dorsal/arxiv
View SchemaStability Results for Steady, Spatially--Periodic Planforms
| Authors | B. Dionne, M. Silber, A. C. Skeldon |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9509004 |
| URL | https://arxiv.org/abs/patt-sol/9509004 |
Abstract
We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a finite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of 4 complex Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator consists of 6 complex Fourier modes, also parameterized by an integer pair (a,b). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, a countable set of rhombs, and a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying a and b. For the hexagonal lattice, we analyze the degenerate bifurcation problem obtained by setting the coefficient of the quadratic term to zero. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.
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"abstract": "We consider the symmetry-breaking steady state bifurcation of a\nspatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict\nthe space of solutions to those that are doubly-periodic with respect to a\nsquare or hexagonal lattice, and consider the bifurcation problem restricted to\na finite-dimensional center manifold. For the square lattice we assume that the\nkernel of the linear operator, at the bifurcation point, consists of 4 complex\nFourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and\nK_4=(-a,b), where a\u003eb\u003e0 are integers. For the hexagonal lattice, we assume that\nthe kernel of the linear operator consists of 6 complex Fourier modes, also\nparameterized by an integer pair (a,b). We derive normal forms for the\nbifurcation problems, which we use to compute the linear, orbital stability of\nthose solution branches guaranteed to exist by the equivariant branching lemma.\nThese solutions consist of rolls, squares, hexagons, a countable set of rhombs,\nand a countable set of planforms that are superpositions of all of the Fourier\nmodes in the kernel. Since rolls and squares (hexagons) are common to all of\nthe bifurcation problems posed on square (hexagonal) lattices, this framework\ncan be used to determine their stability relative to a countable set of\nperturbations by varying a and b. For the hexagonal lattice, we analyze the\ndegenerate bifurcation problem obtained by setting the coefficient of the\nquadratic term to zero. The unfolding of the degenerate bifurcation problem\nreveals a new class of secondary bifurcations on the hexagons and rhombs\nsolution branches.",
"arxiv_id": "patt-sol/9509004",
"authors": [
"B. Dionne",
"M. Silber",
"A. C. Skeldon"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Stability Results for Steady, Spatially--Periodic Planforms",
"url": "https://arxiv.org/abs/patt-sol/9509004"
},
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