1. The Complex Ginzburg-Landau equation II (also real case)
Amplitude-Phase representation
Real Gle (mostly 1D)
Stability of plane waves, BFN instability
Phase equation
Phase and amplitude chaos

2. AmpPhase1: amplitude-phase representation 1

3. AmpPhase2: amplitude-phase representation 2

4. GLe1Potential: Overview over all solutions: Potential

5. GLe2AllStatSol: All static solutions

6. GLe3: quasiperiodic solutions, localized saddle point

7. GLe4: Eckhaus with potential
Simulations!

8. Solutions 1: Plane waves and their stability

9. Solutions 2: Eckhaus instability BFN instability
Simulations!

10. ConvAbsStab: Convectice vs absolute intsability

11. I. Aronson, L. Aronson, LK, A. Weber PRA 1992)
Fig. 1 of Review: convective versus absolute Eckhaus
I. Aronson, L. Aronson, LK, A. Weber PRA 1992)

12. Phase diagram for 1D CGLe (b= )N
1D phase diagram
A I= Absolute stability boundary

13. PhaseEq1

14. PhaseEq2

15. Simulations of phase Chaos in 1D H. Chate, 1994
Phase chaos simulations (Chate)

16. Beyond phase chaos (and coexistenct):
“amplitude chaos” (or defect chaos)
1D: nonzero rate of “phase slips” (A=0 at some x,t)
2D: nonzero density of
zeros of A (topological
point defects)
Beyond phase chaos
3D: further restrictions
for persistent defect lines

17. From Vortex to Spiral

18. Spiral in 3D representation

19. Spiral breakup

20. EI = conv. instab., AI=abs. Instab., BFN= Benjamin-Feir-Newell
line, OR=monotonic/oscillatory interaction (bound states).
L=limit of phase turbulence, T-limit of defect turbulence
(Chate & Manneville, 1996)

21. Spiral-filament chaos in 3D
Snapshots at different times
=50, c=-0.5
Spiral-filament chaos in 3D
Aranson, Bishop, LK (1998)
For more details: Aronson & LK, Rev. Mod. Phys. 74, 99 (2002)