Nonlinear Standing and Rotating Waves on the Sphere

Gugg, Christoph; Healey, Timothy J.; Kielhöfer, Hansjörg; Maier-Paape, Stanislaus

doi:10.1006/jdeq.2000.3791

Cited by 8 publications

(6 citation statements)

References 17 publications

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“…The choice of the maximal p 0 is equivalent to the choice of the minimal period T = 2πq/p 0 . This argument is similar to the argument used in [15], [17] and [16].…”

Section: The Bifurcation Equationsupporting

confidence: 70%

“…The main challenge encountered when proving the existence of the periodic solutions is that the trivial branch u λ and its associated spectrum are not known explicitly. Indeed, while the existence of continuous families of periodic solutions has been obtained before for Hamiltonian PDEs in [14], [15], [16] and [17], in those articles the trivial branch and the spectrum of its linearized equation are known explicitly. We overcome our problem with delicate estimates of the spectrum which depend on an accurate estimate for the steady state u λ * at the critical value λ * .…”

mentioning

confidence: 97%

“…The small divisor problem was avoided in [13], [10] and [14] by imposing restrictions on the temporal period of the solutions of a nonlinear wave equation. By imposing similar restrictions, the articles [15] obtains the existence of continuous branches of periodic solutions for a nonlinear wave equation in a sphere, and in [16] and [17] for a Hamiltonian PDE appearing in the n-vortex filament problem.…”

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confidence: 99%

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Free vibrations in a wave equation modeling MEMS

García-Azpeitia¹,

Lessard²

2020

Preprint

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We study a nonlinear wave equation appearing as a model for a membrane (without viscous effects) under the presence of an electrostatic potential with strength λ. The membrane has a unique stable branch of steady states u λ for λ ∈ [0, λ * ]. We prove that the branch u λ has an infinite number of branches of periodic solutions (free vibrations) bifurcating when the parameter λ is varied. Furthermore, using a functional setting, we compute numerically the branch u λ and their branches of periodic solutions. This approach is useful to validate rigorously the steady states u λ at the critical value λ * .Dedicated to the memory of G. Flores.

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“…The choice of the maximal p 0 is equivalent to the choice of the minimal period T = 2πq/p 0 . This argument is similar to the argument used in [15], [17] and [16].…”

Section: The Bifurcation Equationsupporting

confidence: 70%

mentioning

confidence: 97%

mentioning

confidence: 99%

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Free vibrations in a wave equation modeling MEMS

García-Azpeitia¹,

Lessard²

2020

Preprint

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“…We now illustrate with several concrete examples. In what follows, we employ the subgroup notation of [14]: The planar subgroups, denoted as in [15]. Specifically, we choose 1 2 0 1 0 1 0 0 0 0 1 and 0 1 0 (6.30) as the generators of the tetrahedral group .…”

Section: ∈ ℓ ℕmentioning

confidence: 99%

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

Healey¹,

Dharmavaram²

2017

SIAM J. Math. Anal.

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We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly underdetermined in-plane deformation. The resulting governing equations comprise a quasi-linear elliptic system with nonlinear constraints. We show the equivalence of the problem to that of finding the zeros of compact vector field. The latter is not routine. Using spectral and a-priori estimates together with Fredholm properties, we demonstrate that the principal part of the quasi-linear mapping defines an operator with compact resolvent. We then combine well known group-theoretic ideas for symmetry breaking with global bifurcation theory to obtain our results.

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“…The reason for this is that there are only finitely many 'bad sites' and as a result we can solve the range equation using the standard implicit function theorem. Our work is much more similar in spirit and methodology to the works [7,6] but differs in that we do not explicitly look for periodic solutions having a prescribed spatiotemporal symmetry group. We compensate for this by looking for solutions with a fixed spatial frequency vector, as well as understanding the role of the external bifurcation parameter, which is the time delay.…”

mentioning

confidence: 99%

Self-excited vibrations for damped and delayed higher dimensional wave equations

Kosovalić¹,

Pigott²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In the article [12] it is shown that time delay induces self-excited vibrations in a one dimensional damped wave equation. Here we generalize this result for higher spatial dimensions. We prove the existence of branches of nontrivial time periodic solutions for spatial dimensions d ≥ 2. For d > 2, the bifurcating periodic solutions have a fixed spatial frequency vector, which is the solution of a certain Diophantine equation. The case d = 2 must be treated separately from the others. In particular, it is shown that an arbitrary number of symmetry breaking orbitally distinct time periodic solutions exist, provided d is big enough, with respect to the symmetric group action. The direction of bifurcation is also obtained.

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Nonlinear Standing and Rotating Waves on the Sphere

Cited by 8 publications

References 17 publications

Free vibrations in a wave equation modeling MEMS

Free vibrations in a wave equation modeling MEMS

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

Self-excited vibrations for damped and delayed higher dimensional wave equations

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