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.     

Expansion of a cavity in a rubber block under stress: application of the asymptotic expansion method to the analysis of the stability and bifurcation conditions

In order to analyse the stability and bifurcation phenomena occurring during expansion of a small void in a rubbery material, the behaviour of spherical shells submitted to a combined far-field pressure and uniaxial tension has been investigated, considering a general nonlinear isotropic elastic compressible behaviour of the material and without any restrictions on the shell thickness. A radial solution for the deformation gradient with a spherical symmetry has been exhibited, which is valid for any behaviour law and consists of a homogeneous deformation. The three-dimensional problem is then linearized around this trivial solution, and we show the existence of a pressure interval containing the zero value, in which the solution is reduced to the trivial solution, which is therefore infinitesimally stable. The condition for stability obtained is compared with Hadamard's condition; particularly, it is shown that both are identical when the material is supposed to have a St Venant-Kirchhoff behaviour law. When the applied pressure lies outside the stability interval, we determine the bifurcation points of the shell around the trivial solution, first when only a pressure is applied and secondly when there is an additional far-field tension, much smaller than the applied pressure. The form of the stress distribution on the boundary of the cavity suggests a possible bifurcation of the spherical solution towards a family of axisymmetric solutions. Within this hypothesis, we get a relation between the geometrical parameter of the shell (its radius and thickness), the mechanical properties of the material and the critical load. The analyses provide evidence of the non-uniqueness of the bifurcation behaviour, since we exhibit some peculiar bifurcation points associated with an infinity of branches of axisymmetric solutions.     

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.     

Symmetry and chaos in the complex Ginzburg-Landau equation - I. Reflectional symmetries

The complex Ginzburg-Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries. Solutions of the CGL equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions with some reflectional symmetry to perturbations which break that symmetry. This can be achieved by considering the isotypic decomposition of the space and finding the dominant Lyapunov exponent associated with each isotypic component. Our numerical results indicate that for most parameter values, chaotic solutions that have been restricted to lie in invariant subspaces are unstable to perturbations out of these subspaces, leading us to conclude that for these parameter values arbitrary initial conditions will generically evolve to a solution with the minimum amount of symmetry allowable. We have also found a small region of parameter space in which chaotic solutions that are even are stable with respect to odd perturbations.     

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.     

多自由度含间隙振动系统周期运动的Hopf-pitchfork余维二分岔

建立了多自由度含间隙振动系统对称型周期碰撞运动及Poincaré映射的解析表达式,讨论了该映射不动点的稳定性与局部分岔。应用映射的中心流形和范式方法,研究了映射在Hopf-pitchfork余维二分岔点附近的参数开折,揭示了含间隙振动系统在余维二分岔点附近的动力学行为。在该类余维二分岔点附近,不仅存在对称型周期碰撞运动、Hopf分岔和叉式分岔,还存在非对称型周期碰撞运动及其Hopf分岔。通过数值仿真研究了余维二分岔点附近含间隙振动系统对称型周期碰撞运动经叉式分岔和Hopf分岔向混沌的转迁过程。     

Flow in a pipe driven by suction at an accelerating wall

Summary The axisymmetric flow of a viscous fluid in a pipe driven by suction at the pipe wall and acceleration of the wall is examined using a similarity form of solution: the particular combination of the two effects chosen is one in which there is a simple analytical solution. The temporal stability of solutions (both this analytical one and other numerical ones) is investigated for various values of the Reynolds numberR, and stable and unstable parts of the solution branches are identified. A transcritical bifurcation is found and also a Hopf bifurcation. We consider also the corresponding nonlinear diffusion problem and present some solutions. Finally we examine the continuation from the present problem to the flow in a pipe driven by suction only.     

Uncertainty quantification via codimension‐one partitioning

We consider uncertainty quantification in the context of certification, i.e. showing that the probability of some ‘failure’ event is acceptably small. In this paper, we derive a new method for rigorous uncertainty quantification and conservative certification by combining McDiarmid's inequality with input domain partitioning and a new concentration‐of‐measure inequality. We show that arbitrarily sharp upper bounds on the probability of failure can be obtained by partitioning the input parameter space appropriately; in contrast, the bound provided by McDiarmid's inequality is usually not sharp. We prove an error estimate for the method (Proposition 3.2); we define a codimension‐one recursive partitioning scheme and prove its convergence properties (Theorem 4.1); finally, we apply a new concentration‐of‐measure inequality to give confidence levels when empirical means are used in place of exact ones (Section 5). Copyright © 2010 John Wiley & Sons, Ltd.     

Classification of one-state-variable bifurcation problems up to codimension seven

We present a classification of one-state-variable singularities with distinguished parameter through codimension seven. This extends results previously known up to codimension four on contact equivalence with distinguished parameter. Among other interesting phenomena we present the topology of a modal family in codimension five. The classification uses elementary techniques including the notions of tangent space, restricted tangent space, and intrinsic ideals. For each problem of codimension seven or less, a normal form, criteria for recognizing the singularity and a set of unfolding vectors are given. Subordination relations between the different singularities are also described.     

Dynamic bifurcation in a system of PDES with a conserved first integral

A class of initial-boundary value problems with a conserved first integral is studied. The class of problems includes particular cases of interest. In one parameter limit, the equations arise as a similarity reduction of the Navier-Stokes equations, which has been recently studied for its blow-up properties, and in another, a class of scalar reaction-diffusion equations, modified by the addition of a non-local non-linearity to ensure conservation of the first integral. Once the problem has been stated, it is shown how it may be derived from the Navier-Stokes equations in one parameter limit. Steady-state solutions are then constructed using a rigorous iterative method which we call Crandall Iteration. The steady solution set includes, in a particular parameter limit, those of the Cahn-Hilliard equation. Amplitude expansions and centre manifold theory are employed to analyze the heteroclinic orbits connecting these steady solutions. It is proved that Hopf bifurcation of periodic solutions cannot occur     

Scaled boundary finite‐element analysis of a non‐homogeneous elastic half‐space

As a result of stresses experienced during and after the deposition phase, a soil strata of uniform material generally exhibits an increase in elastic stiffness with depth. The immediate settlement of foundations on deep soil deposits and the resultant stress state within the soil mass may be most accurately calculated if this increase in stiffness with depth is taken into account. This paper presents an axisymmetric formulation of the scaled boundary finite‐element method and incorporates non‐homogeneous elasticity into the method. The variation of Young's modulus (E) with depth (z) is assumed to take the form E=m_Ez^α, where m_E is a constant and αis the non‐homogeneity parameter. Results are presented and compared to analytical solutions for the settlement profiles of rigid and flexible circular footings on an elastic half‐space, under pure vertical load with αvarying between zero and one, and an example demonstrating the versatility and practicality of the method is also presented. Known analytical solutions are accurately represented and new insight regarding displacement fields in a non‐homogeneous elastic half‐space is gained. Copyright © 2003 John Wiley & Sons, Ltd.     

Three-dimensional instabilities of ferromagnetic liquid bridges

The cross section of a ferromagnetic liquid drop held in equilibrium between horizontal plates in a magnetic field loses its circular symmetry past a critical value of the applied field strength. This is caused by instabilities that give way to non-circular cross sectional shapes which, in turn, produce three-dimensional magnetic field distribution inside and outside the drop. Theoretical predictions of equilibrium non-circular shapes and their stability are drawn from the equations governing the magnetohydrostatic equilibrium of the drop. The computational problem is three-dimensional, nonlinear and free boundary and it is solved with the Galerkin/finite element method. Entire branches of circular solutions and non-circular ones are traced by continuation in multi-parameter space. Circular, elliptical and dumbbell-shaped drops have been found. The relative stability of the various shapes is computed by means of computer-implemented bifurcation theory.     

冲击振动成型机周期运动的Hopf-flip余维二分岔与混沌

基于冲击映射方法和数值仿真分析了一类冲击振动成型机单冲击周期运动的稳定性与Hopf-flip分岔。应用映射的中心流形-范式方法将冲击振动成型机的冲击映射降阶为三维映射,分析了相关范式映射的局部分岔特性及参数开折。通过定性分析与数值仿真研究了冲击振动成型机在Hopf-flip余维二分岔条件下的动力学行为,讨论了Hopf-flip分岔点附近周期冲击运动不动点类型的转迁及其向混沌运动的演化过程。     

一类具有交叉扩散的捕食-食饵模型正解的存在性

本文研究一类带有交叉扩散的捕食-食饵模型正解的存在性．首先,利用最大值原理得到了与交叉扩散系数无关的正解的先验估计;其次,建立了当交叉扩散系数充分大时的极限系统;最后,利用局部分歧理论得到了极限系统在半平凡解附近的局部分歧解的存在性,借助全局分歧理论说明了该极限系统的局部分歧解可以延拓为全局分歧解,并且该全局分歧解随着分歧参数在正椎内延伸至无穷．结论表明：当交叉扩散系数充分大时,两物种可以共存．     

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.     

Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation

We study a scalar reaction-diffusion equation which contains a nonlocal term in the form of an integral convolution in the spatial variable and demonstrate, using asymptotic, analytical and numerical techniques, that this scalar equation is capable of producing spatio-temporal patterns. Fisher's equation is a particular case of this equation. An asymptotic expansion is obtained for a travelling wavefront connecting the two uniform steady states and qualitative differences to the corresponding solution of Fisher's equation are noted. A stability analysis combined with numerical integration of the equation show that under certain circumstances nonuniform solutions are formed in the wake of this front. Using global bifurcation theory, we prove the existence of such non-uniform steady state solutions for a wide range of parameter values. Numerical bifurcation studies of the behaviour of steady state solutions as a certain parameter is varied, are also presented.     

Bifurcation buckling from a membrane stress state

A membrane stress state may be considered as a general stress state in which mechanical constraints have been imposed. Therefore, it would appear that there are mathematical subsidiary conditions for bifurcation buckling from such a stress state. In this work, such conditions are derived in the frame of the Finite Element Method. The basic condition follows from disintegration of the second derivative of the mathematical formulation of the so‐called consistently linearized eigenproblem with respect to a dimensionless load parameter. It is used for deriving another condition, characterized by the vanishing of a particular bilinear form. Linear stability analysis and bifurcation buckling from linear prebuckling paths are two special cases for which this condition is satisfied. Sensitivity analysis of bifurcation buckling of a two‐hinged arch, subjected to a uniformly distributed static load, by varying the geometric form of its axis serves the purpose of non‐trivial verification of the derived condition for the special case of a thrust‐line arch. Copyright © 2011 John Wiley & Sons, Ltd.     

A piece wise linear suspension bridge model: nonlinear dynamics and orbit continuation

The effect of harmonic excitation on suspension bridges is examined as a first step towards the understanding of the effect of wind, and possibly certain kinds of earthquake, excitation on such structures. The Lazer-McKenna suspension bridge model is studied completely for the first time by using a methodology that has been successfully applied to models of rocking blocks and other free-standing rigid structures. An unexpectedly rich dynamical structure is revealed in this way. Conditions for the existence of asymptotic periodic responses are established, via a complicated nonlinear transcen- dental equation. A two-part Poincare map is derived to study the orbital stability of such solutions. Numerical results are presented which illustrate the application of the analytical procedure to find and classify stable and unstable solutions, as well as determine bifurcation points accurately. The richness of the possible dynamics is then illustrated by a menagerie of solutions which exhibit fold and flip bifurcations, period doubling, period adding, and sub- and superharmonic coexistence of solutions. The solutions are shown both in the phase plane and as Poincare map fixed points under parameter continuation using the package AUTO. Such results illustrate the possibility of the coexistence of 'dangerous', large-amplitude responses at the same point of parameter space as 'safe' solutions. The feasibility of experimental verification of the results is discussed.