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\begin_body
\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
\end_layout
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References
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