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.     

Origin of subcritical shear-banding instability in a dense two-dimensional sheared granular fluid

The granular plane Couette flow is known to be linearly unstable to shear-banding instability beyond a critical density, and our nonlinear analysis suggests that the nature of bifurcation (supercritical/subcritical) in dense flows depends strongly on the choice of the constitutive model. While the standard Enskog model for nearly elastic hard-disks predicts supercritical bifurcations for moderate to dense systems, a more realistic model with global equation of states for hard-disks (that are likely to hold for the whole range of densities) predicts a subcritical bifurcation in the dense limit. The latter prediction agrees with recent particle simulations of a sheared inelastic hard-disk system.     

Isolas, cusps and global bifurcations in an electronic oscillator

The aim of the present work is to describe the bifurcation behaviour of a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch: that is, their bifurcation diagram showed an eight-shaped isola, with a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its destruction in an isola centre with a cusp of periodic orbits. Finally, the introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifurcation) and to homoclinic connections of the non-trivial equilibria. The isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil?nikov homoclinic and heteroclinic connections, related to the symmetric and asymmetric periodic orbits, emerge from T-points and end at Shil?nikov-Hopf singularities     

Non-darcian effects on double diffusive convection in a sparsely packed porous medium

Summary The linear and non-linear stability of double diffusive convection in a sparsely packed porous layer is studied using the Brinkman model. In the case of linear theory conditions for both simple and Hopf bifurcations are obtained. It is found that Hopf bifurcation always occurs at a lower value of the Rayleigh number than one obtained for simple bifurcation and noted that an increase in the value of viscosity ratio is to delay the onset of convection. Non-linear theory is studied in terms of a simplified model, which is exact to second order in the amplitude of the motion, and also using modified perturbation theory with the help of self-adjoint operator technique. It is observed that steady solutions may be either subcritical or supercritical depending on the choice of physical parameters. Nusselt numbers are calculated for various values of physical parameters and representative streamlines, isotherms and isohalines are presented.     

Hopf bifurcation on a simple cubic lattice

We study Hopf bifurcation for diff erential equations defined on the space of functions on R3 which are triply periodic with respect to a simple (primitive) cubic lattice. The centre manifold theorem reduces the problem to a system of ordinary diff erential equations (ODEs) on the space (C+C)3 and symmetric under the group (O=Zc2) + T3. We abstract this group as the wreath product group O(2) /S3, and we use a general theory of symmetry - breaking bifurcations for wreath product groups to find (up to conjugacy) all branches of periodic solutions with maximal isotropy. The stability of these solutions is calculated . Branches of periodic solutions with sub-maximal isotropy can also exist. Some possibilities for bifurcations to heteroclinic cycles are explored.     

考虑压缩效应的简支弹性杆受压屈曲的理论分析

用奇异性理论研究轴向压缩以及由之引起的横截面积扩大对弹性压杆屈曲的影响。新模型得到的屈曲临界压力大于经典欧拉模型所给之值,且新模型显示,在无量纲特征长度的三个不同取值区间中,弹性杆的一阶屈曲模态表现为三种不同的形式:#x0201c;超临界叉式分支#x0201d;、#x0201c;亚临界叉式分支#x0201d;和#x0201c;不存在#x0201d;。而在经典欧拉模型中,弹性杆的一阶屈曲模态总表现为#x0201c;超临界叉式分支#x0201d;。根据新模型,定性评价了现有实验数据,并通过算例分析,解释了传统材料制作的杆件在较短时将发生屈服破坏而非亚临界叉式分支屈曲的现象。     

Bifurcation sequences in the interaction of resonances in a model deriving from nonlinear rotordynamics: the zipper

Using numerical continuation we show a new bifurcation scenario involving resonant periodic orbits in a parametrized four-dimensional autonomous system deriving from nonlinear rotordynamics. The scenario consists of a carefully orchestrated sequence of transcritical bifurcations in which branches of periodic solutions are exchanged. Collectively, the bifurcations resemble the action of a zipper. An underlying governing mechanism clearly exists but still has to be uncovered. For a range of parameter values the sequence of bifurcations forms a global connection between a Sil'nikov bifurcation and (partial) mode-locking. The homoclinic bifurcation is introduced into the system by a Takens-Bogdanov bifurcation. The system also features an interaction between two chaotic Sil'nikov bifurcations.     

Lorenz attractors in unfoldings of homoclinic-flip bifurcations

Lorenz-like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in ?³ that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination-flip homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two homoclinic bifurcation (for homoclinic loops to equilibria with real different eigenvalues); the orbit-flip homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis.     

Partial classification of heteroclinic behaviour associated with the perturbation of hexagonal planforms

Physical systems often exhibit pattern-forming instabilities. Equivariant bifurcation theory is often used to investigate the existence and stability of spatially doubly periodic solutions with respect to the hexagonal lattice. Previous studies have focused on the six- and twelve-dimensional representation of the hexagonal lattice where the symmetry of the model is perfect. Here, perturbation of group orbits of translation-free axial planforms in the six- and twelve-dimensional representations is considered. This problem is studied via the abstract action of the symmetry group of the perturbation on the group orbit of the planform. A partial classification for the behaviour of the group orbits is obtained, showing the existence of homoclinic and heteroclinic cycles between equilibria.     

Heteroclinic cycles and wreath product symmetries

We consider the existence and stability of heteroclinic cycles arising by local bifurcation in dynamical systems with wreath product symmetry = Z 2 G, where Z 2 acts by ±1 on R and G is a transitive subgroup of the permutation group S N (thus G has degree N). The group acts absolutely irreducibly on R N . We consider primary (codimension one) bifurcations from an equilibrium to heteroclinic cycles as real eigenvalues pass through zero. We relate the possibility of such cycles to the existence of non-gradient equivariant vector fields of cubic order. Using Hilbert series and the software package MAGMA we show that apart from the cyclic groups G (previously studied by other authors) only five groups G of degree ≤7 are candidates for the existence of heteroclinic cycles. We establish the existence of certain types of heteroclinic cycle in these cases by making use of the concept of a subcycle. We also discusss edge cycles, and a generalization of heteroclinic cycles which we call a heteroclinic web. We apply our method to three examples.     

Global dynamics of a family of 3-D Lotka–Volterra systems

In this article, we analyse the flow of a family of three-dimensional Lotka–Volterra systems restricted to an invariant and bounded region. The behaviour of the flow in the interior of this region is simple: either every orbit is a periodic orbit or orbits move from one boundary to another. Nevertheless, the complete study of the limit sets in the boundary allows one to understand the bifurcations which take place in the region as a global bifurcation that we denote by focus-centre-focus bifurcation.     

一类周期系数力学系统的Hopf-Flip分岔

研究了一类周期系数力学系统因周期运动失稳而产生Hopf-Flip分岔的问题.首先根据拉格朗日方程给出了该力学系统的运动微分方程,并确定其周期运动的具有周期系数的扰动运动微分方程,再根据周期系数系统的稳定性理论建立了其给定周期运动的Poincaré映射,进一步根据该系统的特征矩阵的特征值穿越单位圆情况分析判断该Poincaré映射不动点失稳后将发生Hopf-Flip分岔,并用数值计算加以验证.结果表明,非共振条件下,系统的周期运动可通过Hopf-Flip分岔,进而演变成次谐运动,而三阶强共振条件下系统周期运动失稳后形成不稳定的次谐运动.     

Closed trajectories in planar relay feedback systems

A planar system of piecewise linear differential equations with a line of discontinuity, ?₂–symmetry and a linear part having negative determinant is investigated. Using the theory of differential inclusions and an appropriate Poincaré map a complete analysis is provided. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number and stability of periodic trajectories and existence of pairs of heteroclinic trajectories connecting two saddle points forming heteroclinic cycles. A complete bifurcation diagram is given.     

Rotating waves in rings of coupled oscillators

In this paper we discuss the types of stable oscillation created via Hopf bifurcations for a ring of identical nonlinear oscillators, each of which is diffusively and symmetrically coupled to both its neighbours, and which, when uncoupled, undergo a supercritical Hopf bifurcation creating a stable periodic orbit as a parameter, λ is increased. We show that for small enough coupling, the only stable rotating waves produced are either one or a conjugate pair, depending on the parity of the number of oscillators in the ring and the sign of the coupling constant, and that the magnitude of the phase difference between neighbouring oscillators for these rotating waves is either zero (i.e. the oscillators are synchronized) or the maximum possible, depending on the sign of the coupling constant. These brances of rotating waves are produced supercritically.     

Buoyant plane plumes from heated horizontal confined wires and cylinders

Two-dimensional computations are reported for time-dependent laminar buoyancy-induced flows above a horizontal heated source immersed in an air-filled vessel. Two kinds of heated source were considered: a line heat source, modelled as a heat source term in the energy equation, and a heat-flux cylinder of small diameter. First, comparisons are presented for the results obtained for these two heated sources. Rather large discrepencies between the velocity fields appeared in the conduction regime due to the weak plume motion, while close agreements were found in the boundary layer regime. Nevertheless, same types of bifurcations occur with almost identical frequencies, whatever the Rayleigh number. It is concluded that for dimensions of the enclosures, which largely compared with the cylinder radius, the heat source term model is a promising way to study the behaviour of unsteady plumes owing to its simplicity, flexibility, and low computational costs. Second, transitions to unsteady flows were studied through direct flow simulations for various depths of immersion of a line heat source in the central vertical plane of a vessel. Different routes to chaos were shown to occur according to the aspect ratio of the vessel and the depth of immersion of the line source. Three distinct regimes were detected with different underlying physical mechanisms called natural swaying motion, penetrative convection and Rayleigh-Benard-like convection. The first bifurcations associated with these regimes are supercritical Hopf bifurcation, pitchfork bifurcation and subcritical Hopf bifurcation. Comparisons with experimental results of confined buoyant plumes above heated wires show very good agreement with laminar frequency correlations.     

Chaos in the Takens–Bogdanov bifurcation with O(2) symmetry

The Takens–Bogdanov bifurcation is a codimension-two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation-invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and travelling and standing waves in the nonlinear regime and predicts the presence of modulated travelling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated travelling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily close to the codimension-two point.     

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.     

Essentially asymptotically stable homoclinic networks

Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835–844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which has the strong attracting property called essential asymptotic stability. We establish that this phenomenon is possible for homoclinic networks, where all heteroclinic trajectories are symmetry related. Moreover, we study a transverse bifurcation from an asymptotically stable to an essentially asymptotically stable homoclinic network. The essentially asymptotically stable homoclinic network turns out to attract all nearby points except those on codimension-one stable manifolds of equilibria outside the homoclinic network.     

Subcritical and supercritical bifurcations of the first- and second-order Benney equations

The problem of the stability threshold of thin-film dynamics as described by the Benney equation of both first and second orders is revisited. The main result is that the primary Hopf bifurcation of the Benney equation of first order is supercritical for smaller values of Reynolds number and subcritical for its larger values. This result is numerically validated and further investigated analytically to reveal coexisting stable and unstable traveling waves. However, the primary bifurcation of the second-order Benney equation is supercritical for any Reynolds numbers. Sideband instability of traveling-wave regimes whose amplitude and frequency arise from the corresponding complex Ginzburg-Landau equation (CGLE) is found for the Benney equation of both first and second orders.