  The multiple futile cycle is a phosphorylation system in which a molecular
substrate might be phosphorylated sequentially n times by means of an enzymatic
mechanism. The system has been studied mathematically using reaction network
theory and ordinary differential equations. It is known that the system might
have at least as many as 2[n/2]+1 steady states (where [x] is the integer part
of x) for particular choices of parameters. Furthermore, for the simple and
dual futile cycles (n=1,2) the stability of the steady states has been
determined in the sense that the only steady state of the simple futile cycle
is globally stable, while there exist parameter values for which the dual
futile cycle admits two asymptotically stable and one unstable steady state.
For general n, evidence that the possible number of asymptotically stable
steady states increases with $n$ has been given, which has led to the
conjecture that parameter values can be chosen such that [n/2]+1 out of
2[n/2]+1 steady states are asymptotically stable and the remaining steady
states are unstable.
  We prove this conjecture here by first reducing the system to a smaller one,
for which we find a choice of parameter values that give rise to a unique
steady state with multiplicity 2[n/2]+1. Using arguments from geometric
singular perturbation theory, and a detailed analysis of the centre manifold of
this steady state, we achieve the desired result.
