Quantum dynamical signatures of topological flow transitions in limit cycle phases

Quantum self-oscillatory phases are ubiquitous in driven-dissipative systems. Classically, each phase is defined by its flow pattern and how stationary sets organize phase space (e.g., fixed points and limit cycles), with transitions triggered by local bifurcations or global basin rearrangements. In the quantum regime, these reorganizations are often blurred by density-matrix averaging, and spectral indicators such as the Liouvillian gap can miss changes that unfold mainly in the transients. Here, we develop a topological toolkit for flows with limit cycles, rooted in Morse-Smale dynamical-systems theory, to reveal and classify dynamical phase transitions in self-oscillating driven-dissipative systems. We formulate a graph invariant, the , encoding the connectivity and chirality of fixed points and limit cycles and the resulting constraints on basin and flow-region rearrangements during phase transitions. Exploiting these constraints, we uncover a class of global dynamical phase transitions, marked by discrete changes of the molecule, that leave no clear trace in the Liouvillian spectrum (gap or excited eigenvalues) yet induce sharp qualitative changes in transient quantum dynamics and relaxation pathways. Our findings show that flow topology offers a clear and unified way to identify and classify dynamical phases beyond what Liouvillian spectra alone reveal.