Consider a nonlinear system with two locally stable fixed points $s_1$ and $s_2$ which have domains of attraction $D_1$ and $D_2$ respectively.
Let $d_1$ and $d_2$ be domains of attraction of one system variable ($x_i$) for $s_1$ and $s_2$ respectively.
$d_1$ and $d_2$ are of dimension $1$ and in fact, $d_1$ and $d_2$ are the projections of $D_1$ and $D_2$ on $x_i$ space.
Please let me know whether the following argument is possible:
As the system's dimension increases, sizes of $d_1$ and $d_2$ keep increasing simultaneously with a same or different rates. Or if you think that this statement is contradictory, why?