arXiv subjects

M. Silber

Publications and source records attributed to M. Silber.

8 recordsLinked to original sources

arXiv2008

Design of parametrically forced patterns and quasipatterns

The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear three-wave interactions between driven and weakly damped modes play a key role in determining which patterns are favoured. We use this idea to design single and multi-frequency forcing functions that produce examples of superlattice patterns and quasipatterns in a new model PDE with parametric forcing. We make quantitative comparisons between the predicted patterns and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very close to onset. The reason that the range of validity is limited is that the theory requires strong damping of all modes apart from the driven pattern-forming modes. This is in conflict with the requirement for weak damping if three-wave coupling is to influence pattern selection effectively. We distinguish the two different ways that three-wave interactions can be used to stabilise quasipatterns, and present examples of 12-, 14- and 20-fold approximate quasipatterns. We identify which computational domains provide the most accurate approximations to 12-fold quasipatterns, and systematically investigate the Fourier spectra of the most accurate approximations.

nlin.PS↗

arXiv2007

Amplification in the auditory periphery: the effect of coupling tuning mechanisms

A mathematical model describing the coupling between two independent amplification mechanisms in auditory hair cells is proposed and analyzed. Hair cells are cells in the inner ear responsible for translating sound-induced mechanical stimuli into an electrical signal that can then be recorded by the auditory nerve. In nonmammals, two separate mechanisms have been postulated to contribute to the amplification and tuning properties of the hair cells. Models of each of these mechanisms have been shown to be poised near a Hopf bifurcation. Through a weakly nonlinear analysis that assumes weak periodic forcing, weak damping, and weak coupling, the physiologically-based models of the two mechanisms are reduced to a system of two coupled amplitude equations describing the resonant response. The predictions that follow from an analysis of the reduced equations, as well as performance benefits due to the coupling of the two mechanisms, are discussed and compared with published experimental auditory nerve data.

nlin.PS↗

arXiv2006

Quasipatterns in parametrically forced systems

We examine two mechanisms that have been put forward to explain the selection of quasipatterns in single and multi-frequency forced Faraday wave experiments. Both mechanisms can be used to generate stable quasipatterns in a parametrically forced partial differential equation that shares some characteristics of the Faraday wave experiment. One mechanism, which is robust and works with single-frequency forcing, does not select a specific quasipattern: we find, for two different forcing strengths, a 12-fold quasipattern and the first known example of a spontaneously formed 14-fold quasipattern. The second mechanism, which requires more delicate tuning, can be used to select particular angles between wavevectors in the quasipattern.

nlin.PS↗

arXiv2003

Near-Resonant, Steady Mode Interaction: Periodic, Quasi-Periodic and Localized Patterns

Motivated by the rich variety of complex periodic and quasi-periodic patterns found in systems such as two-frequency forced Faraday waves, we study the interaction of two spatially periodic modes that are nearly resonant. Within the framework of two coupled one-dimensional Ginzburg-Landau equations we investigate analytically the stability of the periodic solutions to general perturbations, including perturbations that do not respect the periodicity of the pattern, and which may lead to quasi-periodic solutions. We study the impact of the deviation from exact resonance on the destabilizing modes and on the final states. In regimes in which the mode interaction leads to traveling waves our numerical simulations reveal localized waves in which the wavenumbers are resonant and which drift through a steady background pattern that has an off-resonant wavenumber ratio.

nlin.PS↗

arXiv2003

Feedback Control of Traveling Wave Solutions of the Complex Ginzburg Landau Equation

Through a linear stability analysis, we investigate the effectiveness of a noninvasive feedback control scheme aimed at stabilizing traveling wave solutions of the one-dimensional complex Ginzburg Landau equation (CGLE) in the Benjamin-Feir unstable regime. The feedback control is a generalization of the time-delay method of Pyragas, which was proposed by Lu, Yu and Harrison in the setting of nonlinear optics. It involves both spatial shifts, by the wavelength of the targeted traveling wave, and a time delay that coincides with the temporal period of the traveling wave. We derive a single necessary and sufficient stability criterion which determines whether a traveling wave is stable to all perturbation wavenumbers. This criterion has the benefit that it determines an optimal value for the time-delay feedback parameter. For various coefficients in the CGLE we use this algebraic stability criterion to numerically determine stable regions in the (K,rho) parameter plane, where rho is the feedback parameter associated with the spatial translation and K is the wavenumber of the traveling wave. We find that the combination of the two feedbacks greatly enlarges the parameter regime where stabilization is possible, and that the stability regions take the form of stability tongues in the (K,rho)--plane. We discuss possible resonance mechanisms that could account for the spacing with K of the stability tongues.

nlin.PS↗

arXiv2001

Secondary instabilities of hexagons: a bifurcation analysis of experimentally observed Faraday wave patterns

We examine three experimental observations of Faraday waves generated by two-frequency forcing, in which a primary hexagonal pattern becomes unstable to three different superlattice patterns. We use the symmetry-based approach developed by Tse et al. to analyse the bifurcations involved in creating the three new patterns. Each of the three examples reveals a different situation that can arise in the theoretical analysis.

nlin.PS↗

arXiv2000

Spatial period-multiplying instabilities of hexagonal Faraday waves

A recent Faraday wave experiment with two-frequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales [1]. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called `superlattice-II') the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2sqrt{3} from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale (sqrt{3} larger than the original scale) and apparently has 60 degree rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-II pattern presented in [1] and the subharmonic nature of the secondary instability, we show (a) that the superlattice-II pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in [1]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60 degree rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude.

nlin.PS↗

arXiv1995

Stability Results for Steady, Spatially--Periodic Planforms

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.

patt-sol↗