Forced Symmetry Breaking Research Articles

Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators.

Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked-a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions. Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period-doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos.

Forced symmetry breaking as a mechanism for rogue bursts in a dissipative nonlinear dynamical lattice.

We propose an alternative to the standard mechanisms for the formation of rogue waves in a nonconservative, nonlinear lattice dynamical system. We consider an ordinary differential equation (ODE) system that features regular periodic bursting arising from forced symmetry breaking. We then connect such potentially exploding units via a diffusive lattice coupling and investigate the resulting spatiotemporal dynamics for different types of initial conditions (localized or extended). We find that in both cases, particular oscillators undergo extremely fast and large amplitude excursions, resembling a rogue wave burst. Furthermore, the probability distribution of different amplitudes exhibits bimodality, with peaks at both vanishing and very large amplitude. While this phenomenology arises over a range of coupling strengths, for large values thereof the system eventually synchronizes and the above phenomenology is suppressed. We use both distributed (such as a synchronization order parameter) and individual oscillator diagnostics to monitor the dynamics and identify potential precursors to large amplitude excursions. We also examine similar behavior with amplitude-dependent diffusive coupling.

Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of S k

Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of gradient equivariant dynamics related to neural networks. In the symmetric case, methods depend on the generic equivariant bifurcation theory of irreducible representations of the symmetric group on k symbols, S k ; in particular, the standard representation of S k . It is shown that spurious minima (non-global local minima) do not arise from spontaneous symmetry breaking but rather through a complex deformation of the landscape geometry that can be encoded by a generic S k -equivariant bifurcation. We describe minimal models for forced symmetry breaking that give a lower bound on the dynamic complexity involved in the creation of spurious minima when there is no symmetry. Results on generic bifurcation when there are quadratic equivariants are also proved; this work extends and clarifies results of Ihrig & Golubitsky and Chossat, Lauterbach & Melbourne on the instability of solutions when there are quadratic equivariants.

Imperfect O(2) symmetry in counter-rotating split-cylinder flow

The effect of a small imperfection in the counter-rotating split-cylinder flow is studied numerically. The defect is characterized by a small parameter ϵ, corresponding to the difference in the magnitude of rotations in each half of the cylinder. With the two half cylinders not rotating exactly in counter rotation, the O(2) symmetry of the exact counter-rotating case (invariance to azimuthal rotations as well as to an involution consisting of reflections about the mid-plane composed of reflections about any meridional plane) is weakly broken. This small defect results in relevant variations in the flow. For slow rotations (characterized by a small Reynolds number), the system remains axisymmetric with the imperfection only breaking the reflection symmetry about the cylinder half-height. At larger Reynolds numbers, in the absence of the imperfection, axisymmetry is broken resulting in steady states with azimuthal wavenumber m. When axisymmetry is broken in the presence of the imperfection, a background rotation is introduced. Depending on the case and the level of imperfection, either rotating waves or slow-fast dynamics with mean background rotations are found instead. The interaction between azimuthal wavenumbers m = 2 and 3 plays a crucial role in the flow. The flow is analyzed in detail, varying ϵ from a very small value of 0.01%, typical of a natural imperfection in an experimental setup, to higher values corresponding to forced symmetry breaking. The ramifications of the imperfection on various solution states found in the exact counter-rotating case for a fixed aspect ratio are investigated.

Microfluidic control over topological states in channel-confined nematic flows

Compared to isotropic liquids, orientational order of nematic liquid crystals makes their rheological properties more involved, and thus requires fine control of the flow parameters to govern the orientational patterns. In microfluidic channels with perpendicular surface alignment, nematics discontinuously transition from perpendicular structure at low flow rates to flow-aligned structure at high flow rates. Here we show how precise tuning of the driving pressure can be used to stabilize and manipulate a previously unresearched topologically protected chiral intermediate state which arises before the homeotropic to flow-aligned transition. We characterize the mechanisms underlying the transition and construct a phenomenological model to describe the critical behaviour and the phase diagram of the observed chiral flow state, and evaluate the effect of a forced symmetry breaking by introduction of a chiral dopant. Finally, we induce transitions on demand through channel geometry, application of laser tweezers, and careful control of the flow rate.

Lattice Symmetry-Breaking Perturbations for Spiral Waves

Spiral waves in two-dimensional excitable media have been observed experimentally and studied extensively. It is now well known that the symmetry properties of the medium of propagation drives many of the dynamics and bifurcations which are experimentally observed for these waves. Also, symmetry-breaking induced by boundaries, inhomogeneities, and anisotropy have all been shown to lead to different dynamical regimes from that which is predicted for mathematical models that assume infinite homogeneous and isotropic planar geometry. Recent mathematical analyses incorporating the concept of forced symmetry-breaking from the Euclidean group of all planar translations and rotations have given model-independent descriptions of the effects of media imperfections on spiral wave dynamics. In this paper, we continue this program by considering rotating waves in dynamical systems which are small perturbations of a Euclidean-equivariant dynamical system, but for which the perturbation preserves only the symmetry of a regular square lattice.

Singularity theory and forced symmetry breaking in equations

A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify (O(2), 1) problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.

Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.

Switching near a network of rotating nodes

We study the dynamics of a Z 2 ⊕ Z 2-equivariant vector field in the neighbourhood of a heteroclinic network with a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearization of the vector field has non-real eigenvalues. Trajectories starting near each node of the network turn around in space either following the periodic trajectory or due to the complex eigenvalues near the equilibria. Thus, in a network with rotating nodes, the rotations combine with transverse intersections of two-dimensional invariant manifolds to create switching near the network; close to the network, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections of the network in any given order. Our results are motivated by an example where switching was observed numerically by forced symmetry breaking of an asymptotically stable network with O(2) symmetry.

EXAMPLES OF FORCED SYMMETRY-BREAKING TO HETEROCLINIC CYCLES AND NETWORKS IN THREE-DIMENSIONAL EUCLIDEAN-INVARIANT SYSTEMS

In [Parker et al., 2008a] group theory was employed to prove the existence of homoclinic cycles in forced symmetry-breaking of simple (SC), face-centered (FCC), and body-centered (BCC) cubic planforms. In this paper we extend this classification demonstrating that more elaborate heteroclinic cycles and networks can arise through the same process. Our methods naturally generate graphs that represent possible heteroclinic cycles and networks. The results do not depend on the representation of the symmetry group and are thus quite general. This study is motivated by pattern formation in three dimensions which occur in reaction–diffusion systems, certain nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction. This work extends previous work by Parker et al. [2006, 2008a, 2008b] and Hou and Golubitsky [1997].

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms, below which there can be no persistent switching. Our results are illustrated by a numerical example.

Forced symmetry breaking from SO(3) to SO(2) for rotating waves on a sphere

In this article, we consider a small SO(2)-equivariant perturbation of a reaction–diffusion system on a sphere, which is equivariant with respect to the group SO(3) of all rigid rotations. We consider a normally hyperbolic SO(3)-group orbit of a rotating wave on a sphere that persists to a normally hyperbolic SO(2)-invariant manifold M(ε). We investigate the effects of this forced symmetry breaking by studying the perturbed dynamics induced on M(ε) by the above reaction–diffusion system. We prove that depending on the frequency vectors of the rotating waves that form the relative equilibrium SO(3)u0, these rotating waves will give SO(2)-orbits of rotating waves or SO(2)-orbits of modulated rotating waves (if some transversality conditions hold). The orbital stability of these solutions is determined as well.

Forced Symmetry-Breaking of Square Lattice Planforms

Equivariant bifurcation theory has been used to study pattern formation in various physical systems modelled by E(2)-equivariant partial differential equations. The existence of spatially doubly periodic solutions with respect to the square lattice has been the focus of much research. Previous studies have considered the four- and eight-dimensional representation of the square lattice, where the symmetry of the model is perfect. Here we consider the forced symmetry-breaking of the group orbits of translation free axial plan-forms in the four- and eight-dimensional representations. This problem is abstracted to the study of the action of the symmetry group of the perturbation on the group orbit of solutions. A partial classification for the behaviour of the group orbits is obtained, showing the existence of heteroclinic cycles and networks between equilibria. Possible areas of application are discussed including Faraday waves and Rayleigh-Benard convection. Subsequent studies will discuss other two- and three-dimensional lattices.

Nearly inviscid Faraday waves in containers with broken symmetry

In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system.The coupling to the streaming flow is responsible for various types of drift instabilities of standing waves, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves. Two model systems, obtained by projection of the Navier-Stokes-like equation onto the slowest mode of the domain, are examined to clarify the origin of this behavior. In the first the domain is an elliptically distorted cylinder while in the second it is an almost square rectangle. In both cases the forced symmetry breaking results in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction destabilizes standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems the coupling to the streaming flow triggered by these instabilities leads to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and generates behavior that resembles bursting in excitable systems. The results are compared to experiments.

Dynamics and geometry in forced symmetry breaking: a tetrahedral example

We are interested in the dynamics that can arise close to rotational group orbits after forced symmetry-breaking to discrete symmetries. In particular we ask how simple or complicated the dynamics induced by symmetric {\em linear} vector fields can be. We look in detail at related tetrahedral and dihedral examples, and there we find precise conditions for a linear field to exhibit homoclinic orbits.

Bifurcation and forced symmetry breaking in Hamiltonian systems

We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria. To cite this article: F. Grabsi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Bifurcation on the Visual Cortex with Weakly Anisotropic Lateral Coupling

Mathematical studies of drug induced geometric visual hallucinations include three components: a model (or class of models) that abstracts the structure of the primary visual cortex V1; a mathematical procedure for finding geometric patterns as solutions to the cortical models; and a method for interpreting these patterns as visual hallucinations. Ermentrout and Cowan used the Wilson--Cowan equations to model the evolution of an activity variable a(x) that represents, for example, the voltage potential a of the neuron located at point x in V1. Bressloff, Cowan, Golubitsky, Thomas, and Wiener generalize this class of models to include the orientation tuning of neurons in V1 and the Hubel and Wiesel hypercolumns. In these models, $a({\mathbf x},\phi)$ represents the voltage potential a of the neuron in the hypercolumn located at x and tuned to direction $\phi$. The work of Bressloff et al. assumes that lateral connections between hypercolumns are anisotropic; that is, neurons in neighboring hypercolumns are connected only if they are tuned to the same orientation and then only if the neurons are oriented in the cortex along the direction of their cells' preference. In this work, we first assume that lateral connections are isotropic: neurons in neighboring hypercolumns are connected whenever they have the same orientation tuning. Wolf and Geisel use such a model to study development of the visual cortex. Then we consider the effect of perturbing the lateral couplings to be weakly anisotropic. There are two common features in these models: the models are continuum models (neurons and hypercolumns are idealized as points and circles), and the models all have planar Euclidean E(2)-symmetry (when cortical lateral boundaries are ignored). The approach to pattern formation is also common. It is assumed that solutions are spatially periodic with respect to a fixed planar lattice and that patterns are formed by symmetry-breaking bifurcations (corresponding to wave vectors of shortest length) from a spatially uniform state. There are also substantial differences. In the Ermentrout--Cowan model, E(2) acts in its standard representation on R 2, whereas in the Bressloff et al. model, E(2) acts on R 2 x S 1 via the shift-twist action. In our model, isotropic coupling introduces an additional S 1-symmetry. Weak anisotropy is then thought of as a small forced symmetry-breaking from E(2)\dot{+} S 1 to E(2) in its shift-twist action. The bifurcation analyses in each of these theories proceed along similar lines, but each produces different hallucinatory images---many of which have been reported in the psychophysics literature. The Ermentrout--Cowan model produces spirals and funnels, whereas the Bressloff et al. model produces in addition thin line images including honeycombs and cobwebs. Finally, our model produces three types of time-periodic states: rotating structures such as spirals, states that appear to rush into (or out from) a tunnel with its hole in the center of the visual field, and pulsating images. Although it is known that branches of time-periodic states can emanate from steady-state bifurcations in systems with symmetry, this model provides the first examples of this phenomena in a specific class of models.

𝔻n-forced symmetry breaking of 𝕆(2)-equivariant problems

We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is 𝕆(2)-equivariant and f2 is 𝔻n-equivariant with the orthogonal group actions on z ∈ ℝ2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

Dn-forced symmetry breaking of O(2)-equivariant problems

We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is O(2)-equivariant and f2 is Dn-equivariant with the orthogonal group actions on z ∈ R2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

Symmetry breaking to a rotating wave in a lid-driven cylinder with a free surface: Experimental observation

A systematic experimental investigation of the flow in an open cylinder, driven by the constant rotation of the bottom endwall, shows that axisymmetry is spontaneously broken via a supercritical Hopf bifurcation to a rotating wave with azimuthal wave number 4. The physical mechanism responsible for the symmetry breaking is shown to be due to the instability of the shear layer that is produced by the boundary layer on the bottom rotating endwall being turned into the interior by the stationary sidewall. Comparison with other experiments and numerical studies (restricted to axisymmetric subspaces) sheds new light on disparate observations in the literature and helps distinguish between spontaneous and forced (via imperfections) symmetry breaking.