/mnt/data/Targeted_Hopf_detection__increasing_z_.csv
This negative finding indicates that, with the chosen baseline maps from leadership parameters to model coefficients and with moderate increases of the policy loop gain/lag (up to z=5), the linearization of the augmented system remains in the non-oscillatory regime. In dynamical language: the Routh--Hurwitz conditions for cubic E--P--G subloop stability were not violated for the scanned parameter window; the dominant route to instability remained a saddle-node (fold) in E\mu_EE, as shown in Section 3.1.
Interpretation. The absence of Hopf in the scanned region implies one of three possibilities (or their combination):
1. The effective loop gain lag (the product of kp,G,G,k_p,\phi_G,\chi_G,\taukp,G,G,) required to produce Hopf is larger than the range we scanned (i.e., \zeta must exceed the tested maximum).
2. The baseline damping terms (e.g., c4c_4c4 --- protest damping; dEd_EdE --- economic dissipation) are sufficiently large that even large zzz cannot overcome them.
3. Alternative structural elements are needed to produce Hopf, for example: an extra delay (explicit delay differential equation), stronger nonlinear gain, or an additional state that closes a different feedback loop (media contagion, law enforcement dynamics, etc.).
Because the Hopf detection requires precise eigenvalue crossing, coarse scans and finite-difference multilinear calculations can miss narrow windows. Thus the negative result should be interpreted as no Hopf in the explored, plausible parameter wedge, not a universal impossibility.
3.B.7 Example of a Hopf-generating route (recommendations & tested levers)
Analytical inspection of the linearized E--P--G subblock and the Routh--Hurwitz determinant suggests three straightforward ways to create a Hopf regime in model experiments:
Amplify loop gain: increase kpk_pkp, G\phi_GG, and G\chi_GG simultaneously. Politically, this corresponds to a response policy that is both strong and directly couples to protests and the economy.
Lengthen effective lag: increase \tau (slower policy), or introduce an explicit second lagging state. Sluggish feedback produces phase lag, which together with gain yields oscillation.
Reduce damping: decrease c4c_4c4 and/or dEd_EdE so the loop is less attenuated.
Practical numerical bounds (for reproduction): in exploratory trials we found no Hopf up to z=5z=5z=5 where zzz multiplies kpk_pkp and scales \tau (our mapping used moderate tau scaling). Empirically, to produce Hopf one would likely need to push zzz significantly larger than 5 or set 1\tau\gtrsim 11 with large gains --- parameter windows that must be justified by domain evidence if used in an applied study.
3.B.8 Figures and table (to include)
Table 3. Hopf search summary (targeted z sweep). Include columns: Leader | z tested | Hopf detected (Y/N) | h\mu_hh (if found) | 1\ell_11 (if computed) | 0\omega_00 (if computed). (The CSV created by the run is saved at /mnt/data/Targeted_Hopf_detection__increasing_z_.csv.)
Figure 3. Stability trace around candidate regions. If/when Hopf is found, plot Re() of the most oscillatory eigenpair vs E\mu_EE for several z values to show the crossing; overlay imaginary parts (0).
Figure 4 (representative) --- Limit cycle beyond Hopf. If Hopf emerges for a chosen (z,h)(z,\mu_h)(z,h), integrate the full deterministic system slightly beyond h\mu_hh to produce (i) time series of P,E,TP,E,TP,E,T, and (ii) phase portrait EEE vs PPP. The code attempted to save such a figure to /mnt/data/limit_cycle_example.png if a Hopf was found and a durable cycle was observed.
Note: in the present run the CSV indicates no Hopf was detected (all found = False for the four leaders), so Figure 3/4 are placeholders until a Hopf is located with a refined search.
3.B.9 Robustness and recommended next steps
For a publication-quality treatment of Hopf regimes we recommend the following immediate steps:
