Topological classification of driven-dissipative nonlinear systems

In topology, averaging over local geometrical details reveals robust global features. These are crucial in physics for understanding quantized bulk transport and exotic boundary effects of linear wave propagation in (meta-)materials. Beyond linear Hamiltonian systems, topological physics strives to characterize open (non-Hermitian) and interacting systems. Here, we establish a framework for the topological classification of driven-dissipative nonlinear systems by defining a graph index for their Floquet semiclassical equations of motion. Our index builds upon the topology of vector flows and encodes the particle-hole nature of excitations around all out-of-equilibrium stationary states. Thus, we uncover the topology of nonlinear resonator's dynamics under external and parametric forcing. Our framework sheds light on the topology of driven-dissipative phases, including under- to overdamped responses and symmetry-broken phases linked to population inversion. We therefore expose the pervasive link between topology and nonlinear dynamics, with broad implications for interacting topological insulators, topological solitons, neuromorphic networks, and bosonic codes.