The existence of infinitely many homoclinic doubling bifurcations from some codimension 3 homoclinic orbits

An inclination-flip homoclinic orbit of weak type on ³ is a homoclinic orbit given as the intersection of a special one-dimensionalC ²-weak stable manifold and the one-dimensional unstable manifold of a hyperbolic singularity with three real eigenvalues. In this paper, we show that in a generic unfolding of such a homoclinic orbit, there appear curves in the parameter space that correspond to ordinary inclination-flip homoclinic orbit of orderN for any integerN. As a consequence, there exist infinitely many homoclinic doubling bifurcation curves emanating from the codimension three degenerate point corresponding to the inclination flip homoclinic orbit of weak type.     

Calculation of inviscid radiating gas flow past a blunt body

During hypersonic gas flow past a blunt body with a velocity on the order of the escape velocity or more, the gas radiation in the disturbed region behind the shock wave becomes the primary mechanism for aerodynamic heating and has a significant effect on the distribution of the gasdynamic parameters in the shock layer. This problem has been considered from different points of view by many authors. A rather complete review of these studies is presented in [1–4].In earlier studies [5, 6] the approximation of bulk emission was used. In this approximation, in order to account for the effect of radiative heat transfer a term is added in the energy equation which is equivalent to the body efflux, whose magnitude depends on the local thermodynamic state of the gas. However, the use of this assumption to solve the problem of inviscid flow past a blunt body leads to a singularity at the body [7, 8]. To eliminate the singularity, account is taken of the radiation absorption in a narrow wall layer [7], or the concept of a viscous and heat-conductive shock layer is used [8]. A further refinement was obtained by Rumynskii, who considered radiation selectivity and studied the flow of a radiating and absorbing gas in the vicinity of the forward stagnation point of a blunt body.In the present paper we study the distribution of the gasdynamic parameters in the shock layer over the entire frontal surface of a blunt body in a hypersonic flow of a radiating and absorbing gas with account for radiation selectivity.     

A UNIFORMLY DIFFERENCE SCHEME OF SINGULAR PERTURBATION PROBLEM FOR A SEMILINEAR ORDINARY DIFFERENTIAL EQUATION WITH MIXED BOUNDARY VALUE CONDITION

ＡＵＮＩＦＯＲＭＬＹＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＯＦＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＦＯＲＡＳＥＭＩＬＩＮＥＡＲＯＲＤＩＮＡＲＹＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＷＩＴＨＭＩＸＥＤＢＯＵＮＤＡＲＹＶＡＬＵＥＣＯＮＤＩＴ...     

Zero singularity of codimension two or three in a four-neuron BAM neural network model with multiple delays

In this paper, a four-neuron BAM neural network model with multiple delays is considered. The existence conditions under which that the origin of the system is Bogdanov–Takens (B–T) or triple zero singularity are given. By choosing the connected weights as bifurcation parameters and using the center manifold reduction and the normal form theory and the formula developed by Xu and Huang (J Differ Equ 244:582–598 2008) and Qiao et al. (Chinese Ann Math Ser A 31:59–70 2010), the versal unfoldings and the normal forms for this singularity were given to analyze the behaviors of the system. This paper is a further study of paper Cao and Xiao (IEEE Trans Neural Netw 18:416–430 2007).     

Asymptotic Considerations Shedding Light on Incompressible Shell Models

The incompressible singularity found in 3D elasticity when Poissons ratio approaches 1/2 is not present in classical shell models, nor in the limit models obtained from 3D elasticity when performing an asymptotic analysis with respect to the thickness parameter. However, some specific shell models – such as the 3D-shell model – do retain the incompressible singularity. These observations raise the issue of how adequately shell models can represent incompressible conditions, which this paper aims at investigating. We first perform a combined asymptotic analysis of 3D elasticity with respect to both the thickness parameter and Poissons ratio and we obtain a commuting property, which is very valuable as a justification of the concept of an incompressible shell, and substantiates the use of classical shell models with incompressible materials. We then show that the 3D-shell model does not enjoy a similar commuting property; nevertheless we propose a simple modification of this model for which commuting is obtained, hence consistency with incompressibility is recovered. We also illustrate our discussions with some numerical results. Mathematics Subject Classifications (2000) 74K25, 74G10, 74S05.A. Münch: Current address: Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16 route de Gray, 25030 Besancon Cedex, France.     

Extended graphical representation of rational fractions with applications to cybernetics

In this paper,we discuss the extended graphical representation ofthe fraction of a complex variable sWhere K is confined to be real.Three figures of the above fraction canbe used in feedback systems as well as to study the properties of figuresfor any one coefficient of a characteristic equation as a real parameter.It is easy to prove the following theorem:K_1=f(n)(s)/F(d)(s)and K_2=F(d)(s)/f(n)(s)have the same root locus.By this graphical theory,we find out that if the zeros and polesof a fraction are alternatively placed on the axis x,then there is nocomplex root locus of this fraction,therefore the state of such a sys-tem is always non-oscillatory.Using these figures of this,fraction,wecan discuss its stable interval systematically.     

Independence of the Presence of Cracks or Inclusions of the Net Interaction Force Acting on a Dislocation by a Skewed Line Singularity

Orlov and Indenbom [1] have shown that the net (integrated) interaction force F between two skew dislocations with Burgers vectors separated by a distance h in an infinite anisotropic elastic medium is independent of h. Nix [2] computed numerically the net interaction force F between two skew dislocations that are parallel to the traction-free surface X2=0 of an isotropic elastic half-space. His numerical results showed that F was independent of h; a partial result of what Barnett [3] called Nix"s theorem. The separation-independence portion of Nix"s theorem has been proved to hold for a general anisotropic elastic half-space with a traction-free, rigid, or slippery surface, and for bimaterials [3-5]. In this paper, we show that the net interaction force is independent of the presence of inclusions. We will consider the case in which the line dislocation b is a more general line singularity which can include a coincident line force with strength f per unit length of the line singularity. An inclusion is an infinitely long dissimilar anisotropic elastic cylinder of an arbitrary cross-section whose axis is parallel to the line singularity (f, b). The (skew) line dislocation does not intersect the inclusion. The special cases of an inclusion are a void, crack, or rigid inclusion. There can be more than one inclusion of different cross sections and different materials. The line singularity (f, b) can be outside the inclusions or inside one of the inclusions. The inclusions and the matrix need not have a perfect bonding. One can have a debonding with or without friction. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Direct Approach to Homoclinic Orbits in the Fast Dynamics of Singularly Perturbed Systems

Homoclinic orbits in the fast dynamics of singular perturbation problems are usually analyzed by a combination of Fenichel's invariant manifold theory with general transversality arguments (see Ref. 29 and the Exchange Lemma in Ref. 16). In this paper an alternative direct approach is developed which uses a two-time scaling and a contraction argument in exponentially weighted spaces. Homoclinic orbits with one last transition are treated and it is shown how -expansions can be extracted rigorously from this approach. The result is applied to a singularity perturbed Bogdanov point in the FitzHugh–Nagumo system.Supported by DFG Schwerpunktprogramm Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme..     

Localized shear discontinuities near the tip of a mode I crack

In this paper we study the deformation and stress fields near the tip of a crack under plane strain mode I conditions. A fully nonlinear theory of finite deformations is used and the material, which is assumed to be homogeneous, isotropic, incompressible and elastic, is characterized by its stress-strain behavior in simple shear. For the class of materials considered the governing system of differential equations may lose ellipticity at sufficiently severe strains. The analysis is based on a direct asymptotic calculation. The results involve two curves, issuing from each crack-tip, across which the deformation gradient, the effective shear and the stresses are discontinuous.     

Modified Neuber–Novozhilov criterion of rupture for V‐shaped cuts (antiplane problem)

A modified Neuber–Novozhilov criterion of rupture is proposed to study the fracture in the vicinity of the tip of a Vshaped cut. In this criterion, the limits of averaging stresses at the cut axis depend on the presence, size, and location of defects in the initial material. The lattice parameter of the initial material is taken to be the characteristic linear size. For Vshaped cuts, simple relations between the stressintensity factor of the modified singularity factor, the singularity factor itself, and the theoretical shear strength of a single crystal of the material with allowance for damage in the vicinity of the tip are proposed. These relations admit passage to the limit with respect to the angle from a Vshaped cut to a crack. It is shown that the classical critical stressintensity factor used to estimate the strength of cracked bodies is not a material constant.     

Convective heat exchange in a radiating gas

The problem of radiating gas flow past a blunt body has been studied by many authors. A quite complete review of these studies is presented in [1, 2]. Basic attention has been devoted to determining the radiant fluxes to the body surface and calculation of the parameter distribution across the shock layer.In many studies, particularly the foreign ones, use has been made of the approximation of bulk luminescence. In this approximation a term is added to the energy equation in order to account for the effect of the radiant heat exchange. This term is equivalent to the heat flux, whose intensity depends on the local thermodynamic state of the gas. With the use of this assumption the gas enthalpy in the inviscid flow vanishes at the stagnation point. The singularity resulting from the use of various approximations was revealed in [3–5]. This singularity is caused by the fact that a gas particle moving along the stagnation streamline to the body is retarded over an infinitely long time. Obviously, if we take into account the dissipative processess (for example, thermal conduction) which really take place, the singularity disappears. For a perfect gas it may be shown that the enthalpy and velocity are equal to zero along the entire surface of the body. Then the use of conventional boundary layer theory becomes impossible. The concept of a viscous and heat-conducting shock layer has been used in [5,6] and also by the present authors to eliminate this difficulty. However, this approach leads to unjustified complication of the problem and forces the introduction of more or less rough assumptions in carrying out the calculations.In the present study we have investigated a form of the boundary layer equations and the corresponding boundary conditions for flows with bulk gas luminescence and approximating flow regimes with small optical thickness of the shock layer. The solution was carried out with the aid of the method of inner and outer expansions (for example, [7]). The form of the equations and the boundary conditions differed depending on which of the dissipative processes-thermal conduction or absorption of the radiation by the gas in a narrow layer cooled by luminescence of the near-wall layer, was dominating. (The existence in the inviscid shock layer with small but finite optical thickness of an absorbing near-wall sublayer was discovered by V. N. Zhigulev, and also in [2].) In the present study we have used the Newtonian approximation, analogous to that considered for the nonradiating gas by Shidlovskii [8], This made it possible to obtain most of the results in a simple, easily visualized form. However, the flow regimes considered and the corresponding parameters do, of course, have general significance.     

Experimental investigation of GH singularity field of the growing crack-tip for different ductile metals

The displacementsu _x ,u _y inx, y directions at growing crack tip of the specimens with double edge cracks for four different alumium alloys and two coppers are measured by using moire method and optical spatial filtering technique. From experimental displacement fields, the displacement singularity fields are obtained and compared with GH theoretical field. Unknown constantsA and _yo in theoretical solution are determined from experimental data. The theoretical singularity field thus compared is given for plane-stress, mode-I, strain-hardening materials. The error in both the experimental and theoretical evaluations is within ± 10%. The experiments show that there exists dominant singularity region around a growing crack-tip. In the experiments, the strain hardening indexn amounts from 3.158 to 14. The shape of this dominant region ranges from butterfly wing to oblate or circular shape. The size and shape of GH dominant region depend on the material property, the specimen geometry and loading type. Inside GH-field, there is a 3-D deformed damage zone, where no GH singularity exists. Very near to the crack-tip, there is a fracture process zone.The project is supported by the National Science Foundation of China.     

Domain perturbation method and shape of a bubble in a uniform flow of an inviscid liquid

In many problems with a free boundary there is defined a small parameter, , for which the solution is sometimes known for a particular value, =0, and the general solution is obtained as a series in the parameter. To find this solution, the equations can be written on a reference configuration and solved in a fixed domain. The purpose of this study is to show that this method of domain perturbation is a good one. The range of validity of this method will be studied on the model example of the irrotational flow of a perfect fluid around a bubble. The radius of convergence of the series solution will be determined, as will the nature of the solution in the neighbourhood of the first real singularity.     

Independence of friction self-locking from kinematic singularity

Self-locking analysis in closed kinematic chains is sometimes likened to kinematic singularity. Here a novel approach to tackle self-locking analysis due to joints friction is exploited, that is completely different from the classical kinematic analysis based on the jacobian conditioning. It is shown that an inverse kinematic singularity always entails a self-locking phenomenon because of the general increasing of joints reactions and, then, friction forces; hence, a self-locking domain can be always identified including such a locus. On the other side, this paper is aimed at demonstrating that the aforementioned condition is not necessary: namely, self-locking may occurs also if the mechanism kinematics is well-conditioned. Then, the theoretical result is clarified performing the self-locking analysis on a simple crank-slider mechanism.     

Collapsed 27 node Lagrangian element for three-dimensional stress intensity computations

Considered are the details of a collapsed 27 node (singular) Lagrangian element for use in three-dimensional stress intensity factor calculations. The element is shown to capture both the singularity near the front as required for linear elastic fracture mechanics and a 1/r singularity under a relaxed constraint for plastic fracture. It is shown that the radial and circumferential strains exhibit the singularity, but that direct strain along the front is zero, thus providing plane strain conditions. Of equal relevance is the quality of mesh at the front. For this we adopt a standard suite of elements comprising a number of singular ‘wedge-shaped' elements around the crack tip in the form of a tube, with the major axis placed along the front. A number of three-dimensional fracture problems are analysed to demonstrate the accuracy and efficiency of the collapsed element, even when used with moderate mesh sizes.     

Finite Time Singularity of the Nematic Liquid Crystal Flow in Dimension Three

In this paper, we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity. The first example is constructed within the class of axisymmetric solutions, while the second example is constructed for any generic initial data \({(u_0,d_0)}\) that has sufficiently small energy, and \({d_0}\) has a nontrivial topology.     

Properties of Monodromic Points on Center Manifolds in {\mathbb {R}}^3 via Lie Symmetries

In this work we use Lie symmetries to investigate monodromic points on center manifolds of a singularity of an analytic vector field ${\mathcal {X}}$ in ${\mathbb {R}}^3$ . We investigate how whether the singularity is a focus or a center, is analytically normalizable or not, and is linearizable or not is reflected in the centralizer and normalizer of ${\mathcal {X}}$ .     

Nonlinear shear-layer instability waves

The effects of critical-layer nonlinearity on spatially growing instability waves on shear layers between parallel streams are discussed. In the two-dimensional incompressible case, the flow in the critical layer is governed by a nonequilibrium (unsteady) nonlinear vorticity equation. The initial exponential growth of the instability wave is converted into algebraic growth during the streamwise aging of the critical layer into a quasi-equilibrium state. A uniformly valid composite formula for the instability wave amplitude, accounting for both nonparallel and nonlinear effects, is shown to be in good agreement with available experimental results. Nonlinear effects occur at smaller amplitudes for the three-dimensional and supersonic cases than in the two-dimensional incompressible case. The instability-wave amplitude evolution is then described by one integro-differential equation with a cubic-type nonlinearity, whose inviscid solution always end in a singularity at finite downstream distance.The US Government has the right to retain a nonexclusive royalty-free license in and to any copyright covering this paper.     

Influence of stacking sequence and interlaminar layer on stress singularities at free edge of composite laminates

Summary The topic concerning the singularity in the state of stress existing at the free edges of plane composite laminates with resin interlaminar layer, has been dealth with. The problem has been studied by means of a F.E. approach, founded on a method able to determine the free edge state of stress without aprioristical hypotheses about the presence of a singularity. Then the singularity itself has been studied through a linear-logarithmic formulation which enables, for the stacking sequences considered — all belonging to the [°/°–90°] _s family — and for all meaningful stress components, to compute the power of the singularity and the boundary layer stress intensity factor (or free edge stress intensity factor). Taking into account these parameters, various laminations and stacking sequences have been compared, in order to provide a practical measure of the severity of the singular stress field and its influence on failure modes.Then a comparative analysis has been worked out, in order to evaluate the effects due to the thickness of the resin interlaminar layer between two fiber layers.

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Sommario Si prende in considerazione il problema connesso con l'esistenza di singolarità nel campo di sforzo presente ai bordi liberi di laminati piani in materialxe composito, in presenza di uno strato di resina interlaminare.La questione viene affrontata tramite un approccio numerico agli elementi finiti, basato su di un metodo in grado di determinare il campo di sforzo al bordo libero senza assumere aprioristicamente la presenza di singolarità. La eventuale singolarità viene poi studiata mediante una trattazione logaritmico-lineare che consente di calcolare, per tutte le laminazioni prese in considerazione, (appartenenti alla famiglia [°/°–90°] _s )e per le componenti di sforzo più significative, il valore della potenza della singolarità e quello che viene definito fattore d'intensità degli sforzi al bordo libero. Sulla scorta di tali valori si confrontano le varie laminazioni e sequenze di impaccamento.Per tuttii questi aspetti viene inoltre condotta un'indagine comparativa, per valutare l'effetto dovuto allo spessore dello strato di resina interlaminare.