Catalytic distillation integrates the reaction and separation into a single unit. Nevertheless, this integration leads to complex open-loop behaviors, as a consequence of nonlinear interactions. As a result, multiplicities and trajectories with complex attracting sets may arise. Multiplicities must be considered during design, operation, and control, because a process with multiplicities may be difficult to control and operate. In this work, the changes in the topological structure of the solutions of a light-gas-oil deep-hydrodesulfurization one-bed catalytic distillation column are studied. A cost-optimal operating point is chosen as a base case. It is obtained by minimizing the total annual cost, while keeping the total sulfur concentration less than 15 ppm. This design is studied to identify the operating region where multiplicities arise. A connection between operability (pressure) and design (catalyst mass) was made and a rich nonlinear behavior was found (i.e., isola branches, criss-cross patterns, cusp bifurcations, input multiplicities, output multiplicities, and input-output multiplicities). Bifurcation diagrams are presented for the most important design and control/operating variables. Yet, no multiplicity was found near the cost-optimal operating point; thus, its operation and control should not be a problem.