\begin{displaymath}\partial_t \eta (t) = g(\eta(t),\varphi(t))\end{displaymath}

$\eta$ $\varphi$

\begin{displaymath}
\varphi(t) = f(\eta(t),\varphi(t))
\end{displaymath}

$t$ $t + \delta t$ $\delta \eta$ $\delta \varphi$

\begin{displaymath}\pmatrix{A & B\cr
-C^+ & I-D\cr} \pmatrix{\delta \eta\cr
\delta \varphi\cr} = \pmatrix{\Gamma\cr
\Omega\cr}\end{displaymath}

$f$ $g$ $\delta t$

\begin{displaymath}\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
}\end{displaymath}


\begin{displaymath}\left\{\eqalign{\partial_t \eta _{prey} &= a \eta _{prey} - a...
... \eta _{pred} &= -c \eta _{pred} + c \varphi _{meet}\cr}\right.\end{displaymath}


\begin{displaymath}\varphi _{meet} = \eta _{prey}\eta _{pred}\end{displaymath}

$a$ $c$ $\varphi_{meet}$ $\eta_{prey}$ $\eta_{pred}$ $\delta \eta$ $\varphi=f(\eta(t-dt)+d\varphi$ $d\varphi$ $\varphi=f(\eta),\varphi)$ $10^{-3}$

\begin{displaymath}\partial_{\eta} g(\eta(t),\varphi(t));
\end{displaymath}


\begin{displaymath}\partial_{\varphi} g(\eta(t),\varphi(t));
\end{displaymath}


\begin{displaymath}\partial_{\eta} f(\eta(t),\varphi(t));
\end{displaymath}


\begin{displaymath}\partial_{\varphi} f(\eta(t),\varphi(t));
\end{displaymath}

$(I-D)$ $k$

\begin{displaymath}\left\{\eqalign{\partial_t \eta _{k} ^{pos} &= \eta _{k} ^{ve...
... + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k \cr}\right.\end{displaymath}


\begin{displaymath}\left\{\eqalign{
\varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- ...
...spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\cr}\right.\end{displaymath}


\begin{displaymath}\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
\eta ^{vel}_{0} &= 0\...
...arphi ^{spr}_{N+1} &= 0\cr
\varphi ^{dmp}_{N+1} &= 0\cr}\right.\end{displaymath}

$m_k$ $r_k$ $d_k$ $N$ $N+1$

\begin{displaymath}\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f + f^g \log f\partial_x g\cr
&= f^{g-1}(g\partial_x f + f\partial_x g)\cr}\end{displaymath}

$x/\Delta$ $\omega$ $h$

\begin{displaymath}
\omega = h ( \eta , \varphi)
\end{displaymath}

$\eta_1(t)$ $\eta_1(t=0)$ $t=0$ $\phi_{j}(t)$ $\eta_i(t=0)$ $h(t)$ $J$ $T$

\begin{displaymath}
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
\end{displaymath}

$\psi$ $l$

\begin{displaymath}\eqalign{
\partial_t \eta (t) &= g(\eta(t),\varphi(t)) + W(t)...
...varphi(t))\cr
\omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
}\end{displaymath}

$t=s_i$ $\mu$ $Q$ $\nu$ $R$ $W$

\begin{displaymath}\left\{\eqalign{
\partial_t \eta_1 &= a_{11} \eta_1 + a_{12} ...
...varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
}\right.\end{displaymath}


\begin{displaymath}\left\{\eqalign{
\varphi _1 &= \eta _1\cr
\varphi _2 &= \eta _2\cr
\varphi _3 &= \eta _3
}\right.\end{displaymath}


\begin{displaymath}\left\{\eqalign{
\omega _1 &= \varphi _1 + \nu_1\cr
\omega _2 &= \eta _2 + \nu_2 \cr
\omega _3 &= \eta _3 + \nu_3
}\right.\end{displaymath}

$g(\tau)$ $\tau$ $g(\tau;t)$ $2$ $\tau$ $0.2$ $\sqrt{\sqrt{2}}$ $g(t,\tau)$ $A_{st}$ $A_{st} + A_{st}^\dagger$

\begin{displaymath}
U w V^\dagger
\end{displaymath}

$U$ $w$ $V$ $\Phi(t+\delta t,t)=\exp{A_{st} \delta t}$ $\Phi$ $\Phi(t,0)$