Next, we will study the lag module-phase synchronization for complex-valued neural networks. The module vector and phase vector of neural networks are considered: \(M(z_{i}^{r})=\sqrt {{{(x_{i}^{r})}^2}+{{(y_{i}^{r})}^2}}\) and \(P(z_{i}^{r})={\tan ^{ - 1}}\)\((y_{i}^{r}/x_{i}^{r})\), \(x_{i}^{r} \ne 0,i=1,...,n\), where\(x_{i}^{r}\) is the real component and \(y_{i}^{r}\) is the imaginary part. The module-phase matrix is described as\(\Theta (.)={(M(.),P(.))^T}\).
From the system (2) and (4), we conclude the lag module-phase error of system as:
$$\left\{ \begin{gathered} e_{i}^{M}(t)=M({y_i}(t)) - M\left( {{x_i}(t - \tau )} \right) \hfill \\ e_{i}^{P}(t)=P({y_i}(t)) - P\left( {{x_i}(t - \tau )} \right) \hfill \\ \end{gathered} \right.$$
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Where\(i=1,...,n\). The error of complex system is constructed:
$${\dot {e}_i}(t)=D\Theta (Y)\dot {Y} - D\Theta (X)\dot {X}$$
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\(D\Theta (X)\) is the Jacobian matrix of\(\Theta (X)\). And\(D\Theta (Y)\)is the Jacobian matrix of \(\Theta (Y)\), Then
\(D\Theta (Y(t))=\left( {\begin{array}{*{20}{c}} {\frac{{\partial {r_{n1}}(y)}}{{y_{1}^{r}}}}&{\frac{{\partial {r_{n1}}(y)}}{{y_{1}^{p}}}}& \cdots &{\frac{{\partial {r_{n1}}(y)}}{{y_{n}^{r}}}}&{\frac{{\partial {r_{n1}}(y)}}{{y_{n}^{p}}}} \\ {\frac{{\partial {\phi _{n1}}(y)}}{{y_{1}^{r}}}}&{\frac{{\partial {\phi _{n1}}(y)}}{{y_{1}^{p}}}}& \cdots &{\frac{{\partial {\phi _1}(y)}}{{y_{n}^{r}}}}&{\frac{{\partial {\phi _1}(y)}}{{y_{n}^{p}}}} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\frac{{\partial {r_{nn}}(y)}}{{y_{1}^{r}}}}&{\frac{{\partial {r_{nn}}(y)}}{{y_{1}^{p}}}}& \cdots &{\frac{{\partial {r_{nn}}(y)}}{{y_{n}^{r}}}}&{\frac{{\partial {r_{nn}}(y)}}{{y_{n}^{p}}}} \\ {\frac{{\partial {\phi _{nn}}(y)}}{{y_{1}^{r}}}}&{\frac{{\partial {\phi _{nn}}(y)}}{{y_{1}^{p}}}}& \cdots &{\frac{{\partial {\phi _{nn}}(y)}}{{y_{n}^{r}}}}&{\frac{{\partial {\phi _{nn}}(y)}}{{y_{n}^{p}}}} \end{array}} \right)\)
\(=\left( {\begin{array}{*{20}{c}} {\frac{{\partial {r_{n1}}(y)}}{{y_{1}^{r}}}}&{\frac{{\partial {r_{n1}}(y)}}{{y_{1}^{p}}}}& \cdots &0&0 \\ {\frac{{\partial {\phi _{n1}}(y)}}{{y_{1}^{r}}}}&{\frac{{\partial {\phi _{n1}}(y)}}{{y_{1}^{p}}}}& \cdots &0&0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0&0& \cdots &{\frac{{\partial {r_{nn}}(y)}}{{y_{n}^{r}}}}&{\frac{{\partial {r_{nn}}(y)}}{{y_{n}^{p}}}} \\ 0&0& \cdots &{\frac{{\partial {\phi _{nn}}(y)}}{{y_{n}^{r}}}}&{\frac{{\partial {\phi _{nn}}(y)}}{{y_{n}^{p}}}} \end{array}} \right)\)
$$=\left( {\begin{array}{*{20}{c}} {\frac{{y_{1}^{r}}}{{{r_1}}}}&{\frac{{y_{1}^{p}}}{{{r_1}}}}& \cdots &0&0 \\ {\frac{{ - y_{1}^{p}}}{{r_{1}^{2}}}}&{\frac{{y_{1}^{r}}}{{r_{1}^{2}}}}& \cdots &0&0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0&0& \cdots &{\frac{{y_{n}^{r}}}{{{r_n}}}}&{\frac{{y_{n}^{p}}}{{{r_n}}}} \\ 0&0& \cdots &{\frac{{ - y_{n}^{p}}}{{r_{n}^{2}}}}&{\frac{{y_{n}^{r}}}{{r_{n}^{2}}}} \end{array}} \right)$$
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Similarly:
$$D\Theta (X(t))=\left( {\begin{array}{*{20}{c}} {\frac{{x_{1}^{r}}}{{{{\bar {r}}_1}}}}&{\frac{{x_{1}^{p}}}{{{{\bar {r}}_1}}}}& \cdots &0&0 \\ {\frac{{ - x_{1}^{p}}}{{\bar {r}_{1}^{2}}}}&{\frac{{x_{1}^{r}}}{{\bar {r}_{1}^{2}}}}& \cdots &0&0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0&0& \cdots &{\frac{{x_{n}^{r}}}{{{{\bar {r}}_n}}}}&{\frac{{x_{n}^{p}}}{{{{\bar {r}}_n}}}} \\ 0&0& \cdots &{\frac{{ - x_{n}^{p}}}{{\bar {r}_{n}^{2}}}}&{\frac{{x_{n}^{r}}}{{\bar {r}_{n}^{2}}}} \end{array}} \right)$$
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The lag module- phase error of complex system is constructed as:
\({\dot {e}_i}(t)=D\Theta (Y(t))\dot {Y}(t) - D\Theta (X(t))X(t - \tau )\)
\(=\left( {\begin{array}{*{20}{c}} {\frac{{y_{i}^{r}}}{{M({y_i})}}}&{\frac{{y_{i}^{p}}}{{M({y_i})}}} \\ {\frac{{ - y_{i}^{p}}}{{{M^2}({y_i})}}}&{\frac{{y_{i}^{r}}}{{{M^2}({y_i})}}} \end{array}} \right)\left( \begin{gathered} - Cy_{i}^{r}(i)+Af(y_{i}^{r}(t))+Bh(y_{i}^{r}(t - {\tau _1} - \tau ))+{I^r}+U_{{is}}^{r} \hfill \\ - Cy_{i}^{p}+Af(y_{i}^{p}(t))+Bh(y_{i}^{p}(t - {\tau _1} - \tau ))+{I^p}+U_{{is}}^{p} \hfill \\ \end{gathered} \right)\)
\(- \left( {\begin{array}{*{20}{c}} {\frac{{x_{i}^{r}}}{{M({x_i})}}}&{\frac{{x_{i}^{p}}}{{M({x_i})}}} \\ {\frac{{ - x_{i}^{p}}}{{{M^2}({x_i})}}}&{\frac{{x_{i}^{r}}}{{{M^2}({x_i})}}} \end{array}} \right)\left( \begin{gathered} - Cx_{i}^{r}(i)+Af(x_{i}^{r}(t))+Bh(x_{i}^{r}(t - 2\tau ))+{I^r} \hfill \\ - Cx_{i}^{p}+Af(x_{i}^{p}(t))+Bh(x_{i}^{p}(t - 2\tau ))+{I^p} \hfill \\ \end{gathered} \right)\)
$$=\left( \begin{gathered} - Ce_{i}^{M}(t)+A(\frac{{{f^r}({y_i}(t))}}{{M({y_i})}} - \frac{{{f^r}({x_i}(t - {\tau _1} - \tau ))}}{{M({x_i})}})+B(\frac{{{h^r}({y_i}(t - \tau )}}{{M({y_i})}} - \frac{{{h^r}({x_i}(t - 2\tau ))}}{{M({x_i})}})+\frac{{U_{y}^{r}}}{{M({y_i})}} \hfill \\ - {C_0}e_{i}^{P}(t)+A(\frac{{{f^p}({y_i}(t))}}{{{M^2}({y_i})}} - \frac{{{f^p}({x_i}(t - {\tau _1} - \tau ))}}{{{M^2}({x_i})}})+B(\frac{{{h^p}({y_i}(t - \tau )}}{{{M^2}({y_i})}} - \frac{{{h^p}({x_i}(t - 2\tau ))}}{{{M^2}({x_i})}})+\frac{{U_{x}^{p}}}{{{M^2}({x_i})}} \hfill \\ \end{gathered} \right)$$
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We command:
\({F_r}=\frac{{{f^r}({y_i}(t))}}{{M({y_i})}} - \frac{{{f^r}({x_i}(t - {\tau _1} - \tau ))}}{{M({x_i})}}\) , \({H_r}=\frac{{{h^r}({y_i}(t - \tau )}}{{M({y_i})}} - \frac{{{h^r}({x_i}(t - 2\tau ))}}{{M({x_i})}}\)
\({F_p}=\frac{{{f^p}({y_i}(t))}}{{{M^2}({y_i})}} - \frac{{{f^p}({x_i}(t - {\tau _1} - \tau ))}}{{{M^2}({x_i})}}\) ,\({H_p}=\frac{{{h^p}({y_i}(t - \tau )}}{{{M^2}({y_i})}} - \frac{{{h^p}({x_i}(t - 2\tau ))}}{{{M^2}({x_i})}}\)
In order to study the lag module-phase synchronization for complex system, the adaptive controller are constructed as:
$${u_i}=\left\{ \begin{gathered} \left\{ {[ - k_{i}^{M}M({y_i})e_{i}^{M}x - {\alpha _1}M({y_i})(\left\| A \right\|\left\| {{F_r}} \right\|+\left\| B \right\|\left\| {{H_r}} \right\|)x]} \right. \\ \left. {+[k_{i}^{p}{M^2}({y_i})e_{i}^{M}y - {\alpha _2}{M^2}({y_i})(\left\| A \right\|\left\| {{F_p}} \right\|+\left\| B \right\|\left\| {{H_p}} \right\|)y]} \right\}/M{({y_i})^2} \\ [\left\{ {[ - k_{i}^{M}M({y_i})e_{i}^{M}x - {\alpha _1}M({y_i})(\left\| A \right\|\left\| {{F_r}} \right\|+\left\| B \right\|\left\| {{H_r}} \right\|)x]} \right. \\ \left. { - [k_{i}^{p}{M^2}({y_i})e_{i}^{M}y - {\alpha _2}{M^2}({y_i})(\left\| A \right\|\left\| {{F_p}} \right\|+\left\| B \right\|\left\| {{H_p}} \right\|)y]} \right\}/M{({y_i})^2} \\ \end{gathered} \right.$$
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where\(\dot {k}_{i}^{M}={\delta _i}{(e_{i}^{M}(t))^T}e_{i}^{M}(t)\),\(\dot {k}_{i}^{P}={\delta _i}{(e_{i}^{P}(t))^T}e_{i}^{P}(t)\),\({\alpha _1}>0\), \({\alpha _2}>0\).\({K^M}{\text{=}}diag(k_{1}^{M},k_{2}^{M},...,k_{n}^{M})\), \({K^P}{\text{=}}diag(k_{1}^{P},k_{2}^{P},...,k_{n}^{P})\). \(\left\| . \right\|\)represents the norm relationship. At the same time the controller is also described as:
$$U=\left\{ \begin{gathered} - k_{i}^{M}M({y_i})e_{i}^{M} - {\alpha _1}M({y_i})(\left\| A \right\|\left\| {{F_r}} \right\|{\text{+}}\left\| B \right\|\left\| {{H_r}} \right\|) \hfill \\ - k_{i}^{P}M{({y_i})^2}e_{i}^{P} - {\alpha _1}M{({y_i})^2}(\left\| A \right\|\left\| {{F_q}} \right\|{\text{+}}\left\| B \right\|\left\| {{H_q}} \right\|) \hfill \\ \end{gathered} \right.$$
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The Lyapunov theory, the function is constructed as:
$$\begin{gathered} V(t,{e_i}(t))=\sum\limits_{{i=1}}^{n} {{{(e_{i}^{M}(t))}^T}e_{i}^{M}(t)} +\sum\limits_{{i=1}}^{n} {{{(e_{i}^{P}(t))}^T}e_{i}^{P}(t)} \\ +\sum\limits_{{i=1}}^{n} {\frac{1}{{{\delta _i}}}{{(k_{i}^{M} - {s_i})}^2}} +\sum\limits_{{i=1}}^{n} {\frac{1}{{{\delta _i}}}{{(k_{i}^{P} - {s_i})}^2}} \\ \end{gathered}$$
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Then:
\(\begin{gathered} \dot {V}(t,e(t))=2\sum\limits_{{i=1}}^{n} {{{(e_{i}^{M}(t))}^T}\dot {e}_{i}^{M}(t)} +2\sum\limits_{{i=1}}^{n} {{{(e_{i}^{P}(t))}^T}\dot {e}_{i}^{P}(t)} \\ +2\sum\limits_{{i=1}}^{n} {k_{i}^{M}{{(e_{i}^{r}(t))}^T}e_{i}^{M}(t)} - 2\sum\limits_{{i=1}}^{n} {{s_i}{{(e_{i}^{M}(t))}^T}e_{i}^{M}(t)} \\ +2\sum\limits_{{i=1}}^{n} {k_{i}^{P}{{(e_{i}^{P}(t))}^T}e_{i}^{P}(t)} - 2\sum\limits_{{i=1}}^{n} {{s_i}{{(e_{i}^{P}(t))}^T}e_{i}^{P}(t)} \\ \end{gathered}\)
\(\begin{gathered} =2\sum\limits_{{i=1}}^{n} {{{(e_{i}^{M}(t))}^T}[ - Ce_{i}^{M}(t)+A(\frac{{{f^r}({y_i}(t))}}{{M({y_i})}} - \frac{{{f^r}({x_i}(t - {\tau _1} - \tau ))}}{{M({x_i})}})} \\ +B(\frac{{{h^r}({y_i}(t - \tau )}}{{M({y_i})}} - \frac{{{h^r}({x_i}(t - 2\tau ))}}{{M({x_i})}})+\frac{{U_{y}^{r}}}{{M({y_i})}}] \\ \end{gathered}\)
\(\begin{gathered} +2\sum\limits_{{i=1}}^{n} {{{(e_{i}^{P}(t))}^T}[ - {C_0}e_{i}^{P}(t)+A(\frac{{{f^p}({y_i}(t))}}{{{M^2}({y_i})}} - \frac{{{f^p}({x_i}(t - {\tau _1} - \tau ))}}{{{M^2}({x_i})}})} \\ +B(\frac{{{h^p}({y_i}(t - \tau )}}{{{M^2}({y_i})}} - \frac{{{h^p}({x_i}(t - 2\tau ))}}{{{M^2}({x_i})}})+\frac{{U_{x}^{p}}}{{{M^2}({x_i})}}] \\ \end{gathered}\)
\(\begin{gathered} +2\sum\limits_{{i=1}}^{n} {k_{i}^{M}{{(e_{i}^{r}(t))}^T}e_{i}^{M}(t)} - 2\sum\limits_{{i=1}}^{n} {{s_i}{{(e_{i}^{M}(t))}^T}e_{i}^{M}(t)} \\ +2\sum\limits_{{i=1}}^{n} {k_{i}^{P}{{(e_{i}^{P}(t))}^T}e_{i}^{P}(t)} - 2\sum\limits_{{i=1}}^{n} {{s_i}{{(e_{i}^{P}(t))}^T}e_{i}^{P}(t)} \\ \end{gathered}\)
\(\leqslant 2\sum\limits_{{i=1}}^{n} {{{(e_{i}^{M}(t))}^T}[ - Ce_{i}^{M}(t)+\left\| A \right\|\left\| {{F_r}} \right\|+\left\| B \right\|\left\| {{H_r}} \right\|+\frac{{U_{y}^{r}}}{{M({y_i})}}]}\)
\(+2\sum\limits_{{i=1}}^{n} {{{(e_{i}^{P}(t))}^T}[ - Ce_{i}^{P}(t)+\left\| A \right\|\left\| {{F_p}} \right\|+\left\| B \right\|\left\| {{H_p}} \right\|+\frac{{U_{y}^{p}}}{{{M^2}({y_i})}}]}\)
\(\begin{gathered} +2\sum\limits_{{i=1}}^{n} {k_{i}^{M}{{(e_{i}^{r}(t))}^T}e_{i}^{M}(t)} - 2\sum\limits_{{i=1}}^{n} {{s_i}{{(e_{i}^{M}(t))}^T}e_{i}^{M}(t)} \\ +2\sum\limits_{{i=1}}^{n} {k_{i}^{P}{{(e_{i}^{P}(t))}^T}e_{i}^{P}(t)} - 2\sum\limits_{{i=1}}^{n} {{s_i}{{(e_{i}^{P}(t))}^T}e_{i}^{P}(t)} \\ \end{gathered}\)
$$=\sum\limits_{{i=1}}^{n} {{{(e_{i}^{M}(t))}^T}( - 2C - {s_i})e_{i}^{M}(t)} +\sum\limits_{{i=1}}^{n} {{{(e_{i}^{P}(t))}^T}( - 2C - {s_i})e_{i}^{P}(t)}$$
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If\(c+{s_i}>0\),\(\dot {V}(t)>0\). According to stability theory of neural networks, the lag module-phase synchronization of complex-valued neural networks with mixed delays is realized.