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\begin_layout Standard

The state determinted by two temperatures (refering 
\begin_inset Formula $T_{0}$
\end_inset

as constant heat sink) :

\begin_inset Formula 
$$
\begin{cases}
C\frac{dT}{dt}+K\left(T-T_{0}\right)=K_{h}\left(T_{h}-T\right) & \, 

C_{h}\frac{dT_{h}}{dt}=P_{h}-K_{h}\left(T_{h}-T\right) & \,

\end{cases}
$$

\end_inset


Matrix representation:

\begin_inset Formula 
$$
$$

\end_inset


\end_layout

\begin_layout Standard

Eigenvalues of 
\begin_inset Formula $M$
\end_inset

\begin_inset Formula 
$$\begin{array}{rcl}
\lambda & = & -\frac{CK_{h}+C_{h}K+C_{h}K_{h}\pm\sqrt{C^{2}K_{h}^{2}-2CC_{h}KK_{h}+2CC_{h}K_{h}^{2}+C_{h}^{2}K^{2}+2C_{h}^{2}KK_{h}+C_{h}^{2}K_{h}^{2}}}{2CC_{h}}
\lambda & = & -\frac{1}{2}\left[\frac{K_{h}}{C_{h}}+\frac{K}{C}+\frac{K_{h}}{C}\pm\sqrt{\frac{K_{h}^{2}}{C_{h}^{2}}-2\frac{KK_{h}}{CC_{h}}+2\frac{K_{h}^{2}}{CC_{h}}+\frac{K^{2}}{C^{2}}+2\frac{KK_{h}}{C^{2}}+\frac{K_{h}^{2}}{C^{2}}}\right] 

\, & \, & \mbox{denoting}\;\begin{cases}

\gamma & =\frac{K}{C} 
\gamma_{m} & =\frac{K_{h}}{C} 
\gamma_{h} & =\frac{K_{h}}{C_{h}}
\end{cases} 
\lambda & = & -\frac{1}{2}\left[\gamma_{h}+\gamma+\gamma_{m}\pm\sqrt{\gamma_{h}^{2}-2\gamma\gamma_{h}+2\gamma_{m}\gamma_{h}+\gamma^{2}+2\gamma\gamma_{m}+\gamma_{m}^{2}}\right] 

\lambda & = & -\frac{1}{2}\left[\gamma+\gamma_{m}+\gamma_{h}\pm\sqrt{\left(\gamma+\gamma_{m}+\gamma_{h}\right)^{2}-4\gamma\gamma_{h}}\right]

\, & \, & \mbox{denotinng}\begin{cases}

\gamma+\gamma_{m}+\gamma_{h}=\gamma_{+} & \, 

\gamma\gamma_{h}=\gamma_{*}^{2} & \,

\end{cases} 
\, & \, & \gamma_{+}\gg\gamma_{*} 
\lambda & = & -\frac{1}{2}\left[\gamma_{+}\pm\sqrt{\gamma_{+}^{2}-4\gamma_{*}^{2}}\right] 

\lambda & \approx & -\frac{1}{2}\left[\gamma_{+}\pm\left(\gamma_{+}-2\frac{\gamma_{*}^{2}}{\gamma_{+}}\right)\right]

\, & \, & \begin{cases}

\lambda_{+} & =-\gamma_{+} 
\lambda_{-} & =-\frac{\gamma_{*}^{2}}{\gamma_{+}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}=-\frac{\gamma\gamma_{h}}{\gamma+\gamma_{m}+\gamma_{h}}
\end{cases}
\end{array}$$

\end_inset


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