Consider a smooth one-parameter family of vector fields on a manifold. To identify how parameters can exhibit the maximal dynamical complexity, we introduce the ``Entropy flow``, a vector field on the product of the phase and parameter spaces, that drives trajectories toward increasingly complex states, analogous to the second law of thermodynamics.

In this poster, we analyze the Entropy flow mainly through the lens of Conley index theory. By studying the complement of the periodic orbit set on the extended phase space, we decompose the global behavior of the Entropy flow into a finite collection of Conley indices. This topological framework allows us to construct isolating blocks and rigorously encode transitional dynamics, such as saddle-node bifurcations and period-doubling routes to chaos, as computable topological invariants.

Through this approach, we demonstrate that the Conley index effectively captures Entropy flows, translating the continuous evolution of chaos, such as that observed in the Lorenz, R\{o}ssler, and Shilnikov models, into robust topological invariants.