Theoretical and numerical understanding of the key concept of topological entropy is an important problem in dynamical systems. While most existing studies focus on discrete-time systems (maps), this work examines a continuous-time scenario involving global changes in the structure of an observed attractor. As a representative example, we consider the classical Rössler system. For a specific range of parameters of the system, we prove the existence of a trapping region for a certain Poincaré map, which contains a nonempty, compact, and connected invariant set on which the topological entropy of the Poincaré map is positive. Additionally, we prove the existence of a sequence of periodic orbit bifurcations that lead to an increase in the topological entropy of this Poincaré map. Our results further reveal that the topological structure of the maximal invariant set in the trapping region evolves as the parameters of the system vary. These findings are rigorously supported by computer-assisted proofs, employing interval arithmetic techniques to compute guaranteed bounds on the Poincaré map and its derivatives.