Poincaré described the way in which a dynamical system’s properties depend upon its topology. Topological properties provide detailed information about the fundamental mechanisms — stretching, squeezing, tearing, folding, twisting — that act to shape a dynamical system’s flow in state space.   A topological analysis involves finding a topological representation of the underlying structure and obtaining an algebraic description that allows for the computation of topological invariants. The present work proposes fingerprinting a model’s or system’s nonlinear behaviour using the novel concept of templex. The templex has two components: a cell complex and a directed graph. In the deterministic framework, the cell complex  approximates the branched manifold, and the directed graph  prescribes the allowed  connections between the highest-dimensional cells of the complex according to the flow. The templex properties include  the non-equivalent ways of circulating along the complex, which are essential to provide a full description of the 'topological fingerprints' of the system's  dynamics. [Chaos 2022, doi:10.1063/5.0092933]