Iterated averaging of three-scale oscillatory systems
A theory of iterated averaging is developed for a class of highly oscillatory ordinary differential equations (ODEs) with three well separated time scales \[ \begin{aligned} x'&= \varepsilon^{-1}\tilde f_1(x,y,z)+\tilde f_0(x,y,z), x(0)=x_0,\\ y'&=\varepsilon^{-1}\tilde g_1(x,y,z)+\tilde g_0(x,y,z), y(0)=y_0,\\ z'&=\varepsilon^{-2}\tilde h_2(x,y,z), \qquad z(0)=z_0, \end{aligned} \] where the separation into the three components \(x,y\) and \(z\) is according to the time scale on which the variables evolve. The solutions of these equations are assumed to be (almost) periodic in the fastest time scales. It is proved that the dynamics on the slowest time scale can be approximated by an effective ODE obtained by averaging out oscillations. In particular, the effective dynamics of the considered class of ODEs is always deterministic and does not show any stochastic effects. This seems to be in contrast to systems in which the dynamics on the fastest time scale is mixing. The systems are studied from three perspectives: first, using the tools of averaging theory; second, by formal asymptotic expansions and third, by averaging with respect to fast oscillations using nested convolutions with averaging kernels.
Three-scales oscillatory dynamical system
nested convolutions with averaging kernels
scientific article; zbMATH DE number 6382035