Victor G. LeBlanc

Asymptotic states of magnetic Bianchi I cosmologies

The Einstein--Maxwell field equations for orthogonal Bianchi VI cosmologies with a -law perfect fluid and a pure, homogeneous source-free magnetic field are written as an autonomous differential equation in terms of expansion-normalized variables. The associated dynamical system is studied in order to determine the past, intermediate and future evolution of these models. All asymptotic states of the models, and the likelihood that they will occur, are described. In addition, it is shown that there is a finite probability that an arbitrarily selected model will be close to isotropy during some time interval in its evolution.

Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry

Abstract We consider a general class of first-order nonlinear delay-differential equations (DDEs) with reflectional symmetry, and study completely the bifurcations of the trivial equilibrium under some generic conditions on the Taylor coefficients of the DDE. Our analysis reveals a Hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two Takens–Bogdanov point in parameter space. We compute the normal form coefficients of the reduced vector field on the centre manifold in terms of the Taylor coefficients of the original DDE, and in contrast to many previous bifurcation analyses of DDEs, we also compute the unfolding parameters in terms of these coefficients. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. We illustrate these results using simple model systems relevant to the areas of neural networks and atmospheric physics, and show that the results agree with numerical simulations.

Hopf Bifurcation from Rotating Waves and Patterns in Physical Space

Summary. Hopf bifurcations from time periodic rotating waves to two frequency tori have been studied for a number of years by a variety of authors including Rand and Renardy. Rotating waves are solutions to partial differential equations where time evolution is the same as spatial rotation. Thus rotating waves can exist mathematically only in problems that have at least \bf SO (2) symmetry. In this paper we study the effect on this Hopf bifurcation when the problem has more than \bf SO (2) symmetry. These effects manifest themselves in physical space and not in phase space. We use as motivating examples the experiments of Gorman et al . on porous plug burner flames, of Swinney et al . on the Taylor-Couette system, and of a variety of people on meandering spiral waves in the Belousov-Zhabotinsky reaction. In our analysis we recover and complete Rands classification of modulated wavy vortices in the Taylor-Couette system. It is both curious and intriguing that the spatial manifestations of the two frequency motions in each of these experiments is different, and it is these differences that we seek to explain. In particular, we give a mathematical explanation of the differences between the nonuniform rotation of cellular flames in Gormans experiments and the meandering of spiral waves in the Belousov-Zhabotinsky reaction. Our approach is based on the center bundle construction of Krupa with compact group actions and its extension to noncompact group actions by Sandstede, Scheel, and Wulff.

Resonant Hopf–Hopf Interactions in Delay Differential Equations

A second-order delay differential equation (DDE) which models certain mechanical and neuromechanical regulatory systems is analyzed. We show that there are points in parameter space for which 1:2 resonant Hopf–Hopf interaction occurs at a steady state of the system. Using a singularity theoretic classification scheme [as presented by LeBlanc (1995) and LeBlanc and Langford (1996)], we then give the bifurcation diagrams for periodic solutions in two cases: variation of the delay and variation of the feedback gain near the resonance point. In both cases, period-doubling bifurcations of periodic solutions occur, and it is argued that two tori can bifurcate from these periodic solutions near the period doubling point. These results are then compared to numerical simulations of the DDE.

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.

Translational Symmetry-Breaking for Spiral Waves

Summary. Spiral waves are observed in numerous physical situations, ranging from Belousov-Zhabotinsky (BZ) chemical reactions, to cardiac tissue, to slime-mold aggregates. Mathematical models with Euclidean symmetry have recently been developed to describe the dynamic behavior (for example, meandering) of spiral waves in excitable media. However, no physical experiment is ever infinite in spatial extent, so the Euclidean symmetry is only approximate. Experiments on spiral waves show that inhomogeneities can anchor spirals and that boundary effects (for example, boundary drifting) become very important when the size of the spiral core is comparable to the size of the reacting medium. Spiral anchoring and boundary drifting cannot be explained by the Euclidean model alone.In this paper, we investigate the effects on spiral wave dynamics of breaking the translation symmetry while keeping the rotation symmetry. This is accomplished by introducing a small perturbation in the five-dimensional center bundle equations (describing Hopf bifurcation from one-armed spiral waves) which is SO(2)-equivariant but not equivariant under translations. We then study the effects of this perturbation on rigid spiral rotation, on quasi-periodic meandering and on drifting.

Rotational symmetry breaking for spiral waves

We continue our investigation of the effects of forced symmetry breaking on the dynamics of spiral waves in excitable media. In a previous paper, we have studied the effects of breaking the translation symmetry, while keeping the rotation symmetries in the Euclidean equivariant models for spiral waves. In this paper, we will investigate the effects of breaking the rotational symmetry SO(2) of these Euclidean models to a cyclic subgroup Zl, while keeping the translation symmetries. Thus, we study the effects of forced symmetry breaking from SE(2) = C SO(2) to Σl = C Zl. The goal is to try to obtain a phenomenological explanation of recent experiments on the dynamics of spiral waves in anisotropic media (e.g. numerical simulations of models for electrical activity in heart tissue). Specifically, we show that rotating waves get perturbed to discrete rotating waves. Also, in contrast to the Euclidean case, we show that meandering waves can undergo phase locking, or meander quasi-periodically in such a way that the overall meander path has only discrete spatial rotation symmetry. In the phase-locked case, we give conditions on the rotation number of the periodic solution and on the order l of the cyclic subgroup Zl which lead to a slow linear drifting of the spiral tip superimposed on the periodic epicycle-like motion. These results are strikingly similar to previously mentioned experimental results on spirals in anisotropic media.

Bianchi II magnetic cosmologies

The Einstein-Maxwell field equations for the class of orthogonal Bianchi II cosmologies with a -law perfect fluid and a pure, homogeneous source-free magnetic field are written as an autonomous differential equation in terms of expansion-normalized variables. The future and past asymptotic states for these models are given by the - and -limit sets for the orbits of the resulting dynamical system. These limit sets are studied in detail. As a by-product of the analysis, we find new transitively self-similar solutions to the field equations which act as attractors into the future for a set of models of non-zero measure in this class. The behaviour into the past is described by mixmaster oscillations. Similarities and differences between the asymptotic states of magnetic Bianchi I, II and cosmologies are then discussed from both a mathematical and a physical point of view.

Versal unfoldings for linear retarded functional differential equations

We consider parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds, and address the question of versality of the resulting parametrized family of linear ordinary differential equations. A sufficient criterion for versality is given in terms of readily computable quantities. In the case where the unfolding is not versal, we show how to construct a perturbation of the original linear RFDE (in terms of delay differential operators) whose finite-dimensional projection generates a versal unfolding. We illustrate the theory with several examples, and comment on the applicability of these results to bifurcation analyses of nonlinear RFDEs.

Spiral Anchoring in Media with Multiple Inhomogeneities: A Dynamical System Approach

Abstract The spiral is one of nature’s more ubiquitous shapes: It can be seen in various media, from galactic geometry to cardiac tissue. Mathematically, spiral waves arise as solutions to reaction–diffusion partial differential equations (RDS). In the literature, various experimentally observed dynamical states and bifurcations of spiral waves have been explained using the underlying Euclidean symmetry of the RDS—see for example (Barkley in Phys. Rev. Lett. 68:2090–2093, 1992; Phys. Rev. Lett. 76:164–167, 1994; Sandstede et al. in C. R. Acad. Sci. 324:153–158, 1997; J. Differ. Equ. 141:122–149, 1997; J. Nonlinear Sci. 9:439–478, 1999), or additionally using the concept of forced Euclidean symmetry-breaking for situations where an inhomogeneity or anisotropy is present—see (LeBlanc in Nonlinearity 15:1179–1203, 2002; LeBlanc and Wulff in J. Nonlinear Sci. 10:569–601, 2000). In this paper, we further investigate the role of medium inhomogeneities on spiral wave dynamics by considering the effects of several localized sites of inhomogeneity. Using a model-independent approach based on n>1 simultaneous translational symmetry-breaking perturbations of the dynamics near rotating waves, we fully characterize the local anchoring behavior of the spiral wave in the n-dimensional parameter space of relative “amplitudes” of the individual perturbations. For the case n=2, we supplement the local anchoring results with a classification of the generic one-parameter bifurcation diagrams of anchored states which can be obtained by circling the origin of the two-dimensional amplitude parameter space. Numerical examples are given to illustrate our various results.