{c}}(t) \\ \dot{e}_p(t) &{} = &{} - \dot{\hat{p}}(t) \\ \dot{e}_q(t) &{} = &{} - \dot{\hat{q}}(t) \\ \end{array} \right. \end{aligned}$$

(29)

We consider the quadratic candidate Lyapunov function defined by

$$\begin{aligned} V(\mathbf {x}, e_a, e_b, e_c, e_p, e_q) = {1 \over 2} \left( x_1^2 + x_2^2 + x_3^2 \right) + {1 \over 2} \left( e_a^2 + e_b^2 + e_c^2 + e_p^2 + e_q^2 \right) \end{aligned}$$

(30)

Differentiating V along the trajectories of (28) and (29), we obtain

$$\begin{aligned} \left. \begin{array}{ccl} \dot{V} &{} = &{} - k_1 x_1^2 - k_2 x_2^2 - k_3 x_3^2 + e_a \left[ x_1 (x_2 - x_1) - \dot{\hat{a}} \right] + e_b \left[ x_1 x_2 - \dot{\hat{b}} \right] \\ &{} &{} + e_c \left[ - x_3^2 - \dot{\hat{c}} \right] + e_p \left[ x_1^4 x_3 - \dot{\hat{p}} \right] + e_q \left[ x_1 x_3 - \dot{\hat{q}} \right] \\ \end{array} \right. \end{aligned}$$

(31)

In view of (31), we take the parameter update law as

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{\hat{a}}(t) &{} = &{} x_1 (x_2 - x_1) \\ \dot{\hat{b}}(t) &{} = &{} x_1 x_2 \\ \dot{\hat{c}}(t) &{} = &{} - x_3^2 \\ \dot{\hat{p}}(t) &{} = &{} x_1^4 x_3 \\ \dot{\hat{q}}(t) &{} = &{} x_1 x_3 \\ \end{array} \right. \end{aligned}$$

(32)

Next, we state and prove the main result of this section.

Theorem 1

The novel 3-D highly chaotic system (24) with unknown system parameters is globally and exponentially stabilized for all initial conditions by the adaptive control law (25) and the parameter update law (32), where $k_1, k_2, k_3$ are positive gain constants.

Proof

We prove this result by applying Lyapunov stability theory [24].

We consider the quadratic Lyapunov function defined by (30), which is clearly a positive definite function on ${\mathbf R}^{8}$.

By substituting the parameter update law (32) into (31), we obtain the time-derivative of V as

$$\begin{aligned} \dot{V} = - k_1 x_1^2 - k_2 x_2^2 - k_3 x_3^2 \end{aligned}$$

(33)

From (33), it is clear that $\dot{V}$ is a negative semi-definite function on ${\mathbf R}^{8}$.

Thus, we can conclude that the state vector $\mathbf {x}(t)$ and the parameter estimation error are globally bounded i.e.

$$ \left[ \begin{array}{cccccccc} x_1(t)&x_2(t)&x_3(t)&e_a(t)&e_b(t)&e_c(t)&e_p(t)&e_q(t) \end{array} \right] ^T \in \mathbf {L}_\infty . $$

We define $k = \min \{ k_1, k_2, k_3 \}$.

Then it follows from (33) that

$$\begin{aligned} \dot{V} \le - k ||\mathbf {x}(t) ||^2 \end{aligned}$$

(34)

Thus, we have

$$\begin{aligned} k ||\mathbf {x}(t) ||^2 \le -\dot{V} \end{aligned}$$

(35)

Integrating the inequality (35) from 0 to t, we get

$$\begin{aligned} k \int \limits _0^t \ ||\mathbf {x}(\tau ) ||^2 \ d\tau \, \le \, V(0) - V(t) \end{aligned}$$

(36)

From (36), it follows that $\mathbf {x} \in \mathbf {L}_2$.

Using (28), we can conclude that ${{\dot{\mathbf{x}}}} \in \mathbf {L}_\infty$.

Using Barbalat’s lemma [24], we conclude that $\mathbf {x}(t) \rightarrow 0$ exponentially as $t \rightarrow \infty$ for all initial conditions $\mathbf {x}(0) \in {\mathbf R}^3$.

Hence, the novel highly chaotic system (24) with unknown system parameters is globally and exponentially stabilized for all initial conditions by the adaptive control law (25) and the parameter update law (32).

This completes the proof. $\blacksquare$

For the numerical simulations, the classical fourth-order Runge-Kutta method with step size $h = 10^{-8}$ is used to solve the systems (24) and (32), when the adaptive control law (25) is applied.

The parameter values of the novel 3-D chaotic system (24) are taken as in the chaotic case (2), i.e.

$$\begin{aligned} a = 12, \ \ b = 55, \ \ c = 6, \ \ p = 25, \ \ q = 0.2 \end{aligned}$$

(37)

We take the positive gain constants as $k_i = 8$ for $i = 1, 2, 3$.

Furthermore, as initial conditions of the novel highly chaotic system (24), we take

$$\begin{aligned} x_1(0) = 7.4, \ \ x_2(0) = -10.5, \ \ x_3(0) = 12.1 \end{aligned}$$

(38)

Also, as initial conditions of the parameter estimates, we take

$$\begin{aligned} \hat{a}(0) = 3.2, \ \ \hat{b}(0) = 12.3, \ \ \hat{c}(0) = 13.4, \ \ \hat{p}(0) = 17.8, \ \ \hat{q}(0) = 16.7 \end{aligned}$$

(39)

In Fig. 6, the exponential convergence of the controlled states of the novel 3-D chaotic system (24) is depicted. From Fig. 6, we see that the controlled states $x_1(t)$, $x_2(t)$, $x_3(t)$ converge to zero in just two seconds.

This shows the efficiency of the adaptive controller designed in this section for the novel 3-D chaotic system (24).

Fig. 6

Time-history of the controlled states $x_1, x_2, x_3$

5 Adaptive Synchronization of the Identical Novel 3-D Chaotic Systems

In this section, we apply adaptive control method to derive an adaptive feedback control law for globally synchronizing identical novel 3-D chaotic systems with unknown parameters. The main result is established using Lyapunov stability theory.

As the master system, we consider the novel 3-D chaotic system given by

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{x}_1 &{} = &{} a (x_2 - x_1) + q x_3 + x_2 x_3 \\ \dot{x}_2 &{} = &{} b x_1 - x_1 x_3 \\ \dot{x}_3 &{} = &{} - c x_3 + p x_1^4 \\ \end{array} \right. \end{aligned}$$

(40)

In (40), $x_1, x_2, x_3$ are the states and a, b, c, p, q are unknown system parameters.

As the slave system, we consider the novel 3-D chaotic system given by

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{y}_1 &{} = &{} a (y_2 - y_1) + q y_3 + y_2 y_3 + u_1 \\ \dot{y}_2 &{} = &{} b y_1 - y_1 y_3 + u_2 \\ \dot{y}_3 &{} = &{} - c y_3 + p y_1^4 + u_3 \\ \end{array} \right. \end{aligned}$$

(41)

In (41), $y_1, y_2, y_3$ are the states and $u_1, u_2, u_3$ are the adaptive controls to be determined using estimates of the unknown system parameters.

The synchronization error between the novel chaotic systems is defined by

$$\begin{aligned} \left\{ \begin{array}{ccl} e_1 &{} = &{} y_1 - x_1 \\ e_2 &{} = &{} y_2 - x_2 \\ e_3 &{} = &{} y_3 - x_3 \\ \end{array} \right. \end{aligned}$$

(42)

Then the error dynamics is obtained as

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{e}_1 &{} = &{} a (e_2 - e_1) + q e_3 + y_2 y_3 - x_2 x_3 + u_1 \\ \dot{e}_2 &{} = &{} b e_1 - y_1 y_3 + x_1 x_3 + u_2 \\ \dot{e}_3 &{} = &{} - c e_3 + p (y_1^4 - x_1^4) + u_3 \\ \end{array} \right. \end{aligned}$$

(43)

We consider the adaptive feedback control law

$$\begin{aligned} \left\{ \begin{array}{ccl} u_1 &{} = &{} - \hat{a}(t) (e_2 - e_1) - \hat{q}(t) e_3 - y_2 y_3 + x_2 x_3 - k_1 e_1 \\ u_2 &{} = &{} - \hat{b}(t) e_1 + y_1 y_3 - x_1 x_3 - k_2 e_2 \\ u_3 &{} = &{} \hat{c}(t) e_3 - \hat{p}(t) (y_1^4 - x_1^4) - k_3 e_3 \\ \end{array} \right. \end{aligned}$$

(44)

where $k_1, k_2, k_3$ are positive gain constants.

Substituting (44) into (43), we get the closed-loop error dynamics as

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{e}_1 &{} = &{} \left[ a - \hat{a}(t) \right] (e_2 - e_1) + \left[ q - \hat{q}(t) \right] e_3 - k_1 e_1 \\ \dot{e}_2 &{} = &{} \left[ b - \hat{b}(t) \right] e_1 - k_2 e_2 \\ \dot{e}_3 &{} = &{} - \left[ c - \hat{c}(t) \right] e_3 + \left[ p - \hat{p}(t) \right] (y_1^4 - x_1^4) - k_3 e_3 \\ \end{array} \right. \end{aligned}$$

(45)

The parameter estimation errors are defined as

$$\begin{aligned} \left\{ \begin{array}{ccl} e_a(t) &{} = &{} a - \hat{a}(t) \\ e_b(t) &{} = &{} b - \hat{b}(t) \\ e_c(t) &{} = &{} c - \hat{c}(t) \\ e_p(t) &{} = &{} p - \hat{p}(t) \\ e_q(t) &{} = &{} q - \hat{q}(t) \\ \end{array} \right. \end{aligned}$$

(46)

In view of (46), we can simplify the error dynamics (45) as

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{e}_1 &{} = &{} e_a (e_2 - e_1) + e_q e_3 - k_1 e_1 \\ \dot{e}_2 &{} = &{} e_b e_1 - k_2 e_2 \\ \dot{e}_3 &{} = &{} - e_c e_3 + e_p (y_1^4 - x_1^4) - k_3 e_3 \\ \end{array} \right. \end{aligned}$$

(47)

Differentiating (46) with respect to t, we obtain

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{e}_a(t) &{} = &{} - \dot{\hat{a}}(t) \\ \dot{e}_b(t) &{} = &{} - \dot{\hat{b}}(t) \\ \dot{e}_c(t) &{} = &{} - \dot{\hat{c}}(t) \\ \dot{e}_p(t) &{} = &{} - \dot{\hat{p}}(t) \\ \dot{e}_q(t) &{} = &{} - \dot{\hat{q}}(t) \\ \end{array} \right. \end{aligned}$$

(48)

We consider the quadratic candidate Lyapunov function defined by

$$\begin{aligned} V(\mathbf {e}, e_a, e_b, e_c, e_p, e_q) = {1 \over 2} \left( e_1^2 + e_2^2 + e_3^2 \right) + {1 \over 2} \left( e_a^2 + e_b^2 + e_c^2 + e_p^2 + e_q^2 \right) \end{aligned}$$

(49)

Differentiating V along the trajectories of (47) and (48), we obtain

$$\begin{aligned} \left. \begin{array}{ccl} \dot{V} &{} = &{} - k_1 e_1^2 - k_2 e_2^2 - k_3 e_3^2 + e_a \left[ e_1 (e_2 - e_1) - \dot{\hat{a}} \right] + e_b \left[ e_1 e_2 - \dot{\hat{b}} \right] \\ &{}&{} + e_c \left[ - e_3^2 - \dot{\hat{c}} \right] + e_p \left[ e_3 (y_1^4 - x_1^4) - \dot{\hat{p}} \right] + e_q \left[ e_1 e_3 - \dot{\hat{q}} \right] \\ \end{array} \right. \end{aligned}$$

(50)

In view of (50), we take the parameter update law as

$$\begin{aligned} \left\{ \begin{array}{ccl} \dot{\hat{a}}(t) &{} = &{} e_1 (e_2 - e_1) \\ \dot{\hat{b}}(t) &{} = &{} e_1 e_2 \\ \dot{\hat{c}}(t) &{} = &{} - e_3^2 \\ \dot{\hat{p}}(t) &{} = &{} e_3 (y_1^4 - x_1^4) \\ \dot{\hat{q}}(t) &{} = &{} e_1 e_3 \\ \end{array} \right. \end{aligned}$$

(51)

Next, we state and prove the main result of this section.

Theorem 2

The novel 3-D chaotic systems (40) and (41) with unknown system parameters are globally and exponentially synchronized for all initial conditions by the adaptive control law (44) and the parameter update law (51), where $k_1, k_2, k_3$ are positive gain constants.

Proof

We prove this result by applying Lyapunov stability theory [24].

We consider the quadratic Lyapunov function defined by (49), which is clearly a positive definite function on ${\mathbf R}^{8}$.

By substituting the parameter update law (51) into (50), we obtain

$$\begin{aligned} \dot{V} = - k_1 e_1^2 - k_2 e_2^2 - k_3 e_3^2 \end{aligned}$$

(52)

From (52), it is clear that $\dot{V}$ is a negative semi-definite function on ${\mathbf R}^{8}$.

Thus, we can conclude that the error vector $\mathbf {e}(t)$ and the parameter estimation error are globally bounded, i.e.

$$\begin{aligned} \left[ \begin{array}{cccccccc} e_1(t)&e_2(t)&e_3(t)&e_a(t)&e_b(t)&e_c(t)&e_p(t)&e_q(t) \end{array} \right] ^T \in \mathbf {L}_\infty . \end{aligned}$$

(53)

We define $k = \min \{ k_1, k_2, k_3 \}$.

Then it follows from (52) that

$$\begin{aligned} \dot{V} \le - k ||\mathbf {e}(t) ||^2 \end{aligned}$$

(54)

Thus, we have

$$\begin{aligned} k ||\mathbf {e}(t) ||^2 \le -\dot{V} \end{aligned}$$

(55)

Integrating the inequality (55) from 0 to t, we get

$$\begin{aligned} k \int \limits _0^t \ ||\mathbf {e}(\tau ) ||^2 \ d\tau \, \le \, V(0) - V(t) \end{aligned}$$

(56)

From (56), it follows that $\mathbf {e} \in \mathbf {L}_2$.

Using (47), we can conclude that ${\dot{\mathbf {e}}} \in \mathbf {L}_\infty$.

Using Barbalat’s lemma [24], we conclude that $\mathbf {e}(t) \rightarrow 0$ exponentially as $t \rightarrow \infty$ for all initial conditions $\mathbf {e}(0) \in {\mathbf R}^3$.

This completes the proof. $\blacksquare$

For the numerical simulations, the classical fourth-order Runge-Kutta method with step size $h = 10^{-8}$ is used to solve the systems (40), (41) and (51), when the adaptive control law (44) is applied.

The parameter values of the novel chaotic systems are taken as in the chaotic case (2), i.e. $a = 12$, $b = 55$, $c = 6$, $p = 25$ and $q = 0.2$.

We take the positive gain constants as $k_i = 8$ for $i = 1, 2, 3$.

Furthermore, as initial conditions of the master system (40), we take

$$\begin{aligned} x_1(0) = 12.5, \ \ x_2(0) = 20.7, \ \ x_3(0) = -5.3 \end{aligned}$$

(57)

As initial conditions of the slave system (41), we take

$$\begin{aligned} y_1(0) = 6.8, \ \ y_2(0) = 4.5, \ \ y_3(0) = 11.4 \end{aligned}$$

(58)

Also, as initial conditions of the parameter estimates, we take

$$\begin{aligned} \hat{a}(0) = 3.1, \ \ \hat{b}(0) = 4.3, \ \ \hat{c}(0) = 10.2, \ \ \hat{p}(0) = 5.9, \ \ \hat{q}(0) = 7.5 \end{aligned}$$

(59)

Figures 7, 8 and 9 describe the complete synchronization of the novel 3-D chaotic systems (40) and (41), while Fig. 10 describes the time-history of the complete synchronization errors $e_1, e_2, e_3$.

From Fig. 7, we see that the states $x_1$ and $y_1$ are synchronized in just two seconds. From Fig. 8, we see that the states $x_2$ and $y_2$ are synchronized in just two seconds. From Fig. 9, we see that the states $x_3$ and $y_3$ are synchronized in just two seconds. From Fig. 10, we see that the errors $e_1, e_2, e_3$ converge to zero in just two seconds. This shows the efficiency of the adaptive controller developed in this section for the synchronization of identical 3-D chaotic systems.

Fig. 7

Synchronization of the states $x_1$ and $y_1$

Fig. 8

Synchronization of the states $x_2$ and $y_2$

Fig. 9

Synchronization of the states $x_3$ and $y_3$

Fig. 10

Time-history of the synchronization errors $e_1, e_2, e_3$

6 Conclusions

In this work, we announced a seven-term novel 3-D chaotic system with a quartic nonlinearity and two quadratic nonlinearities. The dynamic properties of the novel 3-D chaotic system were discussed and phase portraits of the novel chaotic system were depicted. The novel 3-D chaotic system has three unstable equilibrium points. We showed that the equilibrium point at the origin is a saddle point, while the other two equilibrium points are saddle foci. The novel 3-D chaotic system has rotation symmetry about the $x_3$ axis, which shows that every non-trivial trajectory of the system must have a twin trajectory. The Lyapunov exponents of the novel 3-D chaotic system have been obtained as $L_1 = 8.6606$, $L_2 = 0$ and $L_3 = -26.6523$, while the Kaplan-Yorke dimension of the novel chaotic system has been obtained as $D_{KY} = 2.3249$. Since the Maximal Lyapunov Exponent (MLE) of the novel chaotic system has a large value, viz. $L_1 = 8.6606$, it follows that the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we derived new results for the global chaos control of the novel chaotic system with unknown parameters via adaptive control method. We also derived new results for the global chaos synchronization of the identical novel chaotic systems with unknown parameters via adaptive control method. The main adaptive control results were proved using Lyapunov stability theory. MATLAB simulations have been shown to illustrate all the main results developed in this work.