foldtimes

Foldtimes is a term used in mathematics and computational geometry to denote the moments in time when a folding operation produces a new layer or when a trajectory intersects the fold locus of a folding map. In dynamical systems and related models, folding refers to the process by which a map maps portions of a space onto overlapping regions, creating noninvertible behavior and potential multi‑valued inverses. Foldtimes are the times or iterations at which the evolving state meets the set where the folding occurs.

Formally, for a smooth map f: M → M on a manifold M, the fold set F consists of points where the map fails to be locally invertible, often characterized by a singular condition such as a vanishing determinant of the Jacobian. A foldtime for a trajectory x0, x1, x2, … under the discrete-time evolution xn+1 = f(xn) is any index n for which xn ∈ F. In continuous-time systems with a flow φt, foldtimes can be defined as times t when the state reaches a fold surface defined by a critical manifold.

Computation of foldtimes typically involves identifying the fold set F first, by solving the fold condition, and then monitoring the trajectory for visits to F during simulation or analysis. The concept is closely related to critical points, bifurcations, and the study of piecewise-smooth maps, where folds drive changes in topology and the emergence of complex dynamics.

Foldtimes have implications for numerical stability, the structure of attractors, and the fractal geometry that can arise from repeated folding. They also inform algorithms for detecting critical events in simulations of folding processes in materials science and computer graphics.