A constructive theory of isolas suported by parabolic cusps, centers and bifurcations points

A technique is developed to construct centered and noncentered isolas supported by parabolic cusps, centers and bifurcation points. Noncentered isolas occur in conjunction with nonisolated solution branches of a nonlinear system. They may merge with or bud from the neighboring branches as an imperfection parameter deviates from zero. The centered isolas studied occur either as single loops or as pairs of loops. A single isola may vanish into a point and then emerge as two isolas when the imperfection has changed sign. Equations that characterize the supporting singularities are given. The imperfection scale(s) of the isola is (are) identified. A perturbation procedure is employed to construct the isola solutions. The theory is applied to the buckling of a cylindrical shell and the analysis of steady-state isolas occurring in a reaction-diffusion system.