Flow topology classification of limit cycles

Abstract

Recent topological tools offer a powerful way to classify how phases of nonlinear bosonic resonators are organized. Yet, they remain incomplete. In particular, self-sustained oscillations in the form of limit cycles act as robust organizing centers in phase space that are not captured by existing fixed-point-based approaches. In this work, we extend the flow topology framework for nonlinear resonators to include limit cycles as fundamental topological elements. Using a graph-based construction, we show how periodic attractors impact the global connectivity of phase-space flows. We illustrate the approach with a minimal nonlinear Van der Pol resonator model, where limit cycles coexist with stationary points. Our results provide a unified topological description of stationary and time-periodic phases in nonlinear bosonic systems, with direct relevance to photonic, superconducting, and optomechanical platforms, and raise new questions on synchronization and the extension of flow topology to the quantum regime.

Figures (3)

• Figure 1: Vector-flow topology phase diagram. (a) Phase diagram of the modified vdP \ref{['eq:vdp']} as a function of $\alpha_3$ and the ration $\mu/\gamma$, with colors indicating distinct vector-flow topologies. Each region corresponds to a different global organization of phase-space trajectories, see legend where $N$ denotes the number of stationary states. (b)–(f) Representative phase-space vector fields [cf. Eqs. \ref{['eq:vflow']}] at the parameter points marked by bullets in panel (a); we observe how fixed points and limit cycles reshape the global flow structure across the diagram. Filled dots, empty circles, and cross markers correspond to attractors, repellers, and saddles, respectively. The color around the dots/circles mark the chirality around the fixed point with blue, red, and white corresponding to CW, CCW and no winding, respectively. Dark blue lines trace CW-rotating limit cycles. Insets show the associated graph indices characterizing the flow topology in each regime. In all panels, $\omega_0=1$ , $\gamma=0.1$, $\alpha_5=0.01$. (b,d) The system does not admit LCs for $\gamma < \mu$. Depending on whether $\alpha_3 \lessgtr -\sqrt{4\alpha_5}$, we either have (b) an elementary flow with only one steady state or (d) a non-elementary flow, with three CW attractors and two saddles. (c,e) As $\gamma > \mu$, the system exhibits a CW attracting LC around the origin (c) as a single feature or (e) coexisting with stationary states. In order for the flow lines of the inner region to converge to the LC, the CW attractor at the origin changes stability and becomes a CW source. (f) As the ratio between $\mu$ and $\gamma$ increases further, the LC is not sustained anymore and vanishes. The source at the origin does not change stability.
• Figure 2: Topological graph invariant without LCs. We explain the construction of the vector-flow graph invariant for the flow in Fig. \ref{['fig:1']}(d), containing only point-like stationary states, with markers as in Fig. \ref{['fig:1']} enumerated by black letters Oshemkov1998villa2025. (a) Since all flow lines are in-coming from infinity, we can perform a one-point compactification step and map infinity to a virtual source (VS). Thus, the whole vector flow can be viewed as lying on the bottom half of a sphere (inset), with all flow lines descending from the top. The separatrices, special trajectories connecting critical points, partition the surface into regions where all flow lines share the same asymptotic behavior. Starting from the saddle points, we use forward and backward time-evolution to draw the separatrices leading to attractors (dashed lines) and originating from VS (dotted), respectively. In a triangulation step, we add the solid lines as representative trajectories connecting VS to the attractors (this choice is arbitrary and has no impact on the topology). The surface is now divided in triangular patches, which we highlight with different colors and identify with orange letters. Crucially, this ensemble of separatrices, critical points and faces is topologically robust and takes the name of Morse-Smale Complex (MSC) Oshemkov1998. The separatrices in (a) are simplified for visual purposes supmat. (b) Schematic representation of the MSC, for better clarity. (c) The topological planar graph invariant is built as the dual of the MSC. The nodes are the faces, while the lines are the common edges between them, e.g, faces a and b share a solid and a dashed edge, hence nodes a and b are connected by a solid and a dashed line. Each region of the planar graph corresponds to a critical point. The chirality around each critical point is encoded in the coloring: here, we use blue shading to highlight that all attractors are CW. Since the VS is also CW, we draw a blue halo around the graph.
• Figure 3: Extension of the graph invariant formalism to LCs. We construct the topological graph invariant for the flow in \ref{['fig:1']}(e), where a CW attracting LC is sustained around the origin. (a) We first subdivide the phase space in regions, each delimited by separatrices of three different dashing styles. Here we omit the VS in the drawing. Separatrices are schematized for graphical purposes supmat. (b) Invariant for a flow with a single limit cycle, according to Ref. Oshemkov1998. The LC separates the phase space in an inner ($r$) and an outer region ($V$). The arrows represent flow directionality with respect to the LC: since it is attracting, the inner and outer regions both point to the LC. The fixed-points $a_1$, $a_2$ and $VS$ represent the elementary regions of $V$, with arrows reflecting the flow asymptotic behaviour. Since we only have a single LC, we omit information on the orientation that the invariant in Ref. Oshemkov1998 introduces. (c) Our proposed graph invariant augments the construction from Ref. villa2025 with the additional information of what is inside and outside the LC. We first draw the graph invariant for the outer region, which in this case results in being topologically equivalent to that of \ref{['fig:1']}(d), when if one replaces the attracting LC with a point-like attractor. We mark the LC region on V with a crossed (double-lined) circle to show that the region inside the LC is repelling (attracting). This uniquely identifies the position of the LC within the graph and the topological behavior of its content. Crucially, the choice of using the colored faces and halo already contain the information on the LC orientation.