| Abstract: |
| In a system $\dot{X}=\Omega X+\epsilon F(X,t)$ where $\Omega$ is a matrix with all imaginary eigenvalues, the weak nonlinearity $F$ can lead to highly nontrivial long time dynamics by interacting with fast oscillations generated by $\Omega$. The method of averaging can characterize this interaction and provide an approximation of the long time effect of the nonlinearity. This talk will describe a simple way for improving the accuracy of 1st-order averaging, without having to resort to the much harder task of 2nd-order averaging. The efficacy of this improvement will also be demonstrated by examples, such as a new engineering device for wireless energy transfer, the Fermi-Pasta-Ulam problem, and a weakly-nonlinear advection-reaction PDE (note the method can be generalized to $\Omega$ being an anti-Hermitian operator). If time permits, numerical averaging will be discussed too. |
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