We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$, $\alpha(0)=(0,0)$. From question 15b, $\tilde{\alpha}'(t) = \omega + x\nu + x\varepsilon(x,t)$. Let $T > 0$ be fixed. Show that there exists a function $\eta : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\alpha ( t ) = ( \omega + x \nu ) t + x ( h ( 0,0 ) - h ( \omega t ) ) + x \eta ( x , t )$$ and $\sup _ { t \in [ 0 , T ] } \| \eta ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.
We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$, $\alpha(0)=(0,0)$. From question 15b, $\tilde{\alpha}'(t) = \omega + x\nu + x\varepsilon(x,t)$.
Let $T > 0$ be fixed. Show that there exists a function $\eta : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that
$$\alpha ( t ) = ( \omega + x \nu ) t + x ( h ( 0,0 ) - h ( \omega t ) ) + x \eta ( x , t )$$
and $\sup _ { t \in [ 0 , T ] } \| \eta ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.