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Connectivity and bifurcation between APC and slalom branches

Determine whether the solution branches corresponding to the active-particle-preceding circular motion (APC) and the slalom (SL) motion are connected within the proposed dynamical system by identifying and characterizing the bifurcation structure between these two motions and locating any critical parameter values at which averaged speed or motion period coincide.

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Background

Numerical simulations reveal multiple motion types (PPS, PPC, APC, SL) and regions of bistability, particularly between SL and APC, depending on the self-propulsion parameter f2.

The authors report they cannot find points where averaged speed or period coincide between the APC and SL branches, suggesting a nontrivial bifurcation structure that remains to be clarified.

References

The solution branches for APC and SL motions do not seem to be connected since we cannot find the point where the averaged speed or the period of motion coincides in Fig.~\ref{fig5}(a) and (c). The bifurcation structures related to these two motions should be clarified in future study.

— Simple mathematical model for a pairing-induced motion of active and passive particles (2501.00411 - Ishikawa et al., 31 Dec 2024) in Section 3 Numerical simulation (discussion after Fig. 5)