Mary Silber

Stability results for steady, spatially periodic planforms

Equivariant bifurcation theory has been used extensively to study pattern formation via symmetry-breaking steady-state bifurcation in various physical systems modelled by E(2)-equivariant partial differential equations. Much attention has been focused on solutions that are doubly periodic with respect to a square or hexagonal lattice, for which the bifurcation problem can be restricted to a finite-dimensional centre manifold. Previous studies have used four- and six-dimensional representations for the square and hexagonal lattice symmetry groups respectively, which in turn allows the relative stability of squares and rolls or hexagons and rolls to be determined. Here we consider the countably infinite set of eight- and 12-dimensional irreducible representations for the square and hexagonal cases, respectively. This extends earlier relative stability results to include a greater variety of bifurcating planforms, and also allows the stability of rolls, squares and hexagons to be established to a countably infinite set of perturbations. In each case we derive the Taylor expansion of the equivariant bifurcation problem and compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. In both cases we find that many of the stability results are established at cubic order in the Taylor expansion, although to completely determine the stability of certain states, higher-order terms are required. 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 Taylor expansion. For this reason we consider two special cases: the degenerate bifurcation problem that is obtained by setting the coefficient of the quadratic term to zero, and the bifurcation problem when an extra reflection symmetry is present.

Simple and superlattice Turing patterns in reaction-diffusion systems: bifurcation, bistability, and parameter collapse

Abstract We use equivariant bifurcation theory to investigate pattern selection at the onset of a Turing instability in a general two-component reaction–diffusion system. The analysis is restricted to patterns that periodically tile the plane in either a square or hexagonal fashion. Both simple periodic patterns (stripes, squares, hexagons, and rhombs) and “superlattice” patterns are considered. The latter correspond to patterns that have structure on two disparate length scales; the short length scale is dictated by the critical wave number from linear theory, while the periodicity of the pattern is on a larger scale. Analytic expressions for the coefficients of the leading nonlinear terms in the bifurcation equations are computed from the general reaction–diffusion system using perturbation theory. We show that no matter how complicated the reaction kinetics might be, the nonlinear reaction terms enter the analysis through just four parameters. Moreover, for hexagonal problems, all patterns bifurcate unstably unless a particular degeneracy condition is satisfied, and at this degeneracy we find that the number of effective system parameters drops to two, allowing a complete characterization of the possible bifurcation results at this degeneracy. For example, we find that rhombs, squares and superlattice patterns always bifurcate unstably. We apply these general results to some specific model equations, including the Lengyel–Epstein CIMA model, to investigate the relative stability of patterns as a function of system parameters, and to numerically test the analytical predictions.

Hopf bifurcation on a square lattice

A complete classification of the generic D4*T2-equivariant Hopf bifurcation problems is presented. This bifurcation arises naturally in the study of extended systems, invariant under the Euclidean group E(2), when a spatially uniform quiescent state loses stability to waves of wavenumber k not=0 and frequency omega not=0. The D4*T2 symmetry group applies when periodic boundary conditions are imposed in two orthogonal horizontal directions. The centre manifold theorem allows a reduction of the infinite dimensional problem to a bifurcation problem on C4. In normal form, the vector field on C4 commutes with an S1 symmetry, which is interpreted as a time translation symmetry. The spatial and spatio-temporal symmetries of all possible solutions are classified in terms of isotropy subgroups of D4*T2*S1.

The Origins of Time-Delay in Template Biopolymerization Processes

Time-delays are common in many physical and biological systems and they give rise to complex dynamic phenomena. The elementary processes involved in template biopolymerization, such as mRNA and protein synthesis, introduce significant time delays. However, there is not currently a systematic mapping between the individual mechanistic parameters and the time delays in these networks. We present here the development of mathematical, time-delay models for protein translation, based on PDE models, which in turn are derived through systematic approximations of first-principles mechanistic models. Theoretical analysis suggests that the key features that determine the time-delays and the agreement between the time-delay and the mechanistic models are ribosome density and distribution, i.e., the number of ribosomes on the mRNA chain relative to their maximum and their distribution along the mRNA chain. Based on analytical considerations and on computational studies, we show that the steady-state and dynamic responses of the time-delay models are in excellent agreement with the detailed mechanistic models, under physiological conditions that correspond to uniform ribosome distribution and for ribosome density up to 70%. The methodology presented here can be used for the development of reduced time-delay models of mRNA synthesis and large genetic networks. The good agreement between the time-delay and the mechanistic models will allow us to use the reduced model and advanced computational methods from nonlinear dynamics in order to perform studies that are not practical using the large-scale mechanistic models.

Stability results for in-phase and splay-phase states of solid-state laser arrays

We present a theoretical study of synchronization in N-element solid-state laser arrays. We carry out the linear stability analysis for three types of solution: the nonlasing state, the in-phase periodic state, and the splayphase state. Both nearest-neighbor (on a ring) coupling and global (all-to-all) coupling are treated; the system symmetries enable us to solve the linear stability problem for arbitrary N. We consider the general case in which the coupling coefficient iκ is complex and find that stability depends crucially on the sign of the imaginary part of κ. In the case of global coupling, we discover a surprising result: the existence of an N − 2 parameter family of frequency-locked neutrally stable states. These states should display substantial phase diffusion in the presence of noise.

Symmetry-breaking Hopf bifurcation in anisotropic systems

Abstract Symmetry-breaking Hopf bifurcation from a spatially uniform steady state of a spatially extended anisotropic system is considered. This work is motivated by the experimental observation of a Hopf bifurcation to oblique traveling rolls in electrohydrodynamic convection in planarly aligned nematic liquid crystals. Symmetry forces four traveling rolls to lose stability simultaneously. Four coupled complex ordinary differential equations describing the nonlinear interaction of the traveling rolls are analyzed using methods of equivariant bifurcation theory. Six branches of periodic solutions always bifurcate from the trivial state at the Hopf bifurcation. These correspond to traveling and standing wave patterns. In an open region of coefficient space there is a primary bifurcation to a quasiperiodic standing wave solution. The Hopf bifurcation can also lead directly to an aperiodic attractor in the form of an asymptotically stable, structurally stable heteroclinic cycle. The theory is applied to a model for the transition from normal to oblique traveling rolls.

Two-frequency forced Faraday waves: weakly damped modes and pattern selection

Abstract Recent experiments [A. Kudrolli, B. Pier, J.P. Gollub, Physica D 123 (1998) 99–111] on two-frequency parametrically excited surface waves produced an intriguing “superlattice” wave pattern near a codimension-two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wave numbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Similar patterns have been shown to exist as primary solution branches of a generic 12-dimensional D 6 + T 2 -equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities [M. Silber, M.R.E. Proctor, Phys. Rev. Lett. 81 (1998) 2450–2453]. Here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equations developed by Zhang and Vinals [J. Fluid Mech. 336 (1997) 301–330] for small amplitude, weakly damped surface waves on a semi-infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2 3 and 6 7 . For the 2 3 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension-two point. Also, we find that the 6 7 case is significantly different from the 2 3 case due to the presence of additional weakly damped harmonic modes. These additional harmonic modes can result in a stabilization of the superpatterns.

Pattern selection in ferrofluids

Pattern formation by surface instabilities in a ferrofluid is studied as a function of an applied vertical magnetic field H. The effects of sidewalls are neglected. The problem is formulated as a bifurcation problem on an appropriately chosen doubly-periodic lattice. Normal forms for both square and hexagonal lattices are given and the necessary coefficients computed from the partial differential equations. On the square lattice solutions in the form of parallel ridges are never stable near onset. For a relative magnetic permeability μ<1.4 stable squares are formed. On the hexagonal lattice a finite amplitude instability produces hexagons. Analysis of a degenerate bifurcation predicts a hysteretic transition to ridges with increasing H.

Travelling wave convection in a rotating layer

Abstract Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.

Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

We show that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [B. Fiedler, V. Flunkert, M. Georgi, P. Hovel, E. Scholl, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett. 98(11) (2007) 114101], who demonstrated that such a feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. The Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, the Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the feedback phase angle satisfies a certain restriction.