How do we characterize bifurcation in nonlinear PDE instead of ODE i.e. ht=f(x,h,hx,hxx,hxxx,...)? For example, study the temporal evolution of a regular pattern into a chaotic one. Can someone please point out the branch of mathematics I should look into? Or how do we apply traditional ODE bifurcation theory to PDEs? And for practical purpose, it seems to me that one can't really do much about it except for the complex Landau-Ginzburg amplitude equation, which is derived via multi-scale expansion.
2026-03-25 23:35:32.1774481732
Bifurcation in PDE
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Pattern formation in nonlinear PDEs has been studied very extensively, using a combination of linearization, bifurcation analysis, weakly nonlinear analysis, group theory, heuristics, approximations, numerical simulation, and experimental labwork. I would highly recommend two textbooks as a good starting point: "Pattern Formation and Dynamics in Nonequilibrium Systems" by Michael Cross and Henry Greenside, and "Pattern Formation: An Introduction to Methods" by Rebecca Hoyle.
A classic example is the Turing bifurcation [1] which occurs in reaction-diffusion equations. It may play a role in the distribution of hair follicles and other biological pattern formation, as suggested by recent experimental evidence [2]. A basic set of references for the analysis amd applications of this pattern-formation mechanism is as follows; I compiled it for a recent project on Turing patterns.
References
[1] A. Turing. The chemical basis of morphogenesis. Bulletin of Mathematical Biology , 52(1-2):153–197, 1990
[2] Luciano Marcon and James Sharpe. Turing patterns in development: what about the horse part? Current Opinion in Genetics Development , 22(6):578 – 584, 2012. Genetics of system biology.
[3] Q. Ouyang and Harry L. Swinney. Transition to chemical turbulence. Chaos: An Inter- disciplinary Journal of Nonlinear Science , 1(4):411–420, 1991.
[4] A. Yochelis, L. Tintut, L. L. Demer, and A. Garfinkel. The formation of labyrinths, spots and stripe patterns in a biochemical approach to cardiovascular calcification. New Journal of Physics , 10(5):1–16, 2008.
[5] Theodore. Kolokolnikov, Wentao. Sun, Michael. Ward, and Juncheng. Wei. The stability of a stripe for the gierer–meinhardt model and the effect of saturation. SIAM Journal on Applied Dynamical Systems , 5(2):313–363, 2006.
[6] M. Seul, L. R. Monar, L. O’Gorman, and R. Wolfe. Morphology and local structure in labyrinthine stripe domain phase. Science , 254(5038):1616–1618, 1991.
[7] Raymond E. Goldstein, David J. Muraki, and Dean M. Petrich. Interface proliferation and the growth of labyrinths in a reaction-diffusion system. Phys. Rev. E , 53:3933–3957, Apr 1996.
[8] Michael Seul and David Andelman. Domain shapes and patterns: The phenomenology of modulated phases. Science , 267(5197):476–483, 1995.