Understanding the Complex Ginzburg-Landau Equation: A Tutorial Introduction to Bifurcation and Chaotic Systems

Jan 04, 2025

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Learn about the mathematical intricacies of the Complex Ginzburg-Landau equation (CGLe) and its applications in spatially extended systems, including wave number bands, coherent structures, and vortices. Explore topics such as phase chaos, defect chaos, and the fascinating insights provided by pioneers like Poincare, Andronov, and Hopf. Delve into the world of bifurcation, chaos, and the evolution of complex systems in this comprehensive tutorial.

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Understanding the Complex Ginzburg-Landau Equation: A Tutorial Introduction to Bifurcation and Chaotic Systems

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The Complex Ginzburg Landau equation (CGLe) Mostly a Tutorial Introduction 0D: The Hopf bifurcation (textbooks!) Spatially extended systems, predominantly (standard!) General discussion: wave number band (phase waves) etc. Solutions in 1D: coherent structures (localised states, fronts, Retracting Fronts, phase chaos and defect chaos…) 6. Solutions in 2D (and maybe 3D) vortices and spirals, chaos… What would Landau have said? Pathological ?
u u v Poincare considered the problem too trivial to write down…. Andronov & Leontovich did it in the plane ( ~1938). Hopf generalized it to many dergrees of freedom(1942). The Hopf Bifurcation. In the plane: Ha-ha-ha! v
Re (A) u Actually a structure for Snapshot!