Non-vanishing basin of attraction with respect to a parametric variation and center manifold

When the origin of a nonlinear system having a parameter is locally asymptotically stable, the size of its basin of attraction may also depend on the parameter. If the parameter variation is confined within a compact interval and if the origin is locally exponentially stable for each parameter, then it is well-known that the radius of basin of attraction does not shrink to zero under the variation of parameter. We relax this condition up to the case where the origin loses exponential stability for a certain parameter. This is done by focusing on the behavior of the system on a parametrized center manifold. In addition, by introducing the concept of stability with respect to a positively invariant subset of the state-space, the proposed analysis is applicable to the case where the system experiences bifurcation of splitting or merging equilibria, which often appears in biological systems. The stability property considered here is called for with the boundary layer system in the singular perturbation context, or with the system having slowly varying inputs. The main results establish that the non-vanishing basin of attraction (relative to a positively invariant set) is guaranteed by confirming the similar property just for the reduced order system on a parametrized center manifold.