Existence of multidimensional patterns

Lecture 1: Inf-Dim dyn. Sys: Center Mfld, normal forms, bifurcation theory (Haragus, Iooss, Ch 2.2, 2.3, 2.4)

1.2 Example 1-d pattern: Existence of roll solutions in SH using center manifold 

Exercises: Center Manifold for Brusselator. 

1.3 Example 2-d pattern: Existence of hexagons in SH using center manifold 

References:

-- Dionne, Benoit, Mary Silber, and Anne C. Skeldon. "Stability results for steady, spatially periodic planforms." Nonlinearity 10.2 (1997): 321.

--Lloyd, David JB, et al. "Localized hexagon patterns of the planar Swift-Hohenberg equation." SIAM Journal on Applied Dynamical Systems 7.3 (2008): 1049-1100.

Exercise: Existence of squares in SH using center manifold.

Exercise with hints: Derivation of CGL using center manifold 

Lecture 2: Lyapunov-Schmidt Reduction 

2.1 Existence of Roll solutions in SH (bounded domain + periodic bc in x)

Reference: Mielke, Alexander. "A new approach to sideband-instabilities using the principle of reduced instability." Nonlinear dynamics and pattern formation in the natural environment. Routledge, 2022. 206-222.

Exercise: Existence of periodic solutions for the Brusselator model following reference,

Sukhtayev, Alim, et al. "Diffusive stability of spatially periodic solutions of the Brusselator model." Communications in Mathematical Physics 358 (2018): 1-43.

2.2 Stability of Roll solutions in SH (bounded domain + periodic bc in x)

Lecture 3:

3.1 Introduction

3.2 Existence of fronts and roll solutions in SH using spatial dynamics 

Reference:Avitabile, Daniele, et al. "To snake or not to snake in the planar Swift-Hohenberg equation." SIAM Journal on Applied Dynamical Systems 9.3 (2010): 704-733.

Exercise: Fronts using spatial dynamics and also center manifold 

(reference/ hint :  Eckmann, J -P., and Clarence Eugene Wayne. "Propagating fronts and the center manifold theorem." Communications in mathematical physics 136 (1991): 285-307.)

3.3 Snaking bifurcation diagrams

Reference:Avitabile, Daniele, et al. "To snake or not to snake in the planar Swift-Hohenberg equation." SIAM Journal on Applied Dynamical Systems 9.3 (2010): 704-733.

Exercise with hints: Existence of grain boundaries in SH, following

Haragus, Mariana, and Arnd Scheel. "Grain boundaries in the Swift–Hohenberg equation." European Journal of Applied Mathematics 23.6 (2012): 737-759.

Lecture 4: Singular Perturbations

4.1 Introduction following Chapters 1 and 2 of C.K.R.T Jones’ notes,

Arnold, Ludwig, et al. "Geometric singular perturbation theory." Dynamical Systems: Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13–22, 1994 (1995): 44-118.

Examples: Rough sketch for construction of pulses

Alternative reference: Hek, Geertje. "Geometric singular perturbation theory in biological practice." Journal of mathematical biology 60.3 (2010): 347-386.

4.2 Matching far field/-near field to construct 2-d patterns like spiral waves, target patterns

References:  

--Scheel, Arnd. "Bifurcation to spiral waves in reaction-diffusion systems."

SIAM journal on mathematical analysis 29.6 (1998): 1399-1418.

--Kollár, Richard, and Arnd Scheel. "Coherent structures generated by inhomogeneities in oscillatory media." SIAM Journal on Applied Dynamical Systems 6.1 (2007): 236-262.

4.3 Connections between singular perturbation and formal match-asymptotics. Revisit construction of spiral waves and target patterns.