Asymptotics of cellular buckling close to the Maxwell load

We study the deformation of ail elastic strut oil a nonlinear Winkler foundation subjected to ail axial compressive load P. Using multi-scale analysis and numerical methods we describe the localized., cellular, post-buckled state of the system when P is removed from the critical load P := 2. The solutions, and their modulation frequencies, differ significantly from those predicted by weakly nonlinear analysis very close to P = 2. In particular, when P approaches the Maxwell load P-M, the localized solutions approach a large-amplitude heteroclinic connection between ail unbuckled solution and a periodic solution. An asymptotic description of P-M in terms of the system parameters is given. The agreement between the numerical calculations and the asymptotic approximations is striking.