William F. Langford

Pattern formation and bistability in flow between counterrotating cylinders

Abstract In the Taykor-Coutte experiment on fluid flow counterrotating cylinders, there is a bicritical point where the onset of instabilities to Taylor vortex flow (a steady-state bifurcation) and spiral vortex flow (a Hopf bifurcation) meet. The nonlinear mode interactions near this bicritical point are analyzed, exploiting the role of symmetry in the bifurcation theory, and with emphasis of the relevance to experiments, for a range of raduis ratios 0.43 ≤η≤0.98. The mechanism of the pattern formation is elucidated, and several new flow patterns and transitions are predicted, including wavy vortices, bistability, hysteresis, and up to 7 quasiperiodic flows.

Normal form for generalized Hopf bifurcation with non-semisimple 1 : 1 resonance

The primary result of this research is the derivation of an explicit formula for the Poincaré-Birkhoff normal form of the generalized Hopf bifurcation with non-semisimple 1:1 resonance. The classical nonuniqueness of the normal form is resolved by the choice of complementary space which yields a unique equivariant normal form. The 4 leading complex constants in the normal form are calculated in terms of the original coefficients of both the quadratic and cubic nonlinearities by two different algorithms. In addition, the universal unfolding of the degenerate linear operator is explicitly determined. The dominant normal forms are then obtained by rescaling the variables. Finally, the methods of averaging and normal forms are compared. It is shown that the dominant terms of the equivariant normal form are, indeed, the same as those of the averaged equations with a particular choice for the constant of integration.

Classification and unfoldings of 1:2 resonant Hopf Bifurcation

In this paper, we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1:2 ratio. This corresponds to a Hopf-Hopf mode interaction with 1:2 resonance, as occurs in the context of dissipative mechanical systems. Using an approach based on Liapunov-Schmidt reduction and singularity theory, we give a framework in which to study these problems and their perturbations in two cases: no distinguished parameter, and one distinguished (bifurcation) parameter. We give a complete classification of the generic cases and their unfoldings.

Pattern Formation: Symmetry Methods and Applications

Towards analyzing the dynamics of flames by D. Armbruster, E. F. Stone, and R. W. Heiland Symmetries in modulated traveling waves in combustion: Jumping ponies on a merry-go-round by A. Bayliss, B. J. Matkowsky, and H. Riecke Modulated traveling waves for the Kuramoto-Sivashinsky equation by H. S. Brown and I. G. Kevrekidis Length scales in phase transition models: Phase field, Cahn-Hilliard, and blow-up problems by G. Caginalp Veronese and the detectives: Finding the symmetry of attractors by D. R. J. Chillingworth

Hopf Meets Hamilton Under Whitney’s Umbrella

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Modulated rotating waves in O(2) mode interactions

-symmetric maps with hidden Euclidean symmetry by J. D. Crawford Modelling travelling waves of spatial patterning in morphogenesis by G. C. Cruywagen Analysis of models of pancreatic

Models of Cheyne-Stokes respiration with cardiovascular pathologies

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Nonlinear delay equations with fluctuating delay: Application to regenerative chatter

-cells exhibiting temporal pattern formation by G. de Vries, R. M. Miura, and M. C. Pemarowski Translating patterns in a generalized Ginzburg-Landau amplitude equation by J. Duan Geometric methods in bifurcation theory by M. J. Field Secondary Hopf bifurcation caused by steady-state steady-state mode interaction by K. Gatermann and B. Werner Symmetry breaking bifurcations in spherical Benard convection. Part I: Results from singularity theory by C. Geiger, G. Dangelmayr, J. D. Rodriguez, and W. Guttinger Symmetry breaking bifurcations in spherical Benard convection. Part II: Numerical results by J. D. Rodriguez, C. Geiger, G. Dangelmayr, and W. Guttinger On a dynamical model for phase transformation in nonlinear elasticity by W. D. Kalies and P. J. Holmes System symmetry breaking and Shilnikov dynamics by E. Knobloch Generalizations of a result on symmetry groups of attractors by I. Melbourne Instabilities induced by differential flows by M. Menzinger and A. B. Rovinsky Self-organized zoning in crystals: Free boundaries, matched asymptotics, and bifurcation by P. Ortoleva Some aspects of successive bifurcations in the Couette-Taylor problem by J. K. Scheurle Bifurcating waves in coupled cells described by delay-differential equations by J. H. Wu.

Symmetry-breaking bifurcations on multidimensional fixed point subspaces

In Hamiltonian mechanics, the classical Hopf bifurcation theorem is not directly applicable. Instead, there is an analogous “Hamiltonian-Hopf bifurcation theorem” in which two pairs of complex conjugate eigenvalues approach the imaginary axis symmetrically from the left and right, then merge in double purely imaginary eigenvalues and separate along the imaginary axis (or the reverse). This phenomenon has codimension one within the class of Hamiltonian systems. In the general case of non-Hamiltonian vector fields, the occurrence of double imaginary eigenvalues has codimension three. This paper presents a first investigation of the interface between these two cases. They meet in an interesting topological singularity known as Whitney’s umbrella. We show that the Hamiltonian case lies on the “handle” of Whitney’s umbrella. This allows us to investigate near-Hamiltonian or weakly dissipative systems that lie in a tubular neigh-borhood of the handle of Whitney’s umbrella.

Hysteresis in a Rotating Differentially Heated Spherical Shell of Boussinesq Fluid

The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reaction-diffusion problems.