Stability results for steady, spatially periodic planforms

Abstract

We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant partial di erential equations (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 nite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of four complex Fourier modes, with wave vectors K 1 = ( ; ); K 2 = ( ; ), K 3 = ( ; ), and K 4 = ( ; ), where > > 0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator at the bifurcation point consists of six complex Fourier modes, also parameterized by an integer pair ( ; ). 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, and a countable set of rhombs, and also 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 and . For the square lattice an O(2( + ) 1) truncation of the normal form is required to completely determine the stability of the planforms, although many of the stability results are established at cubic order. For the hexagonal lattice, all of the solution branches guaranteed by the equivariant branching lemma are, generically, unstable due to the presence of a quadratic term in the normal form. We analyze the degenerate bifurcation problem that is obtained by setting the coe cient of the quadratic term to zero. We must retain terms through O(2 1) in the normal form of the bifurcation problem. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.We also analyze the bifurcation problems for E(2) + Z 2 -equivariant PDEs, which leads to new results in the hexagonal lattice case, only.

Sideband instabilities and defects of quasipatterns

“…The phason modes correspond to relative translations between the two lattices. It is worth mentioning that when the angle between the two lattices satisfies certain conditions, the whole structure becomes periodic with a wavelength larger than that of the individual lattices: this is known as a superlattice [20]. In this case, extra resonance conditions among the modes are satisfied [21], and the phason modes become damped and can, in principle, be eliminated in a long-wave analysis.…”

Design of Parametrically Forced Patterns and Quasipatterns

“…Equivariant bifurcation theory [23] was used to derive the form of bifurcation problem [17]. This bifurcation problem describes the long-time evolution of the twelve modes on the critical circle that are associated with patterns that tile a plane in hexagonal fashion.…”

“…The calculation of their linear stability proceeds in a standard fashion and is summarized in [17]. In fact, due to the presence of the quadratic term, all planforms bifurcate unstably, but because Q is typically very small for multi-frequency forcing and sufficiently weak damping [47], the planforms can be stabilized at small amplitude by secondary bifurcations.…”

Superlattice Patterns in the Complex Ginzburg–Landau Equation with Multiresonant Forcing

“…As expected from the variational character of the amplitude equations and the dependence of the energy of the various patterns on the forcing strength, we find that as the resonant triad interaction is increased more complex patterns dominate over simpler patterns. For the parameters chosen in the numerical simulations we find patterns with 4-fold symmetric elements reminiscent of super-squares and anti-squares [52] as well as 5-fold symmetric elements. The weakly nonlinear analysis (see Fig.5b) suggests that for other system parameters patterns comprised of more modes yet could be stable.…”

“…Indeed, Fig.15c shows 4 modes in the Fourier transform, though they are not quite equally spaced. The corresponding pattern evolution is shown in TimeEvolutionMovie rhoIs1p2.mov; Fig.16c shows the final state, characterized by supersquare (dash-dotted, blue circle) and antisquare (dashed, yellow circles) elements with approximate 4-fold rotational symmetry [52], as well as approximate 8-fold symmetric elements (solid, white circle).…”