The CAPD library has been used in several articles in which chaotic dynamics, bifurcations, heteroclinic/homoclinic solutions and periodic orbits were studied.
 The existence of simple choreographies for the nbody problem
 Homoclinic and heteroclinic solutions
 Chaotic dynamics for various ODE's
 Invariant curves through the KAM theory
 Cocoon bifurcations
 Rigorous verification of period doubling bifurcations for ODE's
 Rigorous numerics for homoclinic tangencies
 Uniformly hyperbolic attractors for Poincare maps
 Rigorous numerics for dissipative PDE's
 Normally Hyperbolic Invariant Manifolds
 Dynamics of the universal areapreserving map associated with period doubling (written by Tomas Johnson)
Normally Hyperbolic Invariant Manifolds
Can one blindly believe numerical simulations? I think that we all know the answer to this. In some cases "numerical evidence" might be very mistakingly interpreted. An example of this is a simple driven logistic map
T : 𝓢^{1}⨯ℝ →𝓢^{1}⨯ℝ, with a=1.31, ε=0.3 and α=g/200, where g is the golden mean g=0.5(√51). The map possesses a global attractor, but when standard (double precision) numerical simulation is performed, this attractor seems to be chaotic. This is not the case. When redoing the computations with multiple precision we find that this attractor is in fact composed of two smooth curves. With the CAPD package we have proved that this is the case. The two curves are normally hyperbolic invariant manifolds. In some parts they are strongly contracting, but in many parts they are expanding. Since the map is nasty enough to mislead standard nonrigorous simulations, in our proof we needed to consider multiple precision (tracking 40 digits), combined with 𝓒^{10} methods. The proof is based on covering relations combined with cone conditions. This description by Maciej Capinski. Reference:

