  A method for solving zero-finding problems is developed by tracking homotopy
paths, which define connecting channels between an auxiliary problem and the
objective problem. Current algorithms' success highly relies on empirical
knowledge, due to manually, inherently selected homotopy paths. This work
introduces a homotopy method based on the Theory of Functional Connections
(TFC). The TFC-based method implicitly defines infinite homotopy paths, from
which the most promising ones are selected. A two-layer continuation algorithm
is devised, where the first layer tracks the homotopy path by monotonously
varying the continuation parameter, while the second layer recovers possible
failures resorting to a TFC representation of the homotopy function. Compared
to pseudo-arclength methods, the proposed TFC-based method retains the
simplicity of direct continuation while allowing a flexible path switching.
Numerical simulations illustrate the effectiveness of the presented method.
