With respect to two coupled chaotic systems, by unidirectional coupling we mean that the behavior of response system is dependent on the behavior of another identical drive system, but the second one is not influenced by the behavior of the first. But the situation is very different to consider the synchronization of three chaotic systems with unidirectional coupling.

Consider the classical Rössler chaotic system

$$ \left\{\begin{array}{l}{\overset{.}{x}}_1=-\left({x}_2+{x}_3\right)\\ {}{\overset{.}{x}}_2={x}_1+a{x}_2\\ {}{\overset{.}{x}}_3={x}_3\left({x}_1-c\right)+b\end{array}\right. $$

where *x*
_{
i
} (*i* = 1, 2, 3) are the state variables and *a*, *b*, *c* are positive real constants, where *a* = 0.2, *b* = 0.2, and *c* = 5.7, so that the Rössler system has chaotic attractors. Here, we consider the unidirectional coupling among three identical Rössler chaotic systems

$$ \left\{\begin{array}{l}{\overset{.}{x}}_1=-\left({x}_2+{x}_3\right)-{d}_1\left({x}_1-{y}_1\right)\\ {}{\overset{.}{x}}_2={x}_1+a{x}_2\\ {}{\overset{.}{x}}_3={x}_3\left({x}_1-c\right)+b\\ {}{\overset{.}{y}}_1=-\left({y}_2+{y}_3\right)-{d}_2\left({y}_1-{z}_1\right)\\ {}{\overset{.}{y}}_2={y}_1+a{y}_2\\ {}{\overset{.}{y}}_3={y}_3\left({y}_1-c\right)+b\\ {}{\overset{.}{z}}_1=-\left({z}_2+{z}_3\right)-{d}_3\left({z}_1-{x}_1\right)\\ {}{\overset{.}{z}}_2={z}_1+a{z}_2\\ {}{\overset{.}{z}}_3={z}_3\left({z}_1-c\right)+b\end{array}\right. $$

(1)

Selecting the synchronization errors as follows:

$$ \begin{array}{l}{e}_{11}(t)={x}_1(t)-{y}_1(t),\ {e}_{21}(t)={y}_1(t)-{z}_1(t),\ {e}_{31}(t)={z}_1(t)-{x}_1(t)\\ {}{e}_{12}(t)={x}_2(t)-{y}_2(t),\ {e}_{22}(t)={y}_2(t)-{z}_2(t),\ {e}_{32}(t)={z}_2(t)-{x}_2(t)\\ {}{e}_{13}(t)={x}_3(t)-{y}_3(t),\ {e}_{23}(t)={y}_3(t)-{z}_3(t),\ {e}_{33}(t)={z}_3(t)-{x}_3(t)\end{array} $$

(2)

And we have *e*
_{31} = − *e*
_{11} − *e*
_{21}, *e*
_{32} = − *e*
_{12} − *e*
_{22}, *e*
_{33} = − *e*
_{13} − *e*
_{23}.

Assuming that

$$ {\overset{.}{d}}_1={k}_1{e}_{11}^2,\ {\overset{.}{d}}_2={k}_2\left({e}_{21}^2-{e}_{11}{e}_{21}\right),\ {\overset{.}{d}}_3={k}_3\left({e}_{21}^2+{e}_{11}{e}_{21}\right).\ {k}_1, {k}_2, {k}_3>0, $$

(3)

According to formulas (2) and (3), the dynamical error system is written as follows:

$$ \left\{\begin{array}{l}{\overset{.}{e}}_{11}=-{e}_{12}-{e}_{13}-{d}_1{e}_{11}+{d}_2{e}_{21}\\ {}{\overset{.}{e}}_{12}={e}_{11}+a{e}_{12}\\ {}{\overset{.}{e}}_{13}={x}_1{e}_{13}+{y}_3{e}_{11}-c{e}_{13}\\ {}{\overset{.}{e}}_{21}=-{e}_{22}-{e}_{23}-{d}_2{e}_{21}-{d}_3{e}_{11}-{d}_3{e}_{21}\\ {}{\overset{.}{e}}_{22}={e}_{21}+a{e}_{22}\\ {}{\overset{.}{e}}_{23}={y}_1{e}_{23}+{z}_3{e}_{21}-c{e}_{23}\end{array}\right. $$

(4)

### Theorem 1

The coupled system (1) will synchronize for any initial values (*x*
_{1}(0), *x*
_{2}(0), *x*
_{3}(0)); (*y*
_{1}(0), *y*
_{2}(0), *y*
_{3}(0)); (*z*
_{1}(0), *z*
_{2}(0), *z*
_{3}(0)); *d*
_{1}(0); *d*
_{2}(0); *d*
_{3}(0). That is, *e*
_{
ij
} → 0 as *t* → ∞, (*i*, *j* = 1, 2, 3).

### Proof

We construct a Lyapunov function

$$ V=\frac{1}{2}\left[{e}_{11}^2+{e}_{12}^2+{e}_{13}^2+{e}_{21}^2+{e}_{22}^2+{e}_{23}^2+\frac{1}{k_1}{\left({d}_1-{d}_1^{\ast}\right)}^2+\frac{1}{k_2}{\left({d}_2-{d}_2^{\ast}\right)}^2+\frac{1}{k_3}{\left({d}_3-{d}_3^{\ast}\right)}^2\right] $$

where \( {d}_1^{\ast },{d}_2^{\ast },{d}_3^{\ast }>0 \) are to be determined. Since the Rössler system is chaotic, we assume |*x*
_{1}|, |*y*
_{1}| ≤ *M*
_{1}, |*y*
_{3}|, |*z*
_{3}| ≤ *M*
_{3}, where *M*
_{1}, *M*
_{3} are positive constants. The time derivative of *V* along the orbits of the system (1) is

$$ \begin{array}{l}\overset{.}{V}={e}_{11}{\overset{.}{e}}_{11}+{e}_{12}{\overset{.}{e}}_{12}+{e}_{13}{\overset{.}{e}}_{13}+{e}_{21}{\overset{.}{e}}_{21}+{e}_{22}{\overset{.}{e}}_{22}+{e}_{23}{\overset{.}{e}}_{23}+\frac{1}{k_1}\left({d}_1-{d}_1^{\ast}\right){\overset{.}{d}}_1+\frac{1}{k_2}\left({d}_2-{d}_2^{\ast}\right){\overset{.}{d}}_2+\frac{1}{k_3}\left({d}_3-{d}_3^{\ast}\right){\overset{.}{d}}_3\\ {}\kern0.5em ={e}_{11}\left(-{e}_{12}-{e}_{13}-{d}_1{e}_{11}+{d}_2{e}_{21}\right)+{e}_{12}\left({e}_{11}+a{e}_{12}\right)+{e}_{13}\left({x}_1{e}_{13}+{y}_3{e}_{11}-c{e}_{13}\right)\\ {}\kern0.5em +{e}_{21}\left(-{e}_{22}-{e}_{23}-{d}_2{e}_{21}-{d}_3{e}_{11}-{d}_3{e}_{21}\right)+{e}_{22}\left({e}_{21}+a{e}_{22}\right)+{e}_{23}\left({y}_1{e}_{23}+{z}_3{e}_{21}-c{e}_{23}\right)\\ {}\kern0.5em +\left({d}_1-{d}_1^{\ast}\right){e}_{11}^2+\left({d}_2-{d}_2^{\ast}\right)\left({e}_{21}^2-{e}_{11}{e}_{21}\right)+\left({d}_3-{d}_3^{\ast}\right)\left({e}_{21}^2+{e}_{11}{e}_{21}\right)\\ {}\kern0.5em =-{d}_1{e}_{11}^2+a{e}_{12}^2+\left({x}_1-c\right){e}_{13}^2-\left({d}_2+{d}_3\right){e}_{21}^2+a{e}_{22}^2+\left({y}_1-c\right){e}_{23}^2-\left(1-{y}_3\right){e}_{11}{e}_{13}+\left({d}_2-{d}_3\right){e}_{11}{e}_{21}\\ {}\kern0.5em -\left(1-{z}_3\right){e}_{21}{e}_{23}+\left({d}_1-{d}_1^{\ast}\right){e}_{11}^2+\left({d}_2-{d}_2^{\ast}\right)\left({e}_{21}^2-{e}_{11}{e}_{21}\right)+\left({d}_2-{d}_2^{\ast}\right)\left({e}_{21}^2+{e}_{11}{e}_{21}\right)\\ {}\kern0.5em \le -{d}_1^{\ast }{e}_{11}^2+a{e}_{12}^2+\left({M}_1-c\right){e}_{13}^2-\left({d}_2^{\ast }+{d}_3^{\ast}\right){e}_{21}^2+a{e}_{22}^2+\left({M}_1-c\right){e}_{23}^2+\left({M}_3-c\right){e}_{11}{e}_{13}\\ {}\kern0.5em +\left({d}_2-{d}_3\right){e}_{11}{e}_{21}+\left({M}_3-1\right){e}_{21}{e}_{23}\\ {}\kern0.5em =-\left(\left|{e}_{11}\right|,\;\left|{e}_{12}\right|,\;\left|{e}_{12}\right|,\;\left|{e}_{21}\right|,\;\left|{e}_{22}\right|,\;\left|{e}_{23}\right|\right)\mathbf{P}{\left(\left|{e}_{11}\right|,\;\left|{e}_{12}\right|,\;\left|{e}_{12}\right|,\;\left|{e}_{21}\right|,\;\left|{e}_{22}\right|,\;\left|{e}_{23}\right|\right)}^{\mathrm{T}}\end{array} $$

where

$$ \mathbf{P}=\left(\begin{array}{cccccc}\hfill {d}_1^{\ast}\hfill & \hfill 0\hfill & \hfill \frac{-{M}_3+1}{2}\hfill & \hfill \frac{d_2^{\ast }+{d}_3^{\ast }}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -a\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \frac{-{M}_3+1}{2}\hfill & \hfill 0\hfill & \hfill -{M}_1+c\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \frac{d_2^{\ast }+{d}_3^{\ast }}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_2^{\ast }+{d}_3^{\ast}\hfill & \hfill 0\hfill & \hfill \frac{-{M}_3+1}{2}\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -a\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{-{M}_3+1}{2}\hfill & \hfill -{M}_1+c\hfill \end{array}\right) $$

(5)

The symmetrical matrix **P** will be positive definite if positive constants \( {d}_1^{\ast } \), \( {d}_2^{\ast } \), and \( {d}_3^{\ast } \) satisfy the following inequalities

$$ {d}_1\left({d}_2+{d}_3\right){a}^2{\left(c-{M}_1\right)}^2-1/4{a}^2\left(C-{M}_1\right)\left({d}_2+{d}_3\right)\left({M}_3^3-2{M}_2+1\right)>0 $$

(6)

Obviously, (6) holds if \( {d}_1^{\ast } \), \( {d}_2^{\ast } \), and \( {d}_3^{\ast } \) are large enough. That is, \( \overset{.}{V} \) is negative semi-definite for *e*
_{11}, *e*
_{12}, *e*
_{13}, *e*
_{21}, *e*
_{22}, *e*
_{23}, *d*
_{1}, *d*
_{2}, and *d*
_{3}. Thus, \( {e}_{11},\;{e}_{12},\;{e}_{13},\;{e}_{21},\;{e}_{22},\;{e}_{23},{d}_1-{d}_1^{\ast },{d}_2-{d}_2^{\ast },{d}_3-{d}_3^{\ast}\in {L}_{\infty } \).

Since \( \overset{.}{V}\le -{e}^{\mathrm{T}}\mathbf{P}e \) and **P** is positive definite, one has

$$ {\displaystyle {\int}_0^t{\lambda}_{\min }}(P)\left({e}_1^2+{e}_1^2+{e}_1^2\right)dt\le {\displaystyle {\int}_0^t{e}^{\mathrm{T}}} Pedt\le -{\displaystyle {\int}_0^t\overset{.}{V}dt}\le V\left({e}_{11}(0),{e}_{12}(0),{e}_{13}(0),{e}_{21}(0),{e}_{22}(0),{e}_{23}(0)\right) $$

where *λ*
_{min}(**P**) is the minimum eigenvalue of the symmetrical positive definite matrix **P**. It follows that *e*
_{11}, *e*
_{12}, *e*
_{13}, *e*
_{21}, *e*
_{22}, *e*
_{23} ∈ *L*
_{2}. We have \( {\overset{.}{e}}_{11},\;{\overset{.}{e}}_{12},\;{\overset{.}{e}}_{13},\;{\overset{.}{e}}_{21},\;{\overset{.}{e}}_{22},\;{\overset{.}{e}}_{23}\in {L}_{\infty } \) from error system (4). Therefore, by Barbalat’s lemma, *e*
_{11}, *e*
_{12}, *e*
_{13}, *e*
_{21}, *e*
_{22}, *e*
_{23} → 0 as *t* → ∞.

In numerical simulations for coupled system with Rössler systems, Fig. 1 shows the synchronization errors of the coupled system with three Rössler systems for *k*
_{1} = 0.1, *k*
_{2} = 0.3, and *k*
_{3} = 0.5. The initial states of three coupled systems are the following: *x*
_{1}(0) = 0.1, *x*
_{2}(0) = 0.1, *x*
_{3}(0) = 0.1; *y*
_{1}(0) = 0.2, *y*
_{2}(0) = 0.2, *y*
_{3}(0) = 0.2; *z*
_{1}(0) = 0.3, *z*
_{2}(0) = 0.3, *z*
_{3}(0) = 0.3; *d*
_{1}(0) = 0.5, *d*
_{2}(0) = 0.4, *d*
_{3}(0) = 0.3.

From Fig. 1, we can see that the three coupled Rössler chaotic systems realize synchronization in a short amount of time, so the above generic criteria of global chaos synchronization are effective.