I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the following necessary bound.
Let $A$ be an $n\times n$-skew-symmetric matrix, $c \in \mathbb{R}^n$, and $M,\epsilon>0$. Define the bump function $\phi(z):= |z-1| |z+1| I_{|z|\leq 1}$. What is the following quantity equal to (or upper-bounded by): $$ \alpha(A,c,M,\epsilon):=\inf_{\|x\|,\|y\|\leq M,\|x-y\|>\epsilon} \left\| e^{A\phi(\|x-c\|)}(x-c) - e^{A\phi(\|y-c\|)}(y-c) \right\|. $$
All, I know, for-sure so far is that this quantity is positive.
