Symmetry-breaking Bifurcation Points Research Articles

A Fractional-Order Sinusoidal Discrete Map.

In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.

Computer-assisted proofs of the existence of a symmetry-breaking bifurcation point for the Kolmogorov problem

We propose a computer-assisted method to prove the existence of a symmetry-breaking bifurcation point for the Kolmogorov problem. First, we numerically show that a symmetry-breaking bifurcation point exists. Then, according to the symmetric property, we define a symmetric operator. Using this operator, we divide the space into a symmetric space and an antisymmetric space. Then, considering the Reynolds number as a variable, we construct an extended system. We confirm the existence of the symmetry-breaking bifurcation point by computer-assisted proofs of the extended system that satisfies both conditions of a bifurcation theorem. The first condition is that the system has an isolated solution and the second is that a linearized operator is bijective. We numerically construct a set containing solutions that satisfy the hypothesis of Banach’s fixed-point theorem in a certain Sobolev space and thus the first condition is satisfied. The second condition is equivalent to an equation having the unique trivial solution zero. We prove that this condition is equivalent to an inequality.

Robustness and timing of cellular differentiation through population-based symmetry breaking.

During mammalian development and homeostasis, cells often transition from a multilineage primed state to one of several differentiated cell types that are marked by the expression of mutually exclusive genetic markers. These observations have been classically explained by single-cell multistability as the dynamical basis of differentiation, where robust cell-type proportioning relies on pre-existing cell-to-cell differences. We propose a conceptually different dynamical mechanism in which cell types emerge and are maintained collectively by cell-cell communication as a novel inhomogeneous state of the coupled system. Differentiation can be triggered by cell number increase as the population grows in size, through organisation of the initial homogeneous population before the symmetry-breaking bifurcation point. Robust proportioning and reliable recovery of the differentiated cell types following a perturbation is an inherent feature of the inhomogeneous state that is collectively maintained. This dynamical mechanism is valid for systems with steady-state or oscillatory single-cell dynamics. Therefore, our results suggest that timing and subsequent differentiation in robust cell-type proportions can emerge from the cooperative behaviour of growing cell populations during development.

Competition instabilities of spike patterns for the 1D Gierer–Meinhardt and Schnakenberg models are subcritical

Spatially localized 1D spike patterns occur for various two-component reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. A competition instability of a steady-state spike pattern is a linear instability that locally preserves the sum of the heights of the spikes. This instability, which results from a zero-eigenvalue crossing of a nonlocal eigenvalue problem at a certain critical value of the inhibitor diffusivity, has been implicated from full PDE numerical simulations of various RD systems of triggering a nonlinear event leading to spike annihilation. As a result, this linear instability is believed to be a key mechanism for initiating a coarsening process of 1D spike patterns. As an extension of the linear theory, we develop and implement a weakly nonlinear theory to analyse competition instabilities associated with symmetric two-boundary spike equilibria on a finite 1D domain for the Gierer–Meinhardt and Schnakenberg RD models. Two symmetric boundary spikes interacting through a long-range bulk diffusion field is the simplest spatial configuration of interacting localized spikes that can undergo a competition instability. Within a neighborhood of the parameter value for the competition instability threshold, a multi-scale asymptotic expansion is used to derive an explicit amplitude equation for the heights of the boundary spikes. This amplitude equation confirms that the competition instability is subcritical and, moreover, it shows that the competition instability threshold corresponds to a symmetry-breaking bifurcation point where an unstable branch of asymmetric two-boundary spike equilibria emerges from the symmetric branch. Results from our weakly nonlinear analysis are confirmed from full numerical solutions of the steady-state problem using numerical bifurcation software.

Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems

Spatially localized two-dimensional spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio. Such localized, far-from-equilibrium patterns are known to exhibit a wide range of different instabilities such as breathing oscillations, spot annihilation, and spot self-replication behavior. Prior numerical simulations of the Schnakenberg and Brusselator systems have suggested that a localized peanut-shaped linear instability of a localized spot is the mechanism initiating a fully nonlinear spot self-replication event. From a development and implementation of a weakly nonlinear theory for shape deformations of a localized spot, it is shown through a normal form amplitude equation that a peanut-shaped linear instability of a steady-state spot solution is always subcritical for both the Schnakenberg and Brusselator reaction-diffusion systems. The weakly nonlinear theory is validated by using the global bifurcation software pde2path [H. Uecker, D. Wetzel, and J. D. Rademacher, Numer. Math. Theory Methods Appl., 7 (2014), pp. 58--106] to numerically compute an unstable, non-radially symmetric, steady-state spot solution branch that originates from a symmetry-breaking bifurcation point.

Symmetry-breaking bifurcation for the one-dimensional Hénon equation

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation [Formula: see text] where [Formula: see text] and [Formula: see text]. Moreover, employing a variant of Rabinowitz’s global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz’s global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz’s global bifurcation) for the problem [Formula: see text] where [Formula: see text] is a specified continuous parametrization function.

Detection of symmetry breaking bifurcations using finite element analysis packages

Trusses with geometric and loading symmetries have been used in many structures to reduce the complexity of the design. Slight asymmetries in geometry and in loading could lead to bifurcation in the structural response. Failure of such structures still occur occasionally causing major damage to the property and to the human lives. A fully nonlinear structural analysis is expected to detect such symmetry breaking bifurcations. In this paper, we conducted the bifurcation analysis of a two-bar truss and a shallow arch structure with seven bars. Two program packages Gesa and Ansys based on finite elements method have been used to detect the symmetry breaking bifurcation points. However, unlike typical bifurcation analysis packages they cannot detect the bifurcation point without inserting a small perturbation in the initial geometry or the loading of the structure. Such a bifurcation can be easily missed in finite element-based analysis. The theoretical analysis reveals that the bifurcation leads to a much lower critical load in the presence of small asymmetry compared to the symmetric case. The results are verified by using Gesa program in Matlab for fully nonlinear analysis and with results by using Ansys commercial program. The two structural examples serve to illustrate the limitations of widely used finite element analysis packages for nonlinear bifurcation analysis.

MIXED FOURIER-LEGENDRE SPECTRAL METHODS FOR THE MULTIPLE SOLUTIONS OF THE SCHRODINGER EQUATION ON THE UNIT DISK

In this paper, we first compute the multiple non-trivial solutions of the Schrodinger equation on a unit disk, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with the mixed Fourier-Legendre spectral and pseudospectral methods. After that, we propose the extended systems, which can detect the symmetry-breaking bifurcation points on the branch of the O(2) symmetric positive solutions. We also compute the multiple positive solutions with various symmetries of the Schrodinger equation by the branch switching method based on the Liapunov-Schmidt reduction. Finally, the bifurcation diagrams are constructed, showing the symmetry/peak breaking phenomena of the Schr¨odinger equation. Numerical results demonstrate the effectiveness of these approaches.

An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation

A new modification of homotopy analysis method (HAM) is proposed in this paper. The auxiliary differential operator is specifically chosen so that more than one secular term must be eliminated. The proposed method can capture asymmetric and period-2 solutions with satisfactory accuracy and hence can be used to predict symmetry-breaking and period-doubling bifurcation points. The variation of accuracy is investigated when different number of frequencies are considered.

Equilibrium states and chaos in an oscillating double-well potential

We investigate numerically parametrically driven coupled nonlinear Schr\"odinger equations modeling the dynamics of coupled wave fields in a periodically oscillating double-well potential. The equations describe, among other things, two coupled periodically curved optical waveguides with Kerr nonlinearity or Bose-Einstein condensates in a double-well potential that is shaken horizontally and periodically in time. In particular, we study the persistence of equilibrium states of the undriven system due to the presence of the parametric drive. Using numerical continuations of periodic orbits and calculating the corresponding Floquet multipliers, we find that the drive can (de)stabilize a continuation of an equilibrium state indicated by the change in the (in)stability of the orbit, showing that parametric drives can provide a powerful control to nonlinear (optical- or matter-wave-) field tunneling. We also discuss the appearance of chaotic regions reported in previous studies that is due to destabilization of a periodic orbit. Analytical approximations based on an averaging method are presented. Using perturbation theory, the influence of the drive on the symmetry-breaking bifurcation point is analyzed.

Bifurcation method for computing the multiple positive solutions to p-Henon equation

In this paper, we proposed the algorithms to solving the D4 symmetric positive solutions to the boundary value problem of p-Henon equation. Taking r in p-Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the D4 symmetric positive solutions is found via the extended systems. Based on the Liapunov–Schmidt reduction, other symmetric positive solutions are computed by the branch switching method.

A bifurcation method for solving multiple positive solutions to the boundary value problem of the Henon equation on a unit disk

Three algorithms based on the bifurcation theory are proposed to compute the O ( 2 ) symmetric positive solutions to the boundary value problem of the Henon equation on the unit disk. Taking l in the Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O ( 2 ) symmetric positive solutions is found via the extended systems. Finally, other symmetric positive solutions are computed by the branch switching method based on the Lyapunov–Schmidt reduction.

Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions

Let A:={a<|x|<1+a}⊂RN and p⩾2. We consider the Neumann problemε2Δu−u+up=0in A,∂νu=0on ∂A. Let λ=1/ε2. When λ is large, we prove the existence of a smooth curve {(λ,u(λ))} consisting of radially symmetric and radially decreasing solutions concentrating on {|x|=a}. Moreover, checking the transversality condition, we show that this curve has infinitely many symmetry breaking bifurcation points from which continua consisting of nonradially symmetric solutions emanate. If N=2, then the closure of each bifurcating continuum is locally homeomorphic to a disk. When the domain is a rectangle (0,1)×(0,a)⊂R2, we show that a curve consisting of one-dimensional solutions concentrating on {0}×[0,a] has infinitely many symmetry breaking bifurcation points. Extending this solution with even reflection, we obtain a new entire solution.

Bifurcation method for solving multiple positive solutions to Henon equation on the unit cube

Three algorithms based on the bifurcation method are applied to solving the D 4(3) symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation points are found via the extended systems on the branch of the D 4(3) symmetric positive solutions. Finally, other symmetric positive solutions are computed by the branch switching method based on the Liapunov–Schmidt reduction.

Equivariant bifurcation index

We consider a bifurcation index BIF G ( ν k 0 − 1 ) ∈ U ( G ) defined in terms of the degree for G -equivariant gradient maps, see Gȩba (1997) [21], Rybicki (1994) [22], Rybicki (2005) [23], where G is a real, compact, connected Lie group and U ( G ) is the Euler ring of G , see tom Dieck (1977) [29], tom Dieck (1987) [30]. The main result of this article is the following: BIF G ( ν k 0 − 1 ) ≠ Θ ∈ U ( G ) iff BIF T ( ν k 0 − 1 ) ≠ Θ ∈ U ( T ) , where T ⊂ G is a maximal torus of G . It is also shown that all the bifurcation points of weak solutions of the following problem { − Δ u = f ( u , λ ) in B n , u = 0 on S n − 1 , are global bifurcation points. Additionally, the global symmetry breaking bifurcation points are characterised.

Bifurcation method for solving multiple positive solutions to boundary value problem of p-Henon equation on the unit disk

In this paper, an algorithm is proposed to solve the O(2) symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking l in the p-Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O(2) symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.

Bifurcation and stability analysis of laminar flow in curved ducts

AbstractThe development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two‐dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio γ (γ=0.8…1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Bifurcation method for solving multiple positive solutions to Henon equation

Three algorithms based on the bifurcation method are applied to solving the D 4 symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bifurcation parameter, the D 4−Σ d (D 4 − Σ1, D 4 − Σ2) symmetry-breaking bifurcation points on the branch of the D 4 symmetric positive solutions are found via the extended systems. Finally, Σ d (Σ1, Σ2) symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.

Forced convection in slightly curved microchannels

To address the effects of curvature, initial conditions and disturbances, a numerical study is made on the fully-developed bifurcation structure and stability of the forced convection in curved microchannels of square cross-section and curvature ratio 5 × 10 −6. No matter how small it is, the channel curvature always generates the secondary flow in the channel cross-plane which increases the mean friction factor moderately and the Nusselt number significantly. Unknown initial conditions of convection lead to the co-existence of multiple steady fully-developed flows of various structures. Ten solution branches (either symmetric or asymmetric) are found with eight symmetry-breaking bifurcation points and thirty-one limit points. Thus a rich solution structure exists with the co-existence of various flow states over certain ranges of governing parameters. Dynamic responses of the multiple steady flows to finite random disturbances are examined by the direct transient computation. It is found that possible physically realizable fully-developed flows under the effect of unknown disturbances evolve, as the Dean number increases, from a stable steady 2-cell state at lower Dean number to a temporal periodic oscillation, another stable steady 2-cell state, a temporal intermittent oscillation, and a chaotic temporal oscillation. There exist no stable steady fully-developed flows in some ranges of governing parameters. Both the mean friction factor and the mean Nusselt number are also obtained and analyzed with their correlating relations listed.

Bifurcation and stability of forced convection in curved ducts of square cross-section

A numerical study is made on the fully developed bifurcation structure and stability of the forced convection in a curved duct of square cross-section (Dean problem). In addition to the extension of three known solution branches to the high Dean number region, three new asymmetric solution branches are found from three symmetry-breaking bifurcation points on the isolated symmetric branch. The flows on these new branches are either an asymmetric two-cell state or an asymmetric seven-cell structure. The linear stability of multiple solutions are conclusively determined by solving the eigenvalue system for all eigenvalues. Only two-cell flows on the primary symmetric branch and on the part of isolated symmetric branch are linearly stable. The symmetric six-cell flow is also linearly unstable to asymmetric disturbances although it was ascertained to be stable to symmetric disturbances in the literature. The linear stability is observed to change along some solution branch even without passing any bifurcation or limit points. Furthermore, dynamic responses of the multiple solutions to finite random disturbances are also examined by the direct transient computation. It is found that possible physically realizable fully developed flows evolve, as the Dean number increases, from a stable steady two-cell state at lower Dean number to a temporal periodic oscillation state, another stable steady two-cell state, a temporal intermittent oscillation, and a chaotic temporal oscillation. Among them, three temporal oscillation states have not been reported in the literature. A temporal periodic oscillation between symmetric/asymmetric two-cell flows and symmetric/asymmetric four-cell flows are found in the range where there are no stable steady fully developed solutions. The symmetry-breaking point on the primary solution branch is determined to be a sub-critical Hopf point by the transient computation.