# American Institute of Mathematical Sciences

May  2021, 20(5): 1821-1831. doi: 10.3934/cpaa.2021042

## A note on the energy transfer in coupled differential systems

 Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

* Corresponding author

Received  January 2021 Revised  January 2021 Published  May 2021 Early access  March 2021

We study the energy transfer in the linear system
 $\begin{cases} \ddot u+u+\dot u = b\dot v\\ \ddot v+v-\epsilon \dot v = -b\dot u \end{cases}$
made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter
 $b$
.
Citation: Monica Conti, Lorenzo Liverani, Vittorino Pata. A note on the energy transfer in coupled differential systems. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1821-1831. doi: 10.3934/cpaa.2021042
##### References:

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##### References:
Plot of ${{\mathtt{E}}}$ for $\epsilon = 1$ and $b = 0.99$ (black), $b = 1$ (blue) and $b = 1.01$ (red)
Parametric plot of $t\mapsto (u(t), \dot u(t))$ for $\epsilon = 1$ and $b = \sqrt{\frac{23}{13}+\frac{13}{23}-1}$
${{\mathtt{E}}}$ for $\epsilon = 1$ and $\boldsymbol{z}_0 = (1, 0, 0, 0)$ with different values of $b$
${{\mathtt{E}}}$ for $\epsilon = 1$ and $\boldsymbol{z}_0 = (1, 0.5, 0, 0)$ with different values of $b$
Numerical $u$ (blue) vs asymptotic $u$ (red) for $\epsilon = 1$ with different values of $b$ (and different time-scales)
Numerical $v$ (blue) vs asymptotic $v$ (red) for $\epsilon = 1$ with different values of $b$ (and different time-scales)
Plot of ${{\mathtt{E}}}$ for $\epsilon = 0.5$ and $b = \sqrt{0.5}-0.1$ (black), $b = \sqrt{0.5}$ (blue) and $b = \sqrt{0.5}+0.1$ (red)
Parametric plot of $t\mapsto (u(t), \dot u(t))$ for $\epsilon = \frac12$ and $b = 1$
Parametric plot of $t\mapsto (u(t), \dot u(t))$ for $\epsilon = \frac12$ and $b = 2$
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