There are many situations in which identical systems are connected in a linear way. We consider such a network, i.e., a network of the following type:

$$\begin{aligned} x_{i}\left( t+1\right) & = f\left( x_{i}\left( t\right) \right) +\sum \limits _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^{n}a_{ij}\cdot \left[ f\left( x_{j}\left( t\right) \right) -f\left( x_{i}\left( t\right) \right) \right] \text {, } \quad \forall _{i=1,...,n} \nonumber \\ & \Leftrightarrow {} \nonumber \\ x_{i}\left( t+1\right) & = (1-\sum \limits _{\begin{array}{c} j=1 \\ j\ne i \end{array}} ^{n}a_{ij})\cdot f\left( x_{i}\left( t\right) \right) +\sum \limits _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^{n}a_{ij}\cdot f\left( x_{j}\left( t\right) \right) \text {, } \quad \forall _{i=1,\ldots,n} \nonumber \\ & \Leftrightarrow {} \nonumber \\ \overrightarrow{x}\left( t+1\right) & = \left( I_{n}+A\right) \cdot \overrightarrow{f}\left( \overrightarrow{x}\left( t\right) \right), \end{aligned}$$

(1)

where *n* is the number of identical systems connected, *f* is their free dynamic, \(\overrightarrow{x}\left( t\right) =\left[ \begin{array}{cccc} x_{1}\left( t\right)&x_{2}\left( t\right)&\ldots&x_{n}\left( t\right) \end{array} \right] ^{T}\), \(\overrightarrow{f}\left( \overrightarrow{x}\left( t\right) \right) = \left[ \begin{array}{cccc} f\left( x_{1}\left( t\right) \right)&f\left( x_{2}\left( t\right) \right)&\ldots&f\left( x_{n}\left( t\right) \right) \end{array} \right] ^{T}\), \(a_{ii}=-\sum \nolimits _{\begin{subarray}{c} j=1 \\ j\ne i \end{subarray}}^{n}a_{ij}\) and \(A=\left[ a_{ij}\right] \). Such a network admits the completely synchronized solution \(s(t)\cdot \overrightarrow{1}\), with \(\overrightarrow{1 }=\left[ 11...1\right] ^{T}\) and *s*(*t*) such that \(s(t+1)=f(s(t))\). In fact, since \(\lambda =0\) is an eigenvalue of the coupling matrix *A*, corresponding to the eigenvector \(\overrightarrow{1}\), we have

$$\begin{aligned} \left( I_{n}+A\right) \cdot f(s(t)\cdot \overrightarrow{1}) & = \left( I_{n}+A\right) \cdot f(s(t))\cdot \overrightarrow{1} \\ & = s(t+1)\cdot \left( \overrightarrow{1}+A\cdot \overrightarrow{1}\right) =s(t+1)\cdot \overrightarrow{1}. \end{aligned}$$

Adding a new identical dynamical system, *y*, to the network, we want to analyze the possibility that, after a transient period, this new node imposes its iterates to all the others. In a coupling, we obtain that using a one-way connection with a coupling strength greater than \(1-e^{-h}\), where *h* is the Lyapunov exponent of the coupled systems [17]. So, we consider that the new node is one-way connected to all nodes of the network, i.e., we consider the new network

$$\begin{array}{c}\left\{ \begin{array}{l} x_{i}\left( t+1\right) = f\left( x_{i}\left( t\right) \right) +\sum \limits _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^{n}a_{ij}\cdot \left[ f\left( x_{j}\left( t\right) \right) -f\left( x_{i}\left( t\right) \right) \right] +\epsilon \cdot \left[ f\left( y(t\right) -f\left( x_{i}\left( t\right) \right) \right] \text {, } \quad \forall _{i=1,...,n} \\ y(t+1) = f\left( y\left( t\right) \right) \end{array} \right. \nonumber \\ \Leftrightarrow \nonumber \\ \left\{ \begin{array}{l} x_{i}\left( t+1\right) = (1-\sum \limits _{\begin{array}{c} j=1 \\ j\ne i \end{array}} ^{n}a_{ij}-\epsilon )\cdot f\left( x_{i}\left( t\right) \right) +\sum \limits _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^{n}a_{ij}\cdot f\left( x_{j}\left( t\right) \right) +\epsilon \cdot f( y( t)) , \ \quad \forall _{i=1,...,n} \\ y(t+1) = f\left( y\left( t\right) \right) \end{array} \right. \nonumber \\ \Leftrightarrow \nonumber \\ \overrightarrow{x_{0}}\left( t+1\right) = \left( I_{n+1}+A_{0}\right) \cdot \overrightarrow{f}\left( \overrightarrow{x_{0}}\left( t\right) \right), \end{array}$$

(2)

where \(\overrightarrow{x_{0}}\left( t\right) =\left[ \begin{array}{ccccc} x_{1}\left( t\right)&x_{2}\left( t\right)&...&x_{n}\left( t\right)&y\left( t\right) \end{array} \right] ^{T}\) and \(A_{0}=\left[ \begin{array}{cc} A-\epsilon I_{n} &{} \overrightarrow{1} \epsilon \\ \overrightarrow{0}^{T} &{} 0 \end{array} \right] \) and we say that the new node *y* is full-commanding the network *A*.

In order that the new node imposes its free evolution to all the others nodes, it is needed that \(y(t)\cdot \overrightarrow{1}\), with *y*(*t*) such that \(y(t+1)=f(y(t))\), be an exponentially stable solution of (2). The following proposition determines conditions for that to happen.

###
**Proposition 1**

*If* \(\left| \epsilon -\left( 1+\lambda _{i}\right) \right| <e^{-h}\)*, for all the eigenvalues* \(\lambda _{i}\) *of the diagonalizable coupling matrix* *A* *of the network* (1)*, then the network is full-commanded by a new node* *y**, i.e.,* \(\overrightarrow{x_{0}}\left( t\right) =y(t)\cdot \overrightarrow{1}\)*, with* *y*(*t*) *such that* \(y(t+1)=f(y(t))\)*, is an exponentially stable solution of* (2). *If* \(\left| \epsilon -\left( 1+\lambda _{i}\right) \right| >e^{-h}\) *for an eigenvalue* \(\lambda _{i}\)*, then network* (1) *is not full-commanded by a new node*.

As proved in [14] this is just the result of applying the following proposition (a variation of similar others [15, 16]) to network (2), i.e., the result of considering that the matrix *A* of the following proposition is \(A_0\).

###
**Proposition 2**

*Considering the dynamical network* (1) *with* *A* *a diagonalizable matrix such that* \(\lambda _{1}=0\) *is an eigenvalue of multiplicity 1, if all the other eigenvalues* \(\lambda _{i}\) (\(i=2,...,n\)) *are such that* \( \left| 1+\lambda _{i}\right| <e^{-h}\)*, where* *h* *is the Lyapunov exponent of the nodes, then the completely synchronized solution* \( \overrightarrow{x}\left( t\right) =x_{1}(t)\cdot \overrightarrow{1}\)*, with* \( x_{1}(t)\) *satisfying* \(x_{1}\left( t+1\right) =f\left( x_{1}\left( t\right) \right) \)*, is exponentially stable. If* \(\left| 1+\lambda _{i}\right| >e^{-h}\) *for a non-zero eigenvalue* \(\lambda _{i}\)*, then there is no exponentially stable completely synchronized solution*.

All the same, we define full-command-window and full-commandable network, in the following way.

###
**Definition 1**

We define full-command-window (*FCW*) of the network (1) as the open set of values of the commanding coupling strength \(\epsilon \in [0,1]\) for which the synchronized solution \(\overrightarrow{x_{0}}\left( t\right) =y(t)\cdot \overrightarrow{1}\) is an exponentially stable solution of (1). If \(FCW\ne \varnothing \), we say that network (1) is full-commandable.

Proposition 1 determines that

$$\begin{aligned} FCW=\underset{i=1}{\overset{n}{\mathop {\displaystyle \bigcap }}}\left\{ \epsilon \in [0,1]:\left| \epsilon -\left( 1+\lambda _{i}\right) \right| <e^{-h}\right\}. \end{aligned}$$

As we noted in [14], there are networks that are not full-commandable, for instance the ones that have a diagonalizable coupling matrix *A* with an eigenvalue such that \({\text {Im}}(\lambda )>e^{-\mu }\) or such that its distance to another eigenvalue is greater than \(2e^{-\mu }\). We also presented in that paper the following results that are useful for obtaining the ones we add in this paper.

###
**Proposition 3**

*Considering a network* (1) *such that* *A* *is diagonalizable, all its eigenvalues are real and* \(\lambda _{n}\) *is the smallest one, then the network is full-commandable if* \(\lambda _{n}>-2e^{-h}\) *and* *FCW* *reduces to*

$$\begin{aligned} FCW=\big ] 1-e^{-h},1+\lambda _{n}+e^{-h}\big [ \cap [0,1]. \end{aligned}$$

###
**Proposition 4**

*A completely disconnected network, i.e., a network* (1) *with a zero matrix* *A**, is full-commandable and its full-command-window is* \(\big ] 1-e^{-h},1 \big ].\).

###
**Proposition 5**

*A completely connected network, i.e., a network* (1) *with*

$$\begin{aligned} A=c\cdot \left[ \begin{array}{ccccc} -(n-1) &{} 1 &{} 1 &{} ... &{} 1 \\ 1 &{} -(n-1) &{} 1 &{} ... &{} 1 \\ 1 &{} 1 &{} -(n-1) &{} ... &{} 1 \\ ... &{} ... &{} ... &{} ... &{} ... \\ 1 &{} 1 &{} 1 &{} ... &{} -(n-1) \end{array} \right] \text {,} \end{aligned}$$

*where* *c* *is the global coupling strength, is full-commandable if* \(nc<2e^{-h} \) *and its full-command-window is* \(\big ] 1-e^{-h},1-nc+e^{-h}\big [ \cap [0,1] \).

We note that for a network satisfying the conditions of any of the these propositions in order to be full-commandable, the smallest commanding coupling constant is \(\epsilon =1-e^{-h} \), a value that does not depend on the structure of the network, it only depends on the dynamic of the nodes. This value is exactly the same that it is needed in a one-way linear coupling for a dynamical system to command the other one [17]. Further, a completely connected network is as more resistant to a full-command as greater is the network (i.e., as greater is *n*) and as stronger are the connections between the dynamical systems (i.e., as greater is *c*).