  This paper introduces a novel computer-assisted method for detecting and
constructively proving the existence of cusp bifurcations in differential
equations. The approach begins with a two-parameter continuation along which a
tool based on the theory of Poincar\'e index is employed to identify the
presence of a cusp bifurcation. Using the approximate cusp location, Newton's
method is then applied to a given augmented system (the cusp map), yielding a
more precise numerical approximation of the cusp. Through a successful
application of a Newton-Kantorovich type theorem, we establish the existence of
a non-degenerate zero of the cusp map in the vicinity of the numerical
approximation. Employing a Gershgorin circles argument, we then prove that
exactly one eigenvalue of the Jacobian matrix at the cusp candidate has zero
real part, thus rigorously confirming the presence of a cusp bifurcation.
Finally, by incorporating explicit control over the cusp's location, a rigorous
enclosure for the normal form coefficient is obtained, providing the explicit
dynamics on the center manifold at the cusp. We show the effectiveness of this
method by applying it to four distinct models.
