6. Eigenvalue sensitivity and transversality derivatives
For transversality checks you often need the derivative of a particular eigenvalue ()\lambda(\mu)() with respect to the parameter =E\mu=\mu_E=E. For a simple eigenvalue with right eigenvector vvv and left www normalized wv=1w^\top v=1wv=1,
dd=wF=wF(X)\boxed{ \; \frac{d\lambda}{d\mu}\;=\; w^\top \frac{\partial F}{\partial \mu}\;=\; w^\top F_{\mu}(X^*) \; }dd=wF=wF(X)
evaluated at the bifurcation point (since F\partial_\mu FF is the parameter derivative of the vector field). In our case F=(0,1,0)F_\mu=(0,1,0)F=(0,1,0), so dd=w2\dfrac{d\lambda}{d\mu} = w_2dd=w2.
For a Hopf, the transversality condition is dd=h0\left.\dfrac{d}{d\mu}\Re\lambda\right|_{\mu=\mu_h} \neq 0dd=h=0. Numerically you can compute the slope by finite differences of \Re\lambda across \mu or use the eigenvector formula above when www and vvv are available.
7. Practical numerical recipe (steps for reproducible computation)
a. Find equilibrium XX^*X for target \mu and \Theta (Newton + continuation is recommended).
b. Assemble Jacobian JJJ using the analytic partials above (preferred) or high-accuracy finite differences (if analytic derivatives are tedious). Use double precision.
c. Compute eigenpairs of JJJ: eigenvalues i\lambda_ii and eigenvectors viv_ivi; compute left eigenvectors wiw_iwi of JJ^\topJ. Normalize each pair so wivi=1w_i^\dagger v_i = 1wivi=1.
d. Stability test: evaluate c1,c2,c3c_1,c_2,c_3c1,c2,c3 and Routh--Hurwitz inequalities.
e. Fold detection: check detJ0\det J \approx 0detJ0 (or min real eigenvalue 0). If a simple zero eigenvalue is found, compute right/left nullvectors v,wv,wv,w. Compute a=12wD2F[v,v]a = \tfrac12 w^\top D^2F[v,v]a=21wD2F[v,v] and b=wFb = w^\top F_\mub=wF. If aaa and bbb are not too small, you have a generic fold; record c\mu_cc, XcX_cXc, a,ba,ba,b.
Compute D2F[v,v]D^2F[v,v]D2F[v,v] analytically or by central finite differences: D2F[v,v]F(X+v)+F(Xv)2F(X)2D^2F[v,v] \approx \dfrac{F(X+\varepsilon v) + F(X-\varepsilon v) - 2F(X)}{\varepsilon^2}D2F[v,v]2F(X+v)+F(Xv)2F(X) for small \varepsilon. Use \varepsilon around 10610^{-6}106--10410^{-4}104 and test convergence.
f. Hopf detection: check for a complex conjugate pair with 0\Re\lambda \approx 00 and 0=>0\omega_0 = |\Im\lambda|>00=>0. If found, verify transversality via d/dd\Re\lambda/d\mud/d (use eigenvector formula or finite differences). Compute 1\ell_11 using the Kuznetsov formula: compute BBB and CCC multilinear forms (via analytic derivatives or finite differences). Use careful finite-difference parameters (e.g., central differences eps ~ 10510^{-5}105--10410^{-4}104).
g. Sanity checks & sensitivity: perturb \Theta slightly to verify smooth dependences and check that computed a,b,1a,b,\ell_1a,b,1 signs are robust.
8. Recommended implementation details and pitfalls
Prefer analytic derivatives whenever possible. The Jacobian and second/third derivatives can be derived symbolically (SymPy, Mathematica) from the model equations and exported to code. This eliminates finite-difference error accumulation.
If using finite differences, use central differences and test convergence by varying step sizes (e.g., 10610^{-6}106 10410^{-4}104 10210^{-2}102) and check that results converge. For higher derivatives (third order) increase step sizes slightly to avoid round-off noise.
Eigenvector normalization: use complex conjugate left eigenvectors properly; compute left eigenvector www as eigenvector of JJ^\topJ corresponding to the conjugate eigenvalue and normalize wv=1w^\dagger v = 1wv=1.
Near-degenerate cases: if the zero eigenvalue at fold is not simple (rare), the normal form degenerates; numerical indicators are very small denominators when normalizing wvw^\top vwv --- treat cautiously.
Continuation: use pseudo-arclength or AUTO/MATCONT for reliable branch tracing. Continuation significantly reduces numerical sensitivity in locating bifurcation points and enables computation of curves c()\mu_c(\Theta)c().
Validation: once you identify a fold or Hopf, perturb \mu slightly and integrate the full nonlinear ODE to confirm the qualitative behavior (e.g., two equilibria coexisting or small amplitude oscillations). For Hopf, check growth/decay of small perturbations to verify super/subcritical nature predicted by 1\ell_11.
9. How leadership parameters \Theta enter the eigenvalue conditions
Leadership parameters \Theta enter JJJ through the coefficient maps (e.g., BL(),BP(),T(),B_L(\Theta), B_P(\Theta), \delta_T(\Theta),BL(),BP(),T(), etc.). Therefore:
All entries Jij=Jij(X,)J_{ij} = J_{ij}(X^*,\Theta)Jij=Jij(X,). Changes in a leadership coordinate (say LLL or RRR) modify the entries and shift the cubic coefficients c1,c2,c3c_1,c_2,c_3c1,c2,c3, thereby moving eigenvalues in the complex plane.
Sensitivity derivatives of a critical eigenvalue with respect to a leadership parameter pp\in\Thetap are given by
p=wFp,\frac{\partial \lambda}{\partial p} = w^\top \frac{\partial F}{\partial p},p=wpF,
evaluated at the equilibrium (for a simple eigenvalue), where pF\partial_p FpF is the parameter derivative of the vector field (this follows the same eigenvalue perturbation formula as for \mu). This is useful for ranking which leadership attributes most strongly shift the stability boundary c()\mu_c(\Theta)c().
Using these formulas you can compute gradients c()\nabla_\Theta \mu_c(\Theta)c() (sensitivity of the fold location to leadership) via implicit differentiation of the fold normal-form equation ay2+b(c)=0a y^2 + b(\mu-\mu_c) =0ay2+b(c)=0 or via differentiating the algebraic discriminant as in Section 4.A.
Short algorithmÂ
Solve F(X;,)=0F(X;\Theta,\mu)=0F(X;,)=0 for XX^*X.
Compute analytic Jacobian JJJ from formulas above.
Compute eigenvalues/eigenvectors of JJJ. Check Routh--Hurwitz.
If min()0\min \Re(\lambda)\approx 0min()0 refine \mu by bisection/continuation to locate the critical c\mu_cc.
At critical point compute nullvectors v,wv,wv,w and normal-form coefficients a,ba,ba,b for fold or 1\ell_11 for Hopf.
Validate by direct time simulation (perturbation and integration) to observe expected dynamics.
