# 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  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:

show all references

##### 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 $b = 2$
 [1] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [2] Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559 [3] Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations & Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019 [4] Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273 [5] Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465 [6] Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 [7] Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599 [8] Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations & Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008 [9] Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773 [10] Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609 [11] Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020082 [12] Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065 [13] Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001 [14] Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521 [15] Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935 [16] Mohammed Aassila. On energy decay rate for linear damped systems. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851 [17] Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017 [18] Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721 [19] Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048 [20] Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583

2019 Impact Factor: 1.105