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.

The Einstein - Maxwell field equations for the class of hypersurface-orthogonal Bianchi I 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 equilibrium points of the associated dynamical system correspond to transitively self-similar cosmologies, some of which appear to be previously undiscovered. It is proven that for each value of the equation of state parameter , there is a unique self-similar cosmology which acts as a late asymptotic state for a set of models of non-zero measure in this class. It is then shown that there exists a flow-invariant compact subset of phase space which generates oscillations on the Kasner circle of equilibrium points. Monotonic functions and numerical evidence are used to support the conjecture that this set is the attractor into the past for generic models.

Meandering of a one-armed spiral tip has been noted in chemical reactions and numerical simulations. Barkley, Kness, and Tuckerman show that meandering can begin by Hopf bifurcation from a rigidly rotating spiral wave (a point that is verified in a B-Z reaction by Li, Ouyang, Petrov, and Swinney). At the codimension-two point where (in an appropriate sense) the frequency at Hopf bifurcation equals the frequency of the spiral wave, Barkley notes that spiral tip meandering can turn to linearly translating spiral tip motion.

Barkley also presents a model showing that the linear motion of the spiral tip is a resonance phenomenon, and this point is verified experimentally by Li et al. and proved rigorously by Wulff. In this paper we suggest an alternative development of Barkley's model extending the center bundle constructions of Krupa from compact groups to noncompact groups and from finite dimensions to function spaces. Our reduction works only under certain simplifying assumptions which are not valid for Euclidean group actions. Recent work of Sandstede, Scheel, and Wulff shows how to overcome these difficulties.

This approach allows us to consider various bifurcations from a rotating wave. In particular, we analyze the codimension-two Barkley bifurcation and the codimension-two Takens-Bogdanov bifurcation from a rotating wave. We also discuss Hopf bifurcation from a many-armed spiral showing that meandering and resonant linear motion of the spiral tip do not always occur.

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 SO(2) symmetry. In this paper we study the effect on this Hopf bifurcation when the problem has more than 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 Rand’s 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 Gorman’s 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.

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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 Z l , while keeping the translation symmetries. Thus, we study the effects of forced symmetry breaking fromThe 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 phaselocked case, we give conditions on the rotation number of the periodic solution and on the order l of the cyclic subgroup Z l 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.

The Einstein-Maxwell field equations for the class of orthogonal Bianchi II cosmologies with a γ -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 α-and ω-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 VI 0 cosmologies are then discussed from both a mathematical and a physical point of view.