Say I have a function $f(x)$ and its decomposition, which consist of alternatively applying $f_1(x)$ and $f_2(x)$ a number of times, i.e.
$$f(x) = (f_2 \circ f_1 \circ \ldots \circ f_2\circ f_1)(x)$$
Now construct a new function, where we have rearranged the components
$$\tilde{f}(x) = (f_2 \circ \ldots \circ f_2\circ f_1 \circ \ldots \circ f_1)(x)$$
Is it now possible to bound the norm $\|f(x)-\tilde{f}(x)\|$ between the two functions? If so, what are the possible requirements on $f(x)$. As I am working in the context of differential equations, linear growth and Lipschitz continuity are the standard assumptions, though You are allowed to assume more.