Realizing Saddle-Node Bifurcations from Finite Data

Abstract

Given a finite set of data generated by an unknown ordinary differential equation it is impossible to exactly determine the associated vector field, and hence, bifurcation theory tells us that it is impossible, in general, to correctly characterize the underlying dynamics. In this paper, we bypass the effort of obtaining an analytic approximation of the vector field, and we adopt an approach based on Occam's razor: identify the simplest robust characterization of the dynamics that is compatible with the given data. Our fundamental assumption is that the data allows for the construction of an isolating block over a parameter space whose homological Conley index is consistent with a saddle-node bifurcation. Our main result establishes that, for phase spaces of dimension greater than or equal to 6, the original vector field can be smoothly deformed into a canonical model exhibiting exactly one structurally stable saddle-node bifurcation. Crucially, this deformation leaves the vector field unaltered outside the isolating block, ensuring strict compatibility with the observed data.