Ginzburg-Landau Spirals

The CGLE ∂A/∂t = A + (1+ib)∇²A − (1+ic)|A|²A governs oscillating media. Colorized by phase arg(A). Stable spirals rotate near b,c small; phase turbulence (Kuramoto-Sivashinsky) appears for large |b|,|c| satisfying 1+bc<0.

b (diffusion imag): -1.0

c (nonlin imag): 2.0

dt: 0.10

Speed: 3x

—

1+bc = —
Turbulent if 1+bc<0

Phase color wheel:
■0 ■π/2 ■π ■3π/2

t = 0

Physics: CGLE is the normal form for Hopf bifurcation in extended systems. The parameter plane (b,c) has three regimes: (1) Benjamin-Feir stable (1+bc>0, small |b|,|c|): stable spiral waves; (2) phase turbulence (intermediate): phase fluctuations grow, weak turbulence; (3) defect turbulence: topological defects (phase vortices) proliferate and interact. The transition at 1+bc=0 is the Benjamin-Feir-Newell criterion. Each colored vortex core is a defect where |A|=0.

Complex Ginzburg-Landau Spirals