The differential equation with k dimensions has the form dFdx=F(x,ϵ), where x is the variable vector and ϵ is a parameter. I focus on the steady state, which yields the solution of F(x,ϵ)=0. Assume that the equation has a perturbation solution x=aϵ+O(ϵ2) for the parameter ϵ→0 and another perturbation solution x=b0−b1ϵ−1+O(ϵ−2) for the parameter ϵ→∞.
Now I already know (from numerical simulations) that the above two kinds of solutions i.e., xi→0 for i∈I and xj→b0 for j∈J, coexists for the parameter ϵ of a specific value interval. (K={1,2,...,k} and the sets I,J⊆K, I∩J=∅).
My question is how to identify the solution sets I and J by perturbation method? or I should consider its Hamiltonian?