Equivariant Bifurcation Theory Research Articles

Bifurcation analysis in the regularized four-sided cavity flow: Equilibrium states in a D2-symmetric fluid system

In this study, we investigate a handful of equilibrium states and their hydrodynamic stability for the incompressible flow in a square cavity driven by the tangential simultaneous motion of all of its lids at the same speed. This problem has been investigated in the recent past, albeit only partially and always disregarding the singular boundary conditions at the corners. The implications of the system symmetries have not been rigorously addressed, either. In our study, we introduce a regularized version of the boundary conditions to avoid the corner singularities, in a way that preserves the main features of the original setup. In addition, the Navier–Stokes equations are discretized by means of highly accurate Chebyshev spectral methods that provide exponential convergence of all computed flows and consistently eliminate any potential source of structural instability of the bifurcation scenario. We employ Newton–Krylov solvers, implemented within continuation algorithms, to accurately compute equilibrium solutions. Linear stability analysis of both the primary symmetric base flow and the secondary asymmetric states uncovers new branches of fully asymmetric steady states. The analysis has allowed identification of six previously undetected bifurcations, all of which are associated with the disruption of either the rotational invariance or the reflection symmetry. Some of these bifurcations have been found to be quite clustered in some regions of the parameter space, which points at the underlying action of higher codimension mechanisms. Notably, all the bifurcations reported occur within the range of low to moderate Reynolds numbers, making the regularized four-sided lid-driven cavity flow a reliable benchmark for assessing different numerical schemes in the context of Navier–Stokes equivariant bifurcation theory.

On the bifurcation of spacelike hypersurfaces with constant mean curvature in spacetimes

In a time-oriented Lorentzian manifold M‾n+1, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants or the local rigidity of family {Ωτ} of open sets of M‾n+1 whose boundaries ∂Ωτ are closed (compact without boundary) spacelike H-hypersurfaces, this is, closed spacelike hypersurfaces with constant mean curvature H. In particular, we do this study for a certain family of open sets in a spatially closed generalized Robertson-Walker spacetime, namely, Lorentz warped products (−I×fMn,−dt2+f2〈,〉M) with 1-dimensional negative definite base (I,−dt2), closed Riemannian fiber (Mn,〈,〉M) and warping function f:I→(0,+∞). In this case, the results are obtained considering some appropriate hypotheses that depend of f and of the behavior of the eigenvalues of Laplacian operator of Mn. Applications to such families in some spacetimes, like the de Sitter space, are also given.

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.

Patterns and Quasipatterns from the Superposition of Two Hexagonal Lattices

When two-dimensional pattern-forming problems are posed on a periodic domain, classical techniques (Lyapunov-Schmidt, equivariant bifurcation theory) give considerable information about what periodic patterns are formed in the transition where the featureless state loses stability. When the problem is posed on the whole plane, these periodic patterns are still present. Recent work on the Swift-Hohenberg equation (an archetypal pattern-forming partial differential equation) has proved the existence of quasipatterns, which are not spatially periodic and yet still have long-range order. Quasipatterns may have 8-fold, 10-fold, 12-fold and higher rotational symmetry, which preclude periodicity. There are also quasipatterns with 6-fold rotational symmetry made up from the superposition of two equal-amplitude hexagonal patterns rotated by almost any angle $\alpha$ with respect to each other. Here, we revisit the Swift-Hohenberg equation (with quadratic as well as cubic nonlinearities) and prove existence of several new quasipatterns. The most surprising are hexa-rolls: periodic and quasiperiodic patterns made from the superposition of hexagons and rolls (stripes) oriented in almost any direction with respect to each other and with any relative translation; these bifurcate directly from the featureless solution. In addition, we find quasipatterns made from the superposition of hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). We consider the periodic case as well, and extend the class of known solutions, including the superposition of hexagons and rolls. While we have focused on the Swift-Hohenberg equation, our work contributes to the general question of what periodic or quasiperiodic patterns should be found generically in pattern-forming problems on the plane.

Symmetry & critical points for a model shallow neural network

Using methods based on the analysis of real analytic functions, symmetry and equivariant bifurcation theory, we obtain sharp results on families of critical points of spurious minima that occur in optimization problems associated with fitting two-layer ReLU networks with k hidden neurons. The main mathematical result proved is to obtain power series representations of families of critical points of spurious minima in terms of 1/k (coefficients independent of k). We also give a path based formulation that naturally connects the critical points with critical points of an associated linear, but highly singular, optimization problem. These critical points closely approximate the critical points in the original problem.The mathematical theory is used to derive results on the original problem in neural nets. For example, precise estimates for several quantities that show that not all spurious minima are alike. In particular, we show that while the loss function at certain types of spurious minima decays to zero like k−1, in other cases the loss converges to a strictly positive constant.

Local rigidity, bifurcation, and stability of $H_f$-hypersurfaces in weighted Killing warped products

In a weighted Killing warped product Mn f ×ρR with warping metric h , iM+ρ2 dt, where the warping function ρ is a real positive function defined on Mn and the weighted function f does not depend on the parameter t ∈ R, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants, or the local rigidity for a family of open sets {Ωγ}γ∈I whose boundaries ∂Ωγ are hypersurfaces with constant weightedmean curvature. For this, we analyze the number of negative eigenvalues of a certain Schr¨odinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface x: Σn ,→ Mn f ×ρ R with constant weighted mean curvature in terms of the first eigenvalue of the f-Laplacian of Σn.

Lattice Solutions in a Ginzburg\u2013Landau Model for a Chiral Magnet

We examine micromagnetic pattern formation in chiral magnets, driven by the competition of Heisenberg exchange, Dzyaloshinskii–Moriya interaction, easy-plane anisotropy and thermodynamic Landau potentials. Based on equivariant bifurcation theory, we prove existence of lattice solutions branching off the zero magnetization state and investigate their stability. We observe in particular the stabilization of quadratic vortex–antivortex lattice configurations and instability of hexagonal skyrmion lattice configurations, and we illustrate our findings by numerical studies.

Connected sets of solutions of symmetric elliptic systems

The purpose of this paper is twofold. First we study bifurcations of connected sets of critical orbits of some invariant functional from a given family of critical orbits. We use techniques of equivariant bifurcation theory to obtain a Rabinowitz type alternative for symmetric gradient operators. The second aim is to apply the abstract results to study orbits of nonconstant solutions of a nonlinear Neumann problem.

Multiple periodic orbits from Hopf bifurcation in a hierarchical neural network with [formula omitted]-symmetry and delays

This paper concerns a model of hierarchical neural network with Dn×Dn-symmetry and time delays. The linear stability of the steady state is obtained via an artful factorization technique of the characteristic equation corresponding to the linearized system. And we also perform a delay-induced Hopf bifurcation analysis involving spatio-temporal patterns of bifurcated periodic solutions relying on the equivariant Hopf bifurcation theory for retarded functional differential equations and the representation method of Lie groups. Some numerical simulations for a simple example are carried out to demonstrate the application of our theoretical conclusion that multiple periodic orbits could occur simultaneously from Hopf bifurcation in the model.

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

Bifurcation and local rigidity of constant second mean curvature hypersurfaces in Riemannian warped products

In a Riemannian warped product I×fMn, where I⊂R is an open interval, f is a positive real function defined on I and Mn is a compact Riemannian manifold without boundary, we use equivariant bifurcation theory in order to establish sufficient conditions, in terms of f and the spectrum of the Laplacian on Mn, that allow us to guarantee the existence of bifurcation instants or the local rigidity of a certain family of open sets whose boundaries are H2-hypersurfaces, namely, whose boundaries are hypersurfaces with constant second mean curvature H2. For each of our results, we have provided a considerable number of examples that verify all the assumptions under consideration.

Symmetries in the Lorenz-96 Model

The Lorenz-96 model is widely used as a test model for various applications, such as data assimilation methods. This symmetric model has the forcing [Formula: see text] and the dimension [Formula: see text] as parameters and is [Formula: see text]-equivariant. In this paper, we unravel its dynamics for [Formula: see text] using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces that play an important role in this model. We exploit them in order to generalize results from a low dimension to all multiples of that dimension. We discuss symmetry for periodic orbits as well. Our analysis leads to proofs of the existence of pitchfork bifurcations for [Formula: see text] in specific dimensions [Formula: see text]: In all even dimensions, the equilibrium [Formula: see text] exhibits a supercritical pitchfork bifurcation. In dimensions [Formula: see text], [Formula: see text], a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one. Furthermore, numerical observations reveal that in dimension [Formula: see text], where [Formula: see text] and [Formula: see text] is odd, there is a finite cascade of exactly [Formula: see text] subsequent pitchfork bifurcations, whose bifurcation values are independent of [Formula: see text]. This structure is discussed and interpreted in light of the symmetries of the model.

Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

Abstract The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in S n {S^{n}} . In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO ⁡ ( n ) {\operatorname{SO}(n)} -symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO ⁡ ( n ) {\operatorname{SO}(n)} -invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].

Symmetric Liapunov center theorem for minimal orbit

Using the techniques of equivariant bifurcation theory we prove the existence of non-stationary periodic solutions of Γ-symmetric systems q¨(t)=−∇U(q(t)) in any neighborhood of an isolated orbit of minima Γ(q0) of the potential U. We show the strength of our result by proving the existence of new families of periodic orbits in the Lennard-Jones two- and three-body problems and in the Schwarzschild three-body problem.

Bifurcation and stability analysis with the role of normal form symmetries on the harmonic streamwise forced oscillation of the cylinder wake

The dynamics of the cylinder wake subjected to harmonic inline oscillation is investigated in this work. Two-dimensional numerical computations are performed for Re=200 to find the effect of inline sinusoidal oscillation on the flow pattern leading to the variation of cylinder lift, drag and wake shedding frequency. Two complex primary modes of the v-velocity field, resulting from a proper orthogonal decomposition (POD) analysis, are considered to model and predict the nonlinear forced wake dynamics governed by the interaction of vortex shedding modes. The equivariant bifurcation theory in the presence of normal form symmetry O(2)×S1 is employed to classify the mode interaction solution branches with respect to lower order symmetries. The coupled amplitude equations are developed with the frequency saturation information included by the addition of complex coefficients. The symmetry-based model is expanded up to 7th power thus including spatio-temporal effects. The coefficients of the model are obtained from CFD simulations. The novelty of this work is that the amplitude equations are derived purely from the mode symmetries and not from a Galerkin reduction of the Navier–Stokes equations.The inline cylinder oscillation effect on the wake dynamics is captured by bifurcation analysis of this model under the variation of the two linear coefficients. As the oscillation amplitude increases, two limit cycles of the model undergo a symmetry-breaking bifurcation leading to a quasi-periodic state. For amplitude-to-diameter ratio A∕D=0.5, the quasi-periodic state undergoes a torus doubling bifurcation. The bifurcated mode S frequency matches the lift coefficient shedding frequency for A∕D=0.5, obtained from the numerical (CFD) computations. The bifurcated modulated traveling waves have mode S as the dominant v-velocity mode, which confirms the symmetric period-doubled quasi-steady v-velocity pattern observed in CFD. The stability behavior of the bifurcated solution branches is also captured in the Poincare plane. The model correctly predicts the bifurcation sequence of the wake dynamic. This knowledge opens the possibility to develop low order controllers for the wake–structure interaction.

Symmetries Constrain Dynamics in a Family of Balanced Neural Networks

We examine a family of random firing-rate neural networks in which we enforce the neurobiological constraint of Dale’s Law—each neuron makes either excitatory or inhibitory connections onto its post-synaptic targets. We find that this constrained system may be described as a perturbation from a system with nontrivial symmetries. We analyze the symmetric system using the tools of equivariant bifurcation theory and demonstrate that the symmetry-implied structures remain evident in the perturbed system. In comparison, spectral characteristics of the network coupling matrix are relatively uninformative about the behavior of the constrained system.

Harmonically free group actions and equivariant bifurcation in the Yamabe problem

We study multiplicity of constant scalar curvature metrics in products of a compact closed manifold and a compact manifold with boundary using equivariant bifurcation theory.

Huygens' clocks revisited.

In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.

Bifurcations of Central Configurations in the Four-Body Problem with Some Equal Masses

We study the bifurcations of central configurations of the Newtonian four-body problem when some of the masses are equal. First, we continue numerically the solutions for the equal mass case, and we find values of the mass parameter at which the number of solutions changes. Then, using the Krawczyk method and some result of equivariant bifurcation theory, we rigorously prove the existence of such bifurcations and classify them.

Topological bifurcations of spatial central configurations in the N-body problem

We study topological bifurcations of classes of spatial central configurations (s.c.c.) from the following highly symmetrical families: two nested regular tetrahedra, octahedra and cubes, two nested rotated regular tetrahedra and two dual regular polyhedra for 14 bodies. We prove the existence of local and global topological bifurcations of s.c.c. from these families. We seek new classes of s.c.c. by using equivariant bifurcation theory. It is worth pointing out that the shapes of the bifurcating families are less symmetrical than the shapes of the considered families of s.c.c.