Solution to Problem 2.2.1
The matrix of error system for the synchronization between two nearly identical Coullete chaotic systems (2) is given as follows:
$$ \left(\begin{array}{c}\hfill {\overset{.}{e}}_1(t)\hfill \\ {}\hfill {\overset{.}{e}}_2(t)\hfill \\ {}\hfill {\overset{.}{e}}_3(t)\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ {}\hfill a\hfill & \hfill  b\hfill & \hfill  c\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\left(\begin{array}{c}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill {x}_1^3(t){x}_2^3(t)\hfill \end{array}\right)+\left(\begin{array}{c}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill {u}_3(t)\hfill \end{array}\right) $$
(4)
where e
_{1}(t) = x
_{2}(t) − x
_{1}(t), e
_{2}(t) = y
_{2}(t) − y
_{1}(t), and e
_{3}(t) = z
_{2}(t) − z
_{1}(t) are the synchronization errors.
Theorem 1.
If the control input vector is designed such that
$$ \left(\begin{array}{c}\hfill {u}_1(t)\hfill \\ {}\hfill {u}_2(t)\hfill \\ {}\hfill {u}_3(t)\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill {x}_2^3(t){x}_1^3(t)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {v}_1(t)\hfill \\ {}\hfill {v}_2(t)\hfill \\ {}\hfill {v}_3(t)\hfill \end{array}\right) $$
(5)
where the subcontroller matrix
v(t) in Eq. (5) :
$$ \left(\begin{array}{c}\hfill {v}_1(t)\hfill \\ {}\hfill {v}_2(t)\hfill \\ {}\hfill {v}_3(t)\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill {k}_{11}\hfill & \hfill {k}_{12}\hfill & \hfill {k}_{13}\hfill \\ {}\hfill {k}_{21}\hfill & \hfill {k}_{22}\hfill & \hfill {k}_{23}\hfill \\ {}\hfill {k}_{31}\hfill & \hfill {k}_{32}\hfill & \hfill {k}_{33}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right) $$
(6)
where k
_{
ij
}, [i, j = 1, 2, 3] is an LCP matrix, and then, the two coupled chaotic systems (2) are globally exponentially synchronized.
Proof of Theorem 1. Substituting Eqs. (5) and (6) into Eq. (4) gives
$$ \left(\begin{array}{c}\hfill {\overset{.}{e}}_1(t)\hfill \\ {}\hfill {\overset{.}{e}}_2(t)\hfill \\ {}\hfill {\overset{.}{e}}_3(t)\hfill \end{array}\right)=\left(\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ {}\hfill a\hfill & \hfill  b\hfill & \hfill  c\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill {k}_{11}\hfill & \hfill {k}_{12}\hfill & \hfill {k}_{13}\hfill \\ {}\hfill {k}_{21}\hfill & \hfill {k}_{22}\hfill & \hfill {k}_{23}\hfill \\ {}\hfill {k}_{31}\hfill & \hfill {k}_{32}\hfill & \hfill {k}_{33}\hfill \end{array}\right)\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right). $$
(7)
Since u
_{1}(t) = u
_{2}(t) = 0, therefore, v
_{1}(t) = v
_{2}(t) = 0 and Eq. (7) becomes
$$ \left(\begin{array}{c}\hfill {\overset{.}{e}}_1(t)\hfill \\ {}\hfill {\overset{.}{e}}_2(t)\hfill \\ {}\hfill {\overset{.}{e}}_3(t)\hfill \end{array}\right)=\left(\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ {}\hfill a\hfill & \hfill  b\hfill & \hfill  c\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {k}_{31}\hfill & \hfill {k}_{32}\hfill & \hfill {k}_{33}\hfill \end{array}\right)\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ {}\hfill a{k}_{31}\hfill & \hfill \left( b+{k}_{32}\right)\hfill & \hfill \left( c+{k}_{33}\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right). $$
(8)
Note that the obtained linearized error system (8) is in the form of \( \overset{.}{e}(t)= A e(t) \), where
$$ A=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ {}\hfill a{k}_{31}\hfill & \hfill \left( b+{k}_{32}\right)\hfill & \hfill \left( c+{k}_{33}\right)\hfill \end{array}\right). $$
(9)
Thus, the remaining problem is that the LCPs k
_{31}, k
_{32}, and k
_{33} are chosen such that the real parts of all eigenvalues of the matrix A ∈ R
^{3 × 3} in (9) are negative with suitable positions of the LCPs in a complex plane, with fast and smooth convergence of the synchronization error signals. In these circumstances (9), if the LCPs satisfy the following condition:
$$ {k}_{31}> a,\ {k}_{32}\ge 0\ \mathrm{and}\kern0.5em {k}_{32}\ge {k}_{33}\ge 0, $$
(10)
then, by the RouthHurwitz criterion and Lyapunov stability theory, the closedloop system (8) is globally exponentially stable. Therefore, the two coupled chaotic systems (2) are globally exponentially synchronized.
Remark 2.
In the following subsection, this study finds numerically the correct balance between the convergence rates of the synchronization error signals to the origin and magnitude of the suitable LCPs.
Numerical Simulation Results and Discussion
The parameters of the Coullete chaotic system (1) are set as a = 5.5, b = 3.5, and c = 1, while the initial values of the state vectors are taken as x
_{1}(0) = 0.145, y
_{1}(0) = 0.625, z
_{1}(0) = 0.925 and x
_{2}(0) = 0.945, y
_{2}(0) = 0.032, z
_{2}(0) = 0.112, respectively. The corresponding numerical results are given as follows.
Case 1: From matrix A in Eq. (9) and the condition (10), it has been observed that the stability of the closedloop system (8) depends on the magnitude of the LCP k
_{31} and the magnitude of k
_{31} depends on the magnitude of the LCPs k
_{32} and k
_{33}. Therefore, let us fix k
_{32} = k
_{33} = 3 and optimize k
_{31}.
If k
_{31} = 5.5, then, the poles of the linearized error system (8) are {−1 ± 1.87083 i, 0}. One can notice that one of the eigenvalues is positive. Hence, the closedloop system (8) is unstable, which is also confirmed from Fig. 2a.
For k
_{31} = 5.6, 6, 9, 15, 25, 31, and 31.49, the closedloop system (8) is globally exponentially stable with the following corresponding poles: {−1.99 ± 1.59i, − 0.015}, {−1.96 ± 1.58i, − 0.08}, {−1.5 ± 1.12i, − 1}, {−0.56 ± 1.73i, − 2.88}, {−0.162 ± 2.298i, − 3.68}, {−0.01 ± 2.53i, − 3.98} and {−0.02 ± 2.58i, − 4.04}, respectively, which can be confirmed from Fig. 2b–f.
If k
_{31} = 31.5, then, the poles of the linearized error system (8) are {1.387 × 10^{− 15} ± 2.55i, − 4}. Hence, the closedloop system (8) is unstable, which is also confirmed from Fig. 2g.
Thus, for 5.5 < k
_{31} < 31.5, the closedloop system (8) is globally exponentially stable and the perfect synchronization behavior is achieved at k
_{31} = 9 after t ≈ 3 s, with underdamped oscillation as shown in Fig. 2c.
Case 2: Let us fix k
_{32} = k
_{33} = 1 and optimize k
_{31}. Then, from the numerical study similar as above, it is observed that the closedloop system (8) is globally exponentially stable at 5.5 < k
_{31} < 14.5 and the perfect synchronization behavior is achieved at k
_{31} = 8 after t ≈ 6 s, with underdamped oscillation as shown in Fig. 3.
Comparative Study
The developed active synchronization controller approach has advantages over the past studies in [16] in terms of the control effort, synchronization transient speed, and suitable position of the LCPs in a complex plane for the GES. For example, in terms of the control effort, only one input feedback controller (5) is utilized to accomplish the GES, while in [16], three control functions are designed. Similarly, in this study, the synchronization speed is 3 s (Fig. 2c), whereas in [15], the synchronization speed is 5 s. Thus, the time difference is 2 s. Furthermore, the proposed AC approach (5) also identifies the correct balance between the converging rates of the synchronization error signals to the origin and magnitude of the LCPs for a fast and smooth synchronization.
The proposed AC function (5) contains a partially nonlinear term and a feedback term. The present study does not only improve the synchronization speed and quality but also decreases the number of feedback controllers. This considerably reduces the amount of energy for the chaos synchronization and establishes the GES. These features give advantages of the current study over the past published works in the literature concerned.
Solution to Problem 2.2.2
The matrix of the error system for the adaptive robust synchronization between two nearly identical Coullete chaotic systems (3) is given as follows:
$$ \begin{array}{l}\left(\begin{array}{c}\hfill {\overset{.}{e}}_1(t)\hfill \\ {}\hfill {\overset{.}{e}}_2(t)\hfill \\ {}\hfill {\overset{.}{e}}_3(t)\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ {}\hfill a\hfill & \hfill  b\hfill & \hfill  c\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {x}_1(t){x}_2(t)\left({x}_1^2(t)+{x}_2^2(t)\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\\ {}\kern2.5em \left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {e}_a(t)\hfill & \hfill {e}_b(t)\hfill & \hfill {e}_c(t)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {x}_2(t)\hfill \\ {}\hfill {y}_2(t)\hfill \\ {}\hfill {z}_2(t)\hfill \end{array}\right)+\left(\begin{array}{l}{g}_1\left({x}_2(t)\right){f}_1\left({x}_1(t)\right)\\ {}{g}_2\left({y}_2(t)\right){f}_2\left({y}_1(t)\right)\\ {}{g}_3\left({z}_2(t)\right){f}_3\left({z}_1(t)\right)\end{array}\right)+\left(\begin{array}{c}\hfill {u}_1(t)\hfill \\ {}\hfill {u}_2(t)\hfill \\ {}\hfill {u}_3(t)\hfill \end{array}\right)\end{array} $$
(11)
where e
_{1}(t) = x
_{2}(t) − x
_{1}(t), e
_{2}(t) = y
_{2}(t) − y
_{1}(t) and e
_{3}(t) = z
_{2}(t) − z
_{1}(t) are the synchronization errors and e
_{
a
}(t) = a − a
_{1}(t), e
_{
b
}(t) = b − b
_{1}(t), and e
_{
c
}(t) = c − c
_{1}(t) are the estimation of unknown timevarying parameters. Note that \( {\overset{.}{e}}_a(t)={\overset{.}{a}}_1(t),\ {\overset{.}{e}}_b(t)={\overset{.}{b}}_1(t) \), and \( {\overset{.}{e}}_c(t)={\overset{.}{c}}_1(t) \). The adaptive synchronization of two coupled chaotic systems (3) is accomplished in the sense that:
$$ \underset{t\to \infty }{ \lim}\left\Vert {e}_i(t)\right\Vert =0,\kern0.5em i=1,\ 2,\ 3, $$
(12)
and the unknown timevarying parameters are estimated from the system parameters in the sense that:
$$ \underset{t\to \infty }{ \lim}\left{e}_a(t)\right=\underset{t\to \infty }{ \lim}\left a{a}_1(t)\right=0,\ \underset{t\to \infty }{ \lim}\left{e}_b(t)\right=\underset{t\to \infty }{ \lim}\left b{b}_1(t)\right\kern0.5em \mathrm{and}\kern0.5em \underset{t\to \infty }{ \lim}\left{e}_c(t)\right=\underset{t\to \infty }{ \lim}\left c{c}_1(t)\right=0\ . $$
(13)
Assumption 1 [33].
It is assumed that the unknown model uncertainties and external disturbances are bounded. Therefore, there exist unknown positive constants
\( {\varDelta}_i^m\ and\kern0.5em {\varDelta}_i^s \)
such that
$$ \left{f}_i(.)\right\le {\varDelta}_i^m\ \mathrm{and}\kern0.5em \left{g}_i(.)\right\le {\varDelta}_i^s,\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em i=1,\ 2,\ 3. $$
$$ \left{g}_i(.){f}_i(.)\right\le {\beta}_i,\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em i=1,\ 2,\ 3, $$
(14)
where β
_{
1
}
is any unknown positive constant such that
\( {\beta}_i={\varDelta}_i^m+{\varDelta}_i^s \)
.
Assumption 2 [34].
Let B ⊂ R
^{n}
be a bounded region containing the whole attractor of the chaotic (or hyperchaotic) system, such that no signal of the chaotic (or hyperchaotic) system ever leaves it. Then, there exist positive constants B
_{
x
} ∈ R, B
_{
y
} ∈ R and B
_{
z
} ∈ R, such that
$$ \left x(t)\right\le {B}_x,\ \left y(t)\right\le {B}_y,\ and\kern0.5em \left z(t)\right\le {B}_z. $$
(15)
Theorem 2.
If the control input u
_{
i
}(t), i = 1, 2, 3, in (3) is designed such that
$$ \begin{array}{l}\left(\begin{array}{c}\hfill {u}_1(t)\hfill \\ {}\hfill {u}_2(t)\hfill \\ {}\hfill {u}_3(t)\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill {\widehat{k}}_1\left( \exp \left(\eta \left{e}_1(t)\right\right)\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\widehat{k}}_2\left( \exp \left(\eta \left{e}_2(t)\right\right)\right)\hfill & \hfill 0\hfill \\ {}\hfill {x}_1^2(t)+{x}_2^2(t)\hfill & \hfill 0\hfill & \hfill {\widehat{k}}_3\left( \exp \left(\eta \left{e}_3(t)\right\right)\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)\\ {}\kern4em \left(\begin{array}{l}{\beta}_1 sgn\left({e}_1(t)\right)\\ {}{\beta}_2 sgn\left({e}_3(t)\right)\\ {}{\beta}_3 sgn\left({e}_3(t)\right)\end{array}\right),\end{array} $$
(16)
and the unknown timevarying parameters a
_{1}(t), b
_{1}(t) and c
_{1}(t) are estimated by the following adaptation laws:
$$ {\overset{.}{a}}_1(t)={x}_2(t){e}_3(t),\ {\overset{.}{b}}_1(t)={y}_2(t){e}_3(t),\ \mathrm{and}\kern0.5em {\overset{.}{c}}_1(t)={z}_2(t){e}_3(t). $$
(17)
where η is any positive constant; exp and sgn, respectively, denote the exponential and signum functions; and
\( {\widehat{k}}_i,\ \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em i=1,\ 2,\ 3 \)
is the estimated LCP, which is updated according to the following adaptation algorithm:
$$ {\overset{.}{\widehat{k}}}_i(t)=\rho \left( \exp \left(\eta \left{e}_i(t)\right\right)\right){e}_i^2(t),\kern0.5em {\widehat{k}}_i(0)=0,\ \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em i=1,\ 2,\ 3, $$
(18)
where ρ is any positive real constant determining the adaptation process. Then, the two coupled chaotic systems (3) are asymptotically synchronized.
Proof of Theorem 2. Substituting Eq. (16) into Eq. (11) gives
$$ \begin{array}{l}\left(\begin{array}{c}\hfill {\overset{.}{e}}_1(t)\hfill \\ {}\hfill {\overset{.}{e}}_2(t)\hfill \\ {}\hfill {\overset{.}{e}}_3(t)\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill {\widehat{k}}_1\left( \exp \left(\eta \left{e}_1(t)\right\right)\right)\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\widehat{k}}_2\left( \exp \left(\eta \left{e}_2(t)\right\right)\right)\hfill & \hfill 1\hfill \\ {}\hfill a\hfill & \hfill  b\hfill & \hfill  c{\widehat{k}}_3\left( \exp \left(\eta \left{e}_3(t)\right\right)\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\\ {}\kern4em \left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {x}_1(t){x}_2(t)\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {e}_a(t)\hfill & \hfill {e}_b(t)\hfill & \hfill {e}_c(t)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {x}_2(t)\hfill \\ {}\hfill {y}_2(t)\hfill \\ {}\hfill {z}_2(t)\hfill \end{array}\right)+\\ {}\kern4em \left(\begin{array}{l}{g}_1\left({x}_2(t)\right){f}_1\left({x}_1(t)\right)\\ {}{g}_2\left({y}_2(t)\right){f}_2\left({y}_1(t)\right)\\ {}{g}_3\left({z}_2(t)\right){f}_3\left({z}_1(t)\right)\end{array}\right)+\left(\begin{array}{l}{d}_1(t){D}_1(t)\\ {}{d}_2(t){D}_2(t)\\ {}{d}_3(t){D}_3(t)\end{array}\right)\left(\begin{array}{l}{\beta}_1 sgn\left({e}_1(t)\right)\\ {}{\beta}_2 sgn\left({e}_3(t)\right)\\ {}{\beta}_3 sgn\left({e}_3(t)\right)\end{array}\right).\end{array} $$
(19)
Consider a Lyapunov function as follows:
$$ V\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)={\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)}^T P\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{2}\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\left({\widehat{k}}_1(t){k}_1\right)}^2\hfill \\ {}\hfill {\left({\widehat{k}}_2(t){k}_2\right)}^2\hfill \\ {}\hfill {\left({\widehat{k}}_3(t){k}_3\right)}^2\hfill \end{array}\right)+\frac{1}{2}\left({e}_a^2(t)+{e}_b^2(t)+{e}_c^2(t)\right)\ge 0, $$
(20)
where
$$ P= diag\left[\frac{1}{2},\ \frac{b}{2},\ \frac{1}{2}\right]. $$
(21)
The time derivative of Eq. (20) is given as:
$$ \begin{array}{l}\overset{.}{V}\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill {e}_1(t)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill b{e}_2(t)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {e}_3(t)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\overset{.}{e}}_1(t)\hfill \\ {}\hfill {\overset{.}{e}}_2(t)\hfill \\ {}\hfill {\overset{.}{e}}_3(t)\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill {e}_a(t)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {e}_b(t)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {e}_c(t)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\overset{.}{e}}_a(t)\hfill \\ {}\hfill {\overset{.}{e}}_b(t)\hfill \\ {}\hfill {\overset{.}{e}}_c(t)\hfill \end{array}\right)+\\ {}\kern4.75em \left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill \left({\widehat{k}}_1(t){k}_1\right){\overset{.}{\widehat{k}}}_1(t)\hfill \\ {}\hfill \left({\widehat{k}}_2(t){k}_2\right){\overset{.}{\widehat{k}}}_2(t)\hfill \\ {}\hfill \left({\widehat{k}}_3(t){k}_3\right){\overset{.}{\widehat{k}}}_3(t)\hfill \end{array}\right).\end{array} $$
(22)
Using Eq. (19) into Eq. (22) yields:
$$ \begin{array}{l}\overset{.}{V}\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)={\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)}^T\left(\begin{array}{ccc}\hfill {\widehat{k}}_1\left( \exp \left(\eta \left{e}_1(t)\right\right)\right)\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{a\left{x}_1(t)\right\left{x}_2(t)\right}{2}\hfill \\ {}\hfill \frac{1}{2}\hfill & \hfill {\widehat{k}}_2\left( \exp \left(\eta \left{e}_2(t)\right\right)\right)\hfill & \hfill 0\hfill \\ {}\hfill \frac{a\left{x}_1(t)\right\left{x}_2(t)\right}{2}\hfill & \hfill 0\hfill & \hfill  c{\widehat{k}}_3\left( \exp \left(\eta \left{e}_3(t)\right\right)\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\\ {}\kern.1em \left(\begin{array}{l}{g}_1\left({x}_2(t)\right){f}_1\left({x}_1(t)\right)+{d}_1(t){D}_1(t)\\ {}{g}_2\left({y}_2(t)\right){f}_2\left({y}_1(t)\right)+{d}_2(t){D}_2(t)\\ {}{g}_3\left({z}_2(t)\right){f}_3\left({z}_1(t)\right)+{d}_3(t){D}_3(t)\end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)\left(\begin{array}{l}{\beta}_1 sgn\left({e}_1(t)\right)\\ {}{\beta}_2 sgn\left({e}_3(t)\right)\\ {}{\beta}_3 sgn\left({e}_3(t)\right)\end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\\ {}\kern.1em {\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)}^T\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {e}_a(t)\hfill & \hfill {e}_b(t)\hfill & \hfill {e}_c(t)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {x}_2(t)\hfill \\ {}\hfill {y}_2(t)\hfill \\ {}\hfill {z}_2(t)\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill \left({\widehat{k}}_1(t){k}_1\right){\overset{.}{\widehat{k}}}_1(t)\hfill \\ {}\hfill \left({\widehat{k}}_2(t){k}_2\right){\overset{.}{\widehat{k}}}_2(t)\hfill \\ {}\hfill \left({\widehat{k}}_3(t){k}_3\right){\overset{.}{\widehat{k}}}_3(t)\hfill \end{array}\right)\\ {}\kern.7em {e}_a(t){\overset{.}{a}}_1(t){e}_b(t){\overset{.}{b}}_1(t)  {e}_c(t){\overset{.}{c}}_1(t)\kern1.25em \end{array} $$
$$ \begin{array}{l}\le {\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)}^T\left(\begin{array}{ccc}\hfill {\widehat{k}}_1\left( \exp \left(\eta \left{e}_1(t)\right\right)\right)\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{a{x}_1(t){x}_2(t)}{2}\hfill \\ {}\hfill \frac{1}{2}\hfill & \hfill {\widehat{k}}_2\left( \exp \left(\eta \left{e}_2(t)\right\right)\right)\hfill & \hfill 0\hfill \\ {}\hfill \frac{a{x}_1(t){x}_2(t)}{2}\hfill & \hfill 0\hfill & \hfill  c{\widehat{k}}_3\left( \exp \left(\eta \left{e}_3(t)\right\right)\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\\ {}\kern0.5em \left(\begin{array}{l}\left{g}_1\left({x}_2(t)\right){f}_1\left({x}_1(t)\right)+{d}_1(t){D}_1(t)\right sgn\left({e}_1(t)\right){\beta}_1 sgn\left({e}_1(t)\right)\\ {}\left{g}_2\left({y}_2(t)\right){f}_2\left({y}_1(t)\right)+{d}_2(t){D}_2(t)\right sgn\left({e}_3(t)\right){\beta}_2 sgn\left({e}_3(t)\right)\\ {}\left{g}_3\left({z}_2(t)\right){f}_3\left({z}_1(t)\right)+{d}_3(t){D}_3(t)\right sgn\left({e}_3(t)\right){\beta}_3 sgn\left({e}_3(t)\right)\end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\\ {}\kern0.5em \left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill \left({\widehat{k}}_1(t){k}_1\right){\overset{.}{\widehat{k}}}_1(t)\hfill \\ {}\hfill \left({\widehat{k}}_2(t){k}_2\right){\overset{.}{\widehat{k}}}_2(t)\hfill \\ {}\hfill \left({\widehat{k}}_3(t){k}_3\right){\overset{.}{\widehat{k}}}_3(t)\hfill \end{array}\right){e}_a(t){x}_2(t){e}_3(t)+{e}_b(t){y}_2(t){e}_3(t)+{e}_c(t){z}_2(t){e}_3(t)\\ {}\kern.1em {e}_a(t){\overset{.}{a}}_1(t){e}_b(t){\overset{.}{b}}_1(t){e}_c(t){\overset{.}{c}}_1(t).\end{array} $$
(23)
Using Assumption 1 and the fact that sgn(e
_{
i
}) ≤ 1 for i = 1, 2, 3, in Eq. (23) yields:
$$ \begin{array}{l}\le {\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)}^T\left(\begin{array}{ccc}\hfill \rho {\widehat{k}}_1(t)\left( \exp \left(\eta \left{e}_1(t)\right\right)\right)\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{a{B}_x}{2}\hfill \\ {}\hfill \frac{1}{2}\hfill & \hfill \rho {\widehat{k}}_2(t)\left( \exp \left(\eta \left{e}_2(t)\right\right)\right)\hfill & \hfill 0\hfill \\ {}\hfill \frac{a{B}_x}{2}\hfill & \hfill 0\hfill & \hfill  c+\rho {\widehat{k}}_3\left( \exp \left(\eta \left{e}_3(t)\right\right)\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)+\\ {}\kern1.5em \left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill \left({\widehat{k}}_1(t){k}_1\right){\overset{.}{\widehat{k}}}_1(t)\hfill \\ {}\hfill \left({\widehat{k}}_2(t){k}_2\right){\overset{.}{\widehat{k}}}_2(t)\hfill \\ {}\hfill \left({\widehat{k}}_3(t){k}_3\right){\overset{.}{\widehat{k}}}_3(t)\hfill \end{array}\right){e}_a(t)\left({\overset{.}{a}}_1(t)+{x}_2(t){e}_3(t)\right){e}_b(t)\left({\overset{.}{b}}_1(t){y}_2(t){e}_3(t)\right)\\ {}\kern1.75em {e}_c(t)\left({\overset{.}{c}}_1(t){z}_2(t){e}_3(t)\right).\end{array} $$
(24)
Using the parameter update laws (17) and (18) into Eq. (24) gives:
$$ \overset{.}{V}\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)\le {\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right)}^T\left(\begin{array}{ccc}\hfill \rho {k}_1(t)\left( \exp \left(\eta \left{e}_1(t)\right\right)\right)\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{B_x a}{2}\hfill \\ {}\hfill \frac{1}{2}\hfill & \hfill \rho {k}_2(t)\left( \exp \left(\eta \left{e}_2(t)\right\right)\right)\hfill & \hfill 0\hfill \\ {}\hfill \frac{B_x a}{2}\hfill & \hfill 0\hfill & \hfill c+\rho {k}_3\left( \exp \left(\eta \left{e}_3(t)\right\right)\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {e}_1(t)\hfill \\ {}\hfill {e}_2(t)\hfill \\ {}\hfill {e}_3(t)\hfill \end{array}\right) $$
$$ \overset{.}{V}\left(\boldsymbol{e}(t)\right)\le \boldsymbol{e}{(t)}^T Q\ \boldsymbol{e}(t)\le 0, $$
(25)
where e(t) = [e
_{1}(t), e
_{2}(t), e
_{3}(t)]^{T} is the absolute state error vector, and
$$ Q=\left(\begin{array}{ccc}\hfill \rho {k}_1(t)\left( \exp \left(\eta \left{e}_1(t)\right\right)\right)\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{B_x a}{2}\hfill \\ {}\hfill \frac{1}{2}\hfill & \hfill \rho {k}_2(t)\left( \exp \left(\eta \left{e}_2(t)\right\right)\right)\hfill & \hfill 0\hfill \\ {}\hfill \frac{B_x a}{2}\hfill & \hfill 0\hfill & \hfill c+\rho {k}_3\left( \exp \left(\eta \left{e}_3(t)\right\right)\right)\hfill \end{array}\right). $$
(26)
At this stage, the remaining problem is that if the estimate of the LCPs k
_{1}, k
_{2}, and k
_{3} and the two positive constants η and ρ are chosen such that the matrix, Q ∈ R
^{3 × 3} (26) becomes a positive definite matrix (PDM). Since V(e(t)) is positive definite, then the equilibrium point \( \left({e}_i(t)=0,\ {\widehat{k}}_i={k}_i,\kern0.5em i=1,\ 2,\ 3\right) \) of the systems (11) and (18) is asymptotically stable. Therefore, the two coupled chaotic systems (3) are asymptotically synchronized. This completes the proof of Theorem 2.
Numerical Simulation Results and Discussion
Numerical simulation results are furnished in order to verify the robustness and performance of the proposed ASC approach. The true value of the parameters of the Coullete chaotic system (1) are set as a = 5.5, b = 3.5, and c = 1, and these values are unknown to the slave system in (3). The initial values of the states vectors are taken as x
_{1}(0) = 0.145, y
_{1}(0) = 0.625, z
_{1}(0) = 0.925 and x
_{2}(0) = 0.945, y
_{2}(0) = 0.032, z
_{2}(0) = 0.112, respectively. The estimated absolute values of the state vectors of the Coullete chaotic system (1) are B
_{
x
} ≤ 3.6, B
_{
y
} ≤ 6, and B
_{
z
} ≤ 12 through numerical simulation. The controlling parameters are considered as k
_{1} = k
_{2} = 5 and k
_{3} = 10, and the two positive constants η and ρ are taken as η = 0.01 and ρ = 1. In numerical simulations, the following model uncertainties and external disturbances are applied to the master and slave systems (3), respectively.
$$ \begin{array}{l}\left(\mathrm{Master}\ \mathrm{system}\right)\kern10em \\ {}{f}_1\left({x}_1(t)\right)+{D}_1(t)=0.3 \sin \left(\frac{\pi}{3}{x}_1(t)\right)0.01 \cos \left(10 t\right),\\ {}{f}_2\left({y}_1(t)\right)+{D}_2(t)=0.25 \cos \left(\frac{\pi}{4}{y}_1(t)\right)0.03 \sin \left(20 t\right),\\ {}{f}_3\left({z}_1(t)\right)+{D}_3(t)=0.3 \sin \left(\frac{\pi}{2}{z}_1(t)\right)+0.04 \cos \left(10 t\right),\\ {}\left(\mathrm{Slave}\ \mathrm{system}\right)\\ {}{g}_1\left({x}_2(t)\right)+{d}_1(t)=0.4 \cos \left(\frac{\pi}{6}{x}_2(t)\right)+0.02 \sin \left(30 t\right),\\ {}{g}_2\left({y}_2(t)\right)+{d}_2(t)=0.25 \sin \left(\frac{5\pi}{6}{y}_2(t)\right)0.01 \cos \left(20 t\right),\\ {}{g}_3\left({z}_2(t)\right)+{d}_3(t)=0.15 \cos \left(\frac{2\pi}{3}{z}_2(t)\right)0.01 \cos \left(15 t\right).\end{array} $$
(27)
Accordingly, β
_{1} = 0.64, β
_{2} = 0.54, and β
_{3} = 0.5.
The corresponding numerical simulation results are given as follows:
Figure 4 displays the result of the synchronized error signals. It is demonstrated that the error signals (11) completely synchronize within a short transient time t ≈ 0.7 s in the presence of external disturbances and model uncertainties under the control action (16). The adaptive process of parameters is shown in Fig. 5. From Fig. 5, it can be observed that the unknown timevarying parameters a
_{1}(t) = a + 0.2 sin(35t), b
_{1}(t) = b + 0.1 sin(25t), and c
_{1}(t) = c + 0.02 cos(90t) with initial values a
_{1}(0) = 7, b
_{1}(0) = 2, c
_{1}(0) = − 1, converge to the true values of a, b, and c as t → ∞, under the parameter update laws (20). Figure 6 shows the time history of the input control signals. The proposed ASC approach (16) is robust against different types of perturbations. The controller response time is short, and the error signals converge to the origin with critically damped oscillation.
Remark 3.

(i)
The proposed ASC approach (16) contains the linear terms, some partially nonlinear terms, and a feedback term. The exponential term (exp(−ηe
_{
i
}(t))) for i = 1, 2, 3, in the controller (16) provides smoothness to the error signals with small amplitude and without disturbing the convergence property, even in the presence of unknown external disturbance and model uncertainties.

(ii)
By selecting a smaller value of η providing the fast convergence rates of the error signals to the origin.
Remark 4.
The proposed ASC approach can be easily utilized for the complete and generalized synchronization for a class of chaotic as well as hyperchaotic systems. For example, the synchronization of two identical hyperchaotic Lu systems [20, 22] can also be achieved by applying the following nonlinear control function:
$$ {u}_i(t)={k}_i\left( exp\left(\eta \left{e}_i(t)\right\right)\right),\ f o r\kern0.5em i=2,\ 4,\ and\ {u}_i(t)=0,\ f o r\kern0.5em i=1,\ 3, $$
(28)
and the parameter update laws are given as:
$$ {\overset{.}{a}}_1(t)=\left({y}_2(t){x}_2(t)\right){e}_1(t),\ {\overset{.}{b}}_1(t)={z}_2(t){e}_3(t),\ {\overset{.}{c}}_1(t)={y}_2(t){e}_2(t),\ and\kern0.5em {\overset{.}{d}}_1(t)={w}_2(t){e}_4(t), $$
(29)
where the true values of parameters are taken as a = 15, b = 5, c = 10, and d = 1.
Numerical simulation results are shown in Figs.
7
and
8
. As compared to the past published works [20, 22], the synchronization speed is faster (0.6 s vs 4 s). Moreover, the synchronized error signals in [20, 22] converged to the origin with underdamped oscillation, while in the current study, the synchronized error signals converged to the origin with critically damped oscillation, which shows the less amount of energy utilized for CS objective.
As a matter of fact, the time variance property of the system parameters and the existence of total disturbances in problem formulation and the controller design procedure for the robust stability of the closedloop system are making the proposed ASC approach to be more effective as compared to the previous results and can be easily implemented in practice.