Keywords

1 Introduction

Chaotic systems are defined as nonlinear dynamical systems which are sensitive to initial conditions, topologically mixing and with dense periodic orbits [10, 11, 13].

Sensitivity to initial conditions of chaotic systems is popularly known as the butterfly effect. Small changes in an initial state will make a very large difference in the behavior of the system at future states.

Poincaré [15] suspected chaotic behaviour in the study of three bodies problem at the end of the 19th century, but chaos was experimentally established by Lorenz [41] only a few decades ago in the study of 3-D weather models.

The Lyapunov exponent is a measure of the divergence of phase points that are initially very close and can be used to quantify chaotic systems. It is common to refer to the largest Lyapunov exponent as the Maximal Lyapunov Exponent (MLE). A positive maximal Lyapunov exponent and phase space compactness are usually taken as defining conditions for a chaotic system.

In the last five decades, there is significant interest in the literature in discovering new chaotic systems [73]. Some popular chaotic systems are Lorenz system [41], Rössler system [63], Arneodo system [2], Henon-Heiles system [27], Genesio-Tesi system [25], Sprott systems [72], Chen system [19], Lü system [42], Rikitake dynamo system [62], Liu system [40], Shimizu system [71], etc.

In the recent years, many new chaotic systems have been found such as Pandey system [46], Qi system [54], Li system [35], Wei-Yang system [171], Zhou system [178], Zhu system [179], Sundarapandian systems [76, 81], Dadras system [21], Tacha system [84], Vaidyanathan systems [92, 93, 95,96,97,98, 101, 112, 126,127,128,129,130,131,132,133, 135,136,137, 139, 148, 150, 159, 161, 163, 165,166,167], Vaidyanathan-Azar systems [142, 143, 145,146,147], Pehlivan system [48], Sampath system [64], Akgul system [1], Pham system [49, 51,52,53], etc.

Chaos theory and control systems have many important applications in science and engineering [3, 10,11,12, 14, 180]. Some commonly known applications are oscillators [115, 119, 121,122,123,124, 134], lasers [37, 174], chemical reactions [102, 103, 107,108,109, 111, 113, 114, 117, 118, 120], biology [22, 33, 100, 104,105,106, 110, 116], ecology [26, 74], encryption [34, 177], cryptosystems [61, 85], mechanical systems [5,6,7,8,9], secure communications [23, 44, 175], robotics [43, 45, 169], cardiology [55, 172], intelligent control [4, 38], neural networks [28, 31, 39], memristors [50, 170], etc.

Synchronization of chaotic systems is a phenomenon that occurs when two or more chaotic systems are coupled or when a chaotic system drives another chaotic system. Because of the butterfly effect which causes exponential divergence of the trajectories of two identical chaotic systems started with nearly the same initial conditions, the synchronization of chaotic systems is a challenging research problem in the chaos literature.

Major works on synchronization of chaotic systems deal with the complete synchronization of a pair of chaotic systems called the master and slave systems. The design goal of the complete synchronization is to apply 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. Active feedback control is used when the system parameters are available for measurement. Adaptive feedback control is used when the system parameters are unknown.

Pecora and Carroll pioneered the research on synchronization of chaotic systems with their seminal papers [18, 47]. The active control method [30, 65, 66, 75, 80, 86, 90, 151, 152, 155] is typically used when the system parameters are available for measurement.

Adaptive control method [67,68,69, 77,78,79, 88, 94, 125, 138, 144, 149, 153, 154, 160, 164, 168] is typically used when some or all the system parameters are not available for measurement and estimates for the uncertain parameters of the systems. Adaptive control method has more relevant for many practical situations for systems with unknown parameters. In the literature, adaptive control method is preferred over active control method due to the wide applicability of the adaptive control method.

Intelligent control methods like fuzzy control method [16, 17] are also used for the synchronization of chaotic systems. Intelligent control methods have advantages like robustness, insensitive to small variations in the parameters, etc.

Sampled-data feedback control method [24, 36, 173, 176] and time-delay feedback control method [20, 29, 70] are also used for synchronization of chaotic systems. Backstepping control method [56,57,58,59,60, 83, 156, 162] is also used for the synchronization of chaotic systems, which is a recursive method for stabilizing the origin of a control system in strict-feedback form.

Another popular method for the synchronization of chaotic systems is the sliding mode control method [82, 87, 89, 91, 99, 140, 141, 157, 158], which is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to “slide” along a cross-section of the system’s normal behavior.

In this research work, we describe an eight-term 3-D novel polynomial chaotic system with three quadratic nonlinearities. Section 2 describes the 3-D dynamical model and phase portraits of the novel chaotic system.

Section 3 describes the dynamic analysis of the novel chaotic system. We show that the novel chaotic system has four equilibrium points of which two equilibrium points are saddle points and the other two equilibrium points are saddle-foci.

The Lyapunov exponents of the eight-term novel chaotic system are obtained as \(L_1 = 0.4715\), \(L_2 = 0\) and \(L_3 = -2.4728\). Since the sum of the Lyapunov exponents of the novel chaotic system is negative, this chaotic system is dissipative. Also, the Lyapunov dimension of the novel chaotic system is obtained as \(D_{L} = 2.1907\).

Section 4 describes the adaptive feedback control of the novel chaotic system with unknown parameters. Section 5 describes the adaptive feedback synchronization of the identical novel chaotic systems with unknown parameters. The adaptive feedback control and synchronization results are proved using Lyapunov stability theory [32].

MATLAB simulations are depicted to illustrate all the main results for the 3-D novel chaotic system. Section 6 concludes this work with a summary of the main results.

2 A Novel 3-D Chaotic System

In this research work, we announce an eight-term 3-D chaotic system described by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} - a x_2 + x_2 x_3 \\ \dot{x}_2 &{} = &{} p x_1 + b x_2 - x_1 x_3 \\ \dot{x}_3 &{} = &{} x_1 - c x_3 + x_1 x_2 \\ \end{array} \right. \end{aligned}$$
(1)

where \(x_1, x_2, x_3\) are the states and a, b, c, p are constant, positive parameters.

The 3-D system (1) is chaotic when the parameter values are taken as

$$\begin{aligned} a = 2.2, \ \ b = 3, \ \ c = 5, \ \ p = 0.1 \end{aligned}$$
(2)

For numerical simulations, we take the initial state of the chaotic system (1) as

$$\begin{aligned} x_1(0) = 0.2, \ \ \ x_2(0) = 0.2, \ \ \ x_3(0) = 0.2 \end{aligned}$$
(3)

The Lyapunov exponents of the novel chaotic system (1) for the parameter values (2) and the initial values (3) are numerically determined as

$$\begin{aligned} L_1 = 0.4715, \ \ L_2 = 0, \ \ L_3 = -2.4728 \end{aligned}$$
(4)

The Lyapunov dimension of the novel chaotic system (1) is calculated as

$$\begin{aligned} D_L = 2 + {L_1 + L_2 \over | L_3 |} = 2.1907, \end{aligned}$$
(5)

which is fractional.

The presence of a positive Lyapunov exponent in (4) shows that the 3-D novel system (1) is chaotic (Fig. 1).

The novel 3-D chaotic system (1) exhibits a strange chaotic attractor. It is interesting to note that the strange chaotic attractor looks like a trumpet. Hence, the novel chaotic system (1) can be also called as a trumpet attractor.

Figure 2 describes the 2-D projection of the strange chaotic attractor of the novel chaotic system (1) on \((x_1, x_2)\)-plane.

Figure 3 describes the 2-D projection of the strange chaotic attractor of the novel chaotic system (1) on \((x_2, x_3)\)-plane.

Figure 4 describes the 2-D projection of the strange chaotic attractor of the novel chaotic system (1) on \((x_1, x_3)\)-plane.

Fig. 1
figure 1

Strange attractor of the novel chaotic system in

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Fig. 2
figure 2

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

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Fig. 3
figure 3

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

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Fig. 4
figure 4

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

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3 Analysis of the 3-D Novel Chaotic System

This section gives the qualitative properties of the novel chaotic system (1).

3.1 Dissipativity

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

$$\begin{aligned} \dot{x} = f(x) = \left[ \begin{array}{c} f_1(x) \\ f_2(x) \\ f_3(x) \\ \end{array} \right] , \end{aligned}$$
(6)

where

$$\begin{aligned} \left. \begin{array}{ccl} f_1(x) &{} = &{} - a x_2 + x_2 x_3 \\ f_2(x) &{} = &{} p x_1 + b x_2 - x_1 x_3 \\ f_3(x) &{} = &{} x_1 - c x_3 + x_1 x_2 \\ \end{array} \right. \end{aligned}$$
(7)

We take the parameter values as

$$\begin{aligned} a = 2.2, \ \ b = 3, \ \ c = 5, \ \ p = 0.1 \end{aligned}$$
(8)

The divergence of the vector field f on is obtained as

$$\begin{aligned} \text{ div } \ f = {\partial f_1(x) \over \partial x_1} + {\partial f_2(x) \over \partial x_2} + {\partial f_3(x) \over \partial x_3} = - (c - b) = -\mu \end{aligned}$$
(9)

where

$$\begin{aligned} \mu = c - b = 2 > 0 \end{aligned}$$
(10)

Let \(\varOmega \) be any region in with a smooth boundary. Let \(\varOmega (t) = \varPhi _t(\varOmega )\), where \(\varPhi _t\) is the flow of the vector field f. Let V(t) denote the volume of \(\varOmega (t)\).

By Liouville’s theorem, it follows that

$$\begin{aligned} {dV(t) \over dt} = \int \limits _{\varOmega (t)} \ (\text{ div } f) dx_1 \, dx_2 \, dx_3 \end{aligned}$$
(11)

Substituting the value of \(\text{ div } f\) in (11) leads to

$$\begin{aligned} {dV(t) \over dt} = - \mu \int \limits _{\varOmega (t)} \ dx_1 \, dx_2 \, dx_3 = - \mu V(t) \end{aligned}$$
(12)

Integrating the linear differential equation (12), V(t) is obtained as

$$\begin{aligned} V(t) = V(0) \exp (-\mu t) \end{aligned}$$
(13)

From Eq. (13), it follows that the volume V(t) shrinks to zero exponentially as \(t \rightarrow \infty \).

Thus, the novel chaotic system (1) is dissipative. Hence, any asymptotic motion of the system (1) settles onto a set of measure zero, i.e. a strange attractor.

3.2 Invariance

It is easily seen that the \(x_3\)-axis is invariant for the flow of the novel chaotic system (1). The invariant motion along the \(x_3\)-axis is characterized by the scalar dynamics

$$\begin{aligned} \dot{x}_3 = - c x_3, \ \ (c > 0) \end{aligned}$$
(14)

which is globally exponentially stable.

3.3 Equilibria

The equilibrium points of the novel chaotic system (1) are obtained by solving the nonlinear equations

$$\begin{aligned} \left. \begin{array}{cclcl} f_1(x) &{} = &{} - a x_2 + x_2 x_3 &{} = &{} 0 \\ f_2(x) &{} = &{} p x_1 + b x_2 - x_1 x_3 &{} = &{} 0 \\ f_3(x) &{} = &{} x_1 - c x_3 + x_1 x_2 &{} = &{} 0 \\ \end{array} \right. \end{aligned}$$
(15)

We take the parameter values as in the chaotic case, viz.

$$\begin{aligned} a = 2.2, \ \ b = 3, \ \ c = 5, \ \ p = 0.1 \end{aligned}$$
(16)

Solving the nonlinear system (15) with the parameter values (16), we obtain four equilibrium points of the novel chaotic system (1), viz.

$$\begin{aligned} E_0 = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right] , \ \ E_1 = \left[ \begin{array}{r} 0.5 \\ 0 \\ 0.1 \\ \end{array} \right] , E_2 = \left[ \begin{array}{r} -4.7422 \\ -3.3196 \\ 2.2000 \end{array} \right] , \ \ E_3 = \left[ \begin{array}{r} 3.3137 \\ 2.3196 \\ 2.2000 \\ \end{array} \right] \end{aligned}$$
(17)

The Jacobian matrix of the novel chaotic system (1) at \((x_1^\star , x_2^\star , x_3^\star )\) is obtained as

$$\begin{aligned} J(x^\star ) = \left[ \begin{array}{ccc} 0 &{} -a + x_3^\star &{} x_2^\star \\ p - x_3^\star &{} b &{} - x_1^\star \\ 1 + x_2^\star &{} x_1^\star &{} - c \\ \end{array} \right] \end{aligned}$$
(18)

The matrix \(J_0 = J(E_0)\) has the eigenvalues

$$\begin{aligned} \lambda _1 = -5, \ \ \lambda _2 = 0.0752, \ \ \lambda _3 = -5 \end{aligned}$$
(19)

This shows that the equilibrium point \(E_0\) is a saddle-point, which is unstable.

The matrix \(J_1 = J(E_1)\) has the eigenvalues

$$\begin{aligned} \lambda _1 = -0.0705, \ \ \lambda _2 = -4.9418, \ \ \lambda _3 = 3.0123 \end{aligned}$$
(20)

This shows that the equilibrium point \(E_1\) is a saddle point, which is unstable.

The matrix \(J_2 = J(E_2)\) has the eigenvalues

$$\begin{aligned} \lambda _1 = -4.6466, \ \ \lambda _{2, 3} = 1.3233 \pm 3.2148 i \end{aligned}$$
(21)

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

The matrix \(J_3 = J(E_3)\) has the eigenvalues

$$\begin{aligned} \lambda _1 = -5.4615, \ \ \lambda _{2, 3} = 1.7307 \pm 2.0469 i \end{aligned}$$
(22)

This shows that the equilibrium point \(E_3\) is a saddle-focus, which is unstable.

Hence, \(E_0, E_1, E_2, E_3\) are all unstable equilibrium points of the 3-D novel chaotic system (1), where \(E_0, E_1\) are saddle points and \(E_3, E_4\) are saddle-foci.

3.4 Lyapunov Exponents and Lyapunov Dimension

We take the initial values of the novel chaotic system (1) as in (3) and the parameter values of the novel chaotic system (1) as in (2).

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

$$\begin{aligned} L_1 = 0.4715, \ \ \ L_2 = 0, \ \ \ L_3 = -2.4728 \end{aligned}$$
(23)

Since \(L_1 + L_2 + L_3 = - 2.0013 < 0\), the system (1) is dissipative.

Also, the Lyapunov dimension of the system (1) is obtained as

$$\begin{aligned} D_{L} = 2 + {L_1 + L_2 \over | L_3 |} = 2.1907, \end{aligned}$$
(24)

which is fractional.

Figure 5 depicts the Lyapunov exponents of the novel chaotic system (1). From this figure, it is seen that the Maximal Lyapunov Exponent (MLE) of the novel chaotic system (1) is \(L_1 = 0.4715\).

Fig. 5
figure 5

Lyapunov exponents of the novel chaotic system

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4 Adaptive Feedback Control of the 3-D Novel Chaotic System

This section derives new results for adaptive feedback controller design in order to stabilize the unstable novel chaotic system with unknown parameters for all initial conditions.

The controlled novel 3-D chaotic system is given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} - a x_2 + x_2 x_3 + u_1 \\ \dot{x}_2 &{} = &{} p x_1 + b x_2 - x_1 x_3 + u_2 \\ \dot{x}_3 &{} = &{} x_1 - c x_3 + x_1 x_2 + u_3 \\ \end{array} \right. \end{aligned}$$
(25)

where \(x_1, x_2, x_3\) are state variables, a, b, c, p are constant, unknown, parameters of the system and \(u_1, u_2, u_3\) are adaptive feedback controls to be designed.

An adaptive feedback control law is taken as

$$\begin{aligned} \left. \begin{array}{ccl} u_1 &{} = &{} \hat{a}(t) x_2 - x_2 x_3 - k_1 x_1 \\ u_2 &{} = &{} - \hat{p}(t) x_1 - \hat{b}(t) x_2 + x_1 x_3 - k_2 x_2 \\ u_3 &{} = &{} - x_1 + \hat{c}(t) x_3 - x_1 x_2 - k_3 x_3 \\ \end{array} \right. \end{aligned}$$
(26)

In (26), \(\hat{a}(t), \hat{b}(t), \hat{c}(t), \hat{p}(t)\) are estimates for the unknown parameters a, b, c, p, respectively, and \(k_1, k_2, k_3\) are positive gain constants.

The closed-loop control system is obtained by substituting (26) into (25) as

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} - [ a - \hat{a}(t) ] x_2 - k_1 x_1 \\ \dot{x}_2 &{} = &{} [ p - \hat{p}(t) ] x_1 + [ b - \hat{b}(t) ] x_2 - k_2 x_2 \\ \dot{x}_3 &{} = &{} - [ c - \hat{c}(t) ] x_3 - k_3 x_3 \\ \end{array} \right. \end{aligned}$$
(27)

To simplify (27), we define the parameter estimation error as

$$\begin{aligned} \left. \begin{array}{ccl} e_a(t) &{} = &{} a - \hat{a}(t) \\ e_b(t) &{} = &{} b - \hat{b}(t) \\ e_c(t) &{} = &{} c - \hat{c}(t) \\ e_p(t) &{} = &{} d - \hat{p}(t) \\ \end{array} \right. \end{aligned}$$
(28)

Using (28), the closed-loop system (27) can be simplified as

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} - e_a x_2 - k_1 x_1 \\ \dot{x}_2 &{} = &{} e_p x_1 + e_b x_2 - k_2 x_2 \\ \dot{x}_3 &{} = &{} - e_c x_3 - k_3 x_3 \\ \end{array} \right. \end{aligned}$$
(29)

Differentiating the parameter estimation error (28) with respect to t, we get

$$\begin{aligned} \left. \begin{array}{ccl} \dot{e}_a &{} = &{} - \dot{\hat{a}} \\ \dot{e}_b &{} = &{} - \dot{\hat{b}} \\ \dot{e}_c &{} = &{} - \dot{\hat{c}} \\ \dot{e}_p &{} = &{} - \dot{\hat{p}} \\ \end{array} \right. \end{aligned}$$
(30)

Next, we find an update law for parameter estimates using Lyapunov stability theory.

Consider the quadratic Lyapunov function defined by

$$\begin{aligned} V(x_1, x_2, x_3, e_a, e_b, e_c, e_p) = {1 \over 2} \, \left( x_1^2 + x_2^2 + x_3^2 + e_a^2 + e_b^2 + e_c^2 + e_p^2 \right) , \end{aligned}$$
(31)

which is positive definite on .

Differentiating V along the trajectories of (29) and (30), we get

$$\begin{aligned} \left. \begin{array}{ccl} \dot{V} &{} = &{} - k_1 x_1^2 - k_2 x_2^2 - k_3 x_3^2 + e_a [ - x_1 x_2 - \dot{\hat{a}} ] + e_b [ x_2^2 - \dot{\hat{b}} ] + e_c [ - x_3^2 - \dot{\hat{c}} ] \\ &{} &{} + e_p [ x_1 x_2 - \dot{\hat{p}} ] \end{array} \right. \end{aligned}$$
(32)

In view of Eq. (32), an update law for the parameter estimates is taken as

$$\begin{aligned} \left. \begin{array}{ccl} \dot{\hat{a}} &{} = &{} - x_1 x_2 \\ \dot{\hat{b}} &{} = &{} x_2^2 \\ \dot{\hat{c}} &{} = &{} - x_3^2 \\ \dot{\hat{p}} &{} = &{} x_1 x_2 \\ \end{array} \right. \end{aligned}$$
(33)

Theorem 1

The novel chaotic system (25) with unknown system parameters is globally and exponentially stabilized for all initial conditions by the adaptive control law (26) and the parameter update law (33), where \(k_i, (i = 1, 2, 3)\) are positive constants.

Proof

The result is proved using Lyapunov stability theory [32]. We consider the quadratic Lyapunov function V defined by (31), which is positive definite on .

Substitution of the parameter update law (33) into (32) yields

$$\begin{aligned} \dot{V} = - k_1 x_1^2 - k_2 x_2^2 - k_3 x_3^2, \end{aligned}$$
(34)

which is a negative semi-definite function on .

Therefore, it can be concluded that the state vector x(t) and the parameter estimation error are globally bounded, i.e.

$$\begin{aligned} \left[ \begin{array}{ccccccc} x_1(t)&x_2(t)&x_3(t)&e_a(t)&e_b(t)&e_c(t)&e_p(t) \end{array} \right] ^T \in \mathbf{L}_\infty . \end{aligned}$$
(35)

Define

$$\begin{aligned} k = \min \left\{ k_1, k_2, k_3 \right\} \end{aligned}$$
(36)

Then it follows from (34) that

$$\begin{aligned} \dot{V} \le - k \Vert \mathbf{x } \Vert ^2 \ \ {\text {or}} \ \ k \Vert \mathbf{x } \Vert ^2 \le - \dot{V} \end{aligned}$$
(37)

Integrating the inequality (37) from 0 to t, we get

$$\begin{aligned} k \int \limits _0^t \ \Vert \mathbf {x}(\tau ) \Vert ^2 \, d\tau \, \le \, - \int \limits _0^t \ \dot{V}(\tau ) \, d\tau = V(0) - V(t) \end{aligned}$$
(38)

From (38), it follows that \(\mathbf {x}(t) \in \mathbf{L}_2\).

Using (29), it can be deduced that \(\dot{x}(t) \in \mathbf{L}_\infty \).

Hence, using Barbalat’s lemma, we can conclude that \(\mathbf {x}(t) \rightarrow 0\) exponentially as \(t \rightarrow \infty \) for all initial conditions .

This completes the proof. \(\qquad \square \)

For numerical simulations, the parameter values of the novel system (25) are taken as in the chaotic case, viz.

$$\begin{aligned} a = 2.2, \ \ b = 3, \ \ c = 5, \ \ p = 0.1 \end{aligned}$$
(39)

The gain constants are taken as \(k_i = 6, (i = 1, 2, 3)\).

The initial values of the parameter estimates are taken as

$$\begin{aligned} \hat{a}(0) = 5.4, \ \ \hat{b}(0) = 12.7, \ \ \hat{c}(0) = 21.3, \ \ \hat{p}(0) = 16.2 \end{aligned}$$
(40)

The initial values of the novel system (25) are taken as

$$\begin{aligned} x_1(0) = 18.3, \ \ x_2(0) = 11.6, \ \ x_3(0) = 7.9 \end{aligned}$$
(41)

Figure 6 shows the time-history of the controlled states \(x_1(t), x_2(t), x_3(t)\).

Figure 6 depicts the exponential convergence of the controlled states and the efficiency of the adaptive controller defined by (26).

Fig. 6
figure 6

Time-history of the states \(x_1(t), x_2(t), x_3(t)\)

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

This section derives new results for the adaptive synchronization of the identical novel chaotic systems with unknown parameters.

The master system is given by the novel chaotic system

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} - a x_2 + x_2 x_3 \\ \dot{x}_2 &{} = &{} p x_1 + b x_2 - x_1 x_3 \\ \dot{x}_3 &{} = &{} x_1 - c x_3 + x_1 x_2 \\ \end{array} \right. \end{aligned}$$
(42)

where \(x_1, x_2, x_3\) are state variables and a, b, c, p are constant, unknown, parameters of the system.

The slave system is given by the controlled novel chaotic system

$$\begin{aligned} \left. \begin{array}{ccl} \dot{y}_1 &{} = &{} - a y_2 + y_2 y_3 + u_1 \\ \dot{y}_2 &{} = &{} p y_1 + b y_2 - y_1 y_3 + u_2 \\ \dot{y}_3 &{} = &{} y_1 - c y_3 + y_1 y_2 + u_3 \\ \end{array} \right. \end{aligned}$$
(43)

where \(y_1, y_2, y_3\) are state variables and \(u_1, u_2, u_3\) are adaptive controls to be designed.

The synchronization error is defined as

$$\begin{aligned} \left. \begin{array}{ccl} e_1 &{} = &{} y_1 - x_1 \\ e_2 &{}= &{} y_2 - x_2 \\ e_3 &{} = &{} y_3 - x_3 \\ \end{array} \right. \end{aligned}$$
(44)

The error dynamics is easily obtained as

$$\begin{aligned} \left. \begin{array}{ccl} \dot{e}_1 &{}= &{} - a e_2 + y_2 y_3 - x_2 x_3 + u_1 \\ \dot{e}_2 &{}= &{} p e_1 + b e_2 - y_1 y_3 + x_1 x_3 + u_2 \\ \dot{e}_3 &{}= &{} e_1 - c e_3 + y_1 y_2 - x_1 x_2 + u_3 \\ \end{array} \right. \end{aligned}$$
(45)

An adaptive control law is taken as

$$\begin{aligned} \left. \begin{array}{ccl} u_1 &{}= &{} \hat{a}(t) e_2 - y_2 y_3 + x_2 x_3 - k_1 e_1 \\ u_2 &{}= &{} - \hat{p}(t) e_1 - \hat{b}(t) e_2 + y_1 y_3 - x_1 x_3 - k_2 e_2 \\ u_3 &{}= &{} - e_1 + \hat{c}(t) e_3 - y_1 y_2 + x_1 x_2 - k_3 e_3 \\ \end{array} \right. \end{aligned}$$
(46)

where \(\hat{a}(t), \hat{b}(t), \hat{c}(t), \hat{p}(t)\) are estimates for the unknown parameters a, b, c, p, respectively, and \(k_1, k_2, k_3\) are positive gain constants.

The closed-loop control system is obtained by substituting (46) into (45) as

$$\begin{aligned} \left. \begin{array}{ccl} \dot{e}_1 &{}= &{} - [ a - \hat{a}(t) ] e_2 - k_1 e_1 \\ \dot{e}_2 &{}= &{} [ p - \hat{p}(t) ] e_1 + [ b - \hat{b}(t) ] e_2 - k_2 e_2 \\ \dot{e}_3 &{}= &{} - [ c - \hat{c}(t) ] e_3 - k_3 e_3 \\ \end{array} \right. \end{aligned}$$
(47)

To simplify (47), we define the parameter estimation error as

$$\begin{aligned} \left. \begin{array}{ccl} e_a(t) &{} = &{} a - \hat{a}(t) \\ e_b(t) &{} = &{} b - \hat{b}(t) \\ e_c(t) &{} = &{} c - \hat{c}(t) \\ e_p(t) &{} = &{} p - \hat{p}(t) \\ \end{array} \right. \end{aligned}$$
(48)

Using (48), the closed-loop system (47) can be simplified as

$$\begin{aligned} \left. \begin{array}{ccl} \dot{e}_1 &{}= &{} - e_a e_2 - k_1 e_1 \\ \dot{e}_2 &{}= &{} e_p e_1 + e_b e_2 - k_2 e_2 \\ \dot{e}_3 &{}= &{} - e_c e_3 - k_3 e_3 \\ \end{array} \right. \end{aligned}$$
(49)

Differentiating the parameter estimation error (48) with respect to t, we get

$$\begin{aligned} \left. \begin{array}{ccl} \dot{e}_a &{} = &{} - \dot{\hat{a}} \\ \dot{e}_b &{} = &{} - \dot{\hat{b}} \\ \dot{e}_c &{} = &{} - \dot{\hat{c}} \\ \dot{e}_p &{} = &{} - \dot{\hat{p}} \\ \end{array} \right. \end{aligned}$$
(50)

Next, we find an update law for parameter estimates using Lyapunov stability theory.

Consider the quadratic Lyapunov function defined by

$$\begin{aligned} V(e_1, e_2, e_3, e_a, e_b, e_c, e_p) = {1 \over 2} \, \left( e_1^2 + e_2^2 +e_3^2 + e_a^2 + e_b^2 + e_c^2 + e_p^2 \right) , \end{aligned}$$
(51)

which is positive definite on .

Differentiating V along the trajectories of (49) and (50), we get

$$\begin{aligned} \left. \begin{array}{ccl} \dot{V} &{} = &{} - k_1 e_1^2 - k_2 e_2^2 - k_3 e_3^2 + e_a \left[ - e_1 e_2 - \dot{\hat{a}} \right] + e_b \left[ e_2^2 - \dot{\hat{b}} \right] \\ &{} &{} + e_c \left[ - e_3^2 - \dot{\hat{c}} \right] + e_p \left[ e_1 e_2 - \dot{\hat{p}} \right] \end{array} \right. \end{aligned}$$
(52)

In view of Eq. (52), an update law for the parameter estimates is taken as

$$\begin{aligned} \left. \begin{array}{ccl} \dot{\hat{a}} &{} = &{} - e_1 e_2 \\ \dot{\hat{b}} &{} = &{} e_2^2 \\ \dot{\hat{c}} &{} = &{} - e_3^2 \\ \dot{\hat{p}} &{} = &{} e_1 e_2 \\ \end{array} \right. \end{aligned}$$
(53)

Theorem 2

The identical novel chaotic systems (42) and (43) with unknown system parameters are globally and exponentially synchronized for all initial conditions by the adaptive control law (46) and the parameter update law (53), where \(k_i, (i = 1, 2, 3)\) are positive constants.

Proof

The result is proved using Lyapunov stability theory [32].

We consider the quadratic Lyapunov function V defined by (51), which is positive definite on .

Substitution of the parameter update law (53) into (52) yields

$$\begin{aligned} \dot{V} = - k_1 e_1^2 - k_2 e_2^2 - k_3 e_3^2, \end{aligned}$$
(54)

which is a negative semi-definite function on .

Therefore, it can be concluded that the synchronization error vector e(t) and the parameter estimation error are globally bounded, i.e.

$$\begin{aligned} \left[ \begin{array}{ccccccc} e_1(t)&e_2(t)&e_3(t)&e_a(t)&e_b(t)&e_c(t)&e_p(t) \end{array} \right] ^T \in \mathbf {L}_\infty . \end{aligned}$$
(55)

Define

$$\begin{aligned} k = \min \left\{ k_1, k_2, k_3 \right\} \end{aligned}$$
(56)

Then it follows from (54) that

$$\begin{aligned} \dot{V} \le - k \Vert e \Vert ^2 \ \ \text{ or } \ \ k \Vert \mathbf {e} \Vert ^2 \le - \dot{V} \end{aligned}$$
(57)

Integrating the inequality (57) from 0 to t, we get

$$\begin{aligned} k \int \limits _0^t \ \Vert \mathbf {e}(\tau ) \Vert ^2 \, d\tau \, \le \, - \int \limits _0^t \ \dot{V}(\tau ) \, d\tau = V(0) - V(t) \end{aligned}$$
(58)

From (58), it follows that \(\mathbf {e}(t) \in \mathbf {L}_2\).

Using (49), it can be deduced that \(\dot{\mathbf {e}}(t) \in \mathbf {L}_\infty \).

Hence, using Barbalat’s lemma, we can conclude that \(\mathbf {e}(t) \rightarrow 0\) exponentially as \(t \rightarrow \infty \) for all initial conditions .

This completes the proof. \(\qquad \square \)

For numerical simulations, the parameter values of the novel systems (42) and (43) are taken as in the chaotic case, viz.

$$\begin{aligned} a = 2.2, \ \ b = 3, \ \ c = 5, \ \ p = 0.1 \end{aligned}$$
(59)

The gain constants are taken as \(k_i = 6\) for \(i = 1, 2, 3\).

The initial values of the parameter estimates are taken as

$$\begin{aligned} \hat{a}(0) = 6.2, \ \ \hat{b}(0) = 12.9, \ \ \hat{c}(0) = 28.5, \ \ \hat{p}(0) = 17.3 \end{aligned}$$
(60)

The initial values of the master system (42) are taken as

$$\begin{aligned} x_1(0) = 5.8, \ \ x_2(0) = 18.3, \ \ x_3(0) = -12.1 \end{aligned}$$
(61)
Fig. 7
figure 7

Synchronization of the states \(x_1\) and \(y_1\)

Full size image

The initial values of the slave system (43) are taken as

$$\begin{aligned} y_1(0) = 16.4, \ \ y_2(0) = 4.5, \ \ y_3(0) = -7.8 \end{aligned}$$
(62)

Figures 7-9 show the complete synchronization of the identical chaotic systems (42) and (43).

Figure 7 shows that the states \(x_1(t)\) and \(y_1(t)\) are synchronized in one second (MATLAB).

Figure 8 shows that the states \(x_2(t)\) and \(y_2(t)\) are synchronized in one second (MATLAB).

Figure 9 shows that the states \(x_3(t)\) and \(y_3(t)\) are synchronized in one second (MATLAB).

Figure 10 shows the time-history of the synchronization errors \(e_1(t), e_2(t), e_3(t)\). From Fig. 10, it is seen that the errors \(e_1(t), e_2(t)\) and \(e_3(t)\) are stabilized in one second (MATLAB).

Fig. 8
figure 8

Synchronization of the states \(x_2\) and \(y_2\)

Full size image
Fig. 9
figure 9

Synchronization of the states \(x_3\) and \(y_3\)

Full size image
Fig. 10
figure 10

Time-history of the synchronization errors \(e_1, e_2, e_3\)

Full size image

6 Conclusions

In this work, we described an eight-term 3-D novel polynomial chaotic system consisting of three quadratic nonlinearities. The qualitative properties of the novel chaotic system have been discussed in detail. We showed that the novel chaotic system has four equilibrium points of which two equilibrium points are saddle points and the other equilibrium points are saddle-foci. The Lyapunov exponents of the novel chaotic system were derived as \(L_1 = 0.4715, L_2 = 0\) and \(L_3 = -2.4728\). The Lyapunov dimension of the novel chaotic system was obtained as \(D_{L} = 2.1907\). Next, we worked on the design of adaptive feedback controller for globally stabilizing the trajectories of the novel chaotic system with unknown parameters. Furthermore, we derived new results for the design of adaptive feedback controller for achieving complete synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results were proved using Lyapunov stability theory. MATLAB simulations were displayed to illustrate all the main results presented in this research work.