LAUSR

Templexes are topological objects that encode the branching organization of a flow in phase space. We build on these objects to introduce the concept of topological modes of variability (TMVs). TMVs are defined as dynamical manifestations of algebraically defined cycles, called generatexes, in the templex; they provide a concrete link between abstract topological invariants and time-dependent behavior in a model or in observations. We apply this approach to a low-order model of the wind-driven ocean circulation, subject to both periodic and aperiodic forcing, and show how TMVs emerge or vanish over time in nonautonomous settings. The analysis reveals that TMVs allow for a qualitatively new understanding of variability in complex systems where linear modes fail to describe the nonlinear dynamics.