conatus.expert.normal_form

conatus.expert.normal_form(system, x0, *, kind, par=None, parameter=None, omega=None, weight=None, symmetry=None, kernel_tol=1e-08)

Normal form at the equilibrium x0 (a located event: fold, branch point or Hopf).

kind: “zero” (fold / transcritical / pitchfork, classified) or “hopf”. parameter: the unfolding parameter (needed for β, μ, λ’). omega: the Hopf frequency guess (default: the critical pair closest to the imaginary axis). weight: diagonal of the norm fixing ‖q‖ = 1 (default Euclidean on the discrete vector, declared). symmetry: optional linear map S (matrix or callable) with S G(x) = G(S x); the kernel parity S q = ±q is reported and a pitchfork is then required to have a2 = 0 up to kernel_tol (else refused). kernel_tol: the critical eigenvalue must satisfy |λ| (zero) or |Re λ| (Hopf) <= kernel_tol·(1+‖J‖), and x0 must be an equilibrium, max|G(x0)| <= kernel_tol·max(1, ‖J‖)·max(1, ‖x0‖) (else refused).

Parameters:

• system (DynamicalSystem)

• x0 (ndarray)

• kind (str)

• par (Mapping[str, float] | None)

• parameter (str | None)

• omega (float | None)

• weight (ndarray | None)

• symmetry (Any)

• kernel_tol (float)

Return type:

Any