SwiftHohenberg

Systems / ODEs / Spatial fields

The prototype pattern-forming PDE — stripes and hexagons from a single scalar field near a finite-wavelength instability.

continuous · ODE1024 dimensionspatterns

spatiotemporal field — plays automatically

Definition

@staticmethod
def _equations(Y, t, *, N, L, r):
    h = L / N
    inv_h2 = 1.0 / (h * h)
    inv_h4 = inv_h2 * inv_h2

    def idx(rr, cc):
        return (rr % N) * N + (cc % N)

    rhs = []
    for rr in range(N):
        for cc in range(N):

            def u(dr, dc, rr=rr, cc=cc):
                return Y(idx(rr + dr, cc + dc))

            u0 = u(0, 0)
            lap = (u(1, 0) + u(-1, 0) + u(0, 1) + u(0, -1) - 4 * u0) * inv_h2
            # Compact 13-point biharmonic (∇⁴) stencil on a periodic grid.
            bih = (
                20 * u0
                - 8 * (u(1, 0) + u(-1, 0) + u(0, 1) + u(0, -1))
                + 2 * (u(1, 1) + u(1, -1) + u(-1, 1) + u(-1, -1))
                + (u(2, 0) + u(-2, 0) + u(0, 2) + u(0, -2))
            ) * inv_h4
            # u_t = r u - (1 + ∇²)² u - u³ = r u - (u + 2 ∇²u + ∇⁴u) - u³
            rhs.append(r * u0 - (u0 + 2.0 * lap + bih) - u0 * u0 * u0)
    return rhs

Parameters

Symbol	Default	Role
N	32	grid points per side (structural — changing it recompiles)
L	40	domain length per side
r	0.3	linear growth control (unstable bandwidth)

Properties

Lyapunov spectrum

TODO — dim 1024 > 6 — full spectrum too slow

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 1024 r - 1024 + \frac{8388608.0}{L^{2}} - \frac{21474836480.0}{L^{4}}$
+ per-cell nonlinear terms (over the 1024-cell field)

Equilibria

TODO — dim 1024 > 8 — equilibrium search skipped

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.SwiftHohenberg()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Swift & Hohenberg (1977), Phys. Rev. A 15, 319-328

BibTeX

@misc{swifthohenberg,
  title = {SwiftHohenberg system},
  note = {Swift & Hohenberg (1977), Phys. Rev. A 15, 319-328}
}