Bifurcation sequence near SL/APC region leading to chaos

Characterize the sequence of bifurcations near the boundary of bistability between slalom and active-particle-preceding circular motion as the self-propulsion parameter f2 decreases, including transitions from slalom to circular slalom, to Lissajous-like trajectories, and finally to chaotic motion, by determining the underlying bifurcation mechanisms and parameter thresholds in the proposed model.

Background

Near the edge of the region where slalom motion is observed, the authors find a progression of increasingly complex dynamics as f2 decreases: slalom along a circle, circular slalom along a Lissajous-like curve, and chaotic motion.

They explicitly state that detailed analyses of this bifurcation sequence are left for future work, indicating an unresolved question about the precise dynamical systems mechanisms and parameter conditions underpinning these transitions.

References

As the parameter f_2 decreased, the SL motion changed to the slalom motion along a circle as shown in Fig.~\ref{fig6}(d), and then to the circular slalom motion along a curve like the Lissajous figure as shown in Fig.~\ref{fig6}(c). With a further decrease in f_2, the chaotic motion was observed as shown in Fig.~\ref{fig6}(a,b). The detailed analyses of these series of bifurcation structure may be interesting from the viewpoint of dynamical systems, though we leave them as future study.

— Simple mathematical model for a pairing-induced motion of active and passive particles  (2501.00411 - Ishikawa et al., 2024) in Section 5 Discussion and Summary (Fig. 6)

An open question here would be the characterization of `chaotic'' motion, perhaps at large P eclet numbers, as seen in experiments~\citep{Hokmabad2021}, 3D simulations of unbounded isotropic active particles~\citep{Hu2022} and 2D simulations ofactive discs'~\citep{Li2022}.

— Steady translation of a chemically active viscous drop along a rigid wall  (2610.10300 - Desai et al., 7 Oct 2026) in Section 7.3, “Perspectives for future work”