We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$: $$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$ Suppose that $\omega$ is not resonant. Show that more generally, if $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$, then $F ( t ) = o ( t )$ when $t \rightarrow + \infty$.
We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$:
$$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$
Suppose that $\omega$ is not resonant.
Show that more generally, if $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$, then $F ( t ) = o ( t )$ when $t \rightarrow + \infty$.