Global Saddles for Planar Maps

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of \(D_2\)-symmetric maps, for which we obtain a similar result for \(C^1\) homeomorphisms. Some applications to differential equations are also given.     

Realization of Period Maps of Planar Hamiltonian Systems

We consider the set of 2π-periodic solutions of the ordinary differential equation u′′ + g(u) = 0 for a nonlinearity , satisfying a dissipative condition of the form for , and under the generic assumption that the potential G, given by , is a Morse function. Under these assumptions, we characterize the period maps realizable by planar Hamiltonian systems of the form . Considering the Morse type of G, the set of periodic orbits in the phase space is decomposed into disks and annular regions. Then, the realizable period maps are described in terms of sets of sequences of positive integers corresponding to the lap numbers of the 2π-periodic solutions. This leads to a characterization of the classes of Morse–Smale attractors that are realizable by dissipative semilinear parabolic equations of the form defined on the circle, .      

Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.     

Fixed Points of Multi-valued Maps and Static Coulomb Friction Problems

The existence of solutions to the Signorini problem with Coulomb friction is a long standing open question. We prove the existence of generalized solutions that satisfy the pointwise Coulomb friction conditions on the entire interface and the normal nonpenetration condition on the complement of a subset with arbitrarily small but possibly positive measure. Furthermore, the penetration itself can also be made arbitrarily small. Although “measure zero” instead of “arbitrarily small measure” would be needed to fully resolve the issue, these generalized solutions seem to be the closest answer available to date. Their existence is proved by a suitable application of Ky Fan's fixed point theorem for multi-valued maps. The same method can be used with a number of variants involving contact of two or more elastic bodies and possible debonding phenomena. This revised version was published online in August 2006 with corrections to the Cover Date.     

Topological Fixed Point Theorems Do Not Hold for Random Dynamical Systems

The aim of this paper is to demonstrate that topological fixed point theorems have no canonical generalization to the case of random dynamical systems. This is done by using tools from algebraic ergodic theory. We give a criterion for the existence of invariant probability measures for group valued cocycles. With that, examples of continuous random dynamical systems on a compact interval without random invariant points, which are an appropriate generalization of fixed points, are constructed.     

A Shilnikov Phenomenon Due to State-Dependent Delay,by Means of the Fixed Point Index

The first part of this paper is a general approach towards chaotic dynamics for a continuous map \(f:X\supset M\rightarrow X\) which employs the fixed point index and continuation. The second part deals with the differential equation

$$\begin{aligned} x'(t)=-\alpha \,x(t-d_{{\varDelta }}(x_t)). \end{aligned}$$

with state-dependent delay. For a suitable parameter \(\alpha \) close to \(5\pi /2\) we construct a delay functional \(d_{{\varDelta }}\), constant near the origin, so that the previous equation has a homoclinic solution, \(h(t)\rightarrow 0\) as \(t\rightarrow \pm \infty \), with certain regularity properties of the linearization of the semiflow along the flowline \(t\mapsto h_t\). The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion.

     

A Robust-to-Noise Deconvolution Algorithm to Enhance Displacement and Strain Maps Obtained with Local DIC and LSA

Experimental Mechanics - Digital Image Correlation (DIC) and Localized Spectrum Analysis (LSA) are two techniques available to extract displacement fields from images of deformed surfaces marked...     

用无网格LRPIM分析中厚板弯曲问题

用径向基函数构造无网格点插值法的形函数,插值函数具有Kronecker delta函数性质,因此可以很方便地施加本质边界条件.利用无网格局部径向点插值方法分别对一个对边固支另对边简支中厚板和一个悬臂中厚板的弯曲进行了分析计算.该方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,是一种真正的无网格方法.算例表明:将无网格局部径向点插值法应用于计算中厚板的弯曲问题,所求得的位移场和应力场都是光滑的;在径向基函数的基础上,附加多项式大大提高了插值精度;所得结果与弹性力学理论解以及有限元解都十分吻合.     

Three-Dimensional Instability of Planar Flows

We study the stability of two-dimensional solutions of the three-dimensional Navier–Stokes equations, in the limit of small viscosity. We are interested in steady flows with locally closed streamlines. We consider the so-called elliptic and centrifugal instabilities, which correspond to the continuous spectrum of the underlying linearized Euler operator. Through the justification of highly oscillating Wentzel–Kramers–Brillouin expansions, we prove the nonlinear instability of such flows. The main difficulty is the control of nonoscillating and nonlocal perturbations issued from quadratic interactions.     

Turning Around a Method of Successive Iterations to Yield Closed-Form Solutions for Vibrating Inhomogeneous Bars

In this study, the method of successive approximations is turned around so as to obtain closed form solutions for vibrating inhomogeneous bars. In particular, the method recently developed by the first author for the homogeneous beams is extended for bars.     

Renormalization of Planar Analytic Saddles

Let ${X = \lambda_{1} x_{1}{\frac{\partial}{\partial {x_1}}} + \lambda_2 x_2 \frac{\partial}{\partial {x_2}} + O(|x|^2)}$ be an analytic vector field near x = 0. We suppose that the linear part of this vector field has real eigenvalues ??₁, ??₂ and that the ratio ${\eta = -\frac{\lambda_1}{\lambda_2}}$ is a positive irrational number. In a previous paper of the first author and P. De Maesschalck, it was shown that any analytic saddle can be conjugated analytically to a form ??as close as desired?? to the formal normal form. In this paper we will iterate and renormalize these conjugacies. The iteration of this procedure will be strongly connected to the diophantine properties of ?? and we will establish the convergence of this process. A consequence of this convergence will be the two dimensional version of the by now classical linearization theorem of Bruno.     

Motion of Planar Link Mechanisms Revisited

The problem of finding the positions of a four-bar linkage at which the coupler link and rocker have extreme angular velocities is solved. A method is proposed for kinematic analysis of class III mechanisms. The method is based on joining a fictitious link to the original mechanism. The kinematic analysis of a class III mechanism is reduced to successive kinematic analyses of two four-bar linkages     

Planar elongation of soft polymeric networks

A new test fixture for the filament stretch rheometer (FSR) has been developed to measure planar elongation of soft polymeric networks with application towards pressure-sensitive adhesives (PSAs). The concept of this new geometry is to elongate a tube-like sample by keeping the perimeter constant. To validate this new technique, soft polymeric networks of poly(propylene oxide) (PPO) were investigated during deformation. Particle tracking and video recording were used to detect to what extent the imposed strain rate and the sample perimeter remained constant. It was observed that, by using an appropriate choice of initial sample height, perimeter, and thickness, the planar stretch ratio will follow l(t) = h(t)/h₀ = exp([(e)\dot] t)\lambda(t) = h(t)/h_0= \exp({\dot{\varepsilon}} t), with h(t) being the height at time t and [(e)\dot]{\dot{\varepsilon}} the imposed constant strain rate. The perimeter would decrease by a few percent only, which is found to be negligible. The ideal planar extension in this new fixture was confirmed by finite element simulations. Analysis of the stress difference, σ _zz  − σ _xx , showed a network response similar to that of the classical neo-Hookean model. As the Deborah number was increased, the stress difference deviated more from the classical prediction due to the dynamic structures in the material. A modified Lodge model using characteristic parameters from linear viscoelastic measurements gave very good stress predictions at all Deborah numbers used in the quasi-linear regime.     

Structural Stability of Planar Polynomial Foliations

We discuss natural notions of structural stability of planar polynomial foliations of fixed degree with respect to perturbation within the same restricted set, within the set of all polynomial vector fields of the same degree, and within the set of smooth vector fields. Characterization theorems for structural stability in the latter two settings are obtained as immediate corollaries of known results. We provide sufficient conditions and separate necessary conditions for structural stability of planar polynomial foliations with respect to perturbation within the set of planar polynomial foliations of the same degree.     

Types of Bifurcations of FitzHugh–Nagumo Maps

The FitzHugh–Nagumo-like systems are of fundamental importance to the understanding of the qualitative nature of nerve impulse propagation. Our work provides a numerical investigation of bifurcations associated with a family of piecewise differentiable canonical maps for a planar FitzHugh–Nagumo system. We describe the bifurcation structure of the maps with the variation of the parameters.     

Biaxial Mechanical Evaluation of Planar Biological Materials

A fundamental goal in constitutive modeling is to predict the mechanical behavior of a material under a generalized loading state. To achieve this goal, rigorous experimentation involving all relevant deformations is necessary to obtain both the form and material constants of a strain-energy density function. For both natural biological tissues and tissue-derived soft biomaterials, there exist many physiological, surgical, and medical device applications where rigorous constitutive models are required. Since biological tissues are generally considered incompressible, planar biaxial testing allows for a two-dimensional stress-state that can be used to characterize fully their mechanical properties. Application of biaxial testing to biological tissues initially developed as an extension of the techniques developed for the investigation of rubber elasticity [43, 57]. However, whereas for rubber-like materials the continuum scale is that of large polymer molecules, it is at the fiber-level (∼1 μm) for soft biological tissues. This is underscored by the fact that the fibers that comprise biological tissues exhibit finite nonlinear stress-strain responses and undergo large strains and rotations, which together induce complex mechanical behaviors not easily accounted for in classic constitutive models. Accounting for these behaviors by careful experimental evaluation and formulation of a constitutive model continues to be a challenging area in biomechanics. The focus of this paper is to describe a history of the application of biaxial testing techniques to soft planar tissues, their relation to relevant modern biomechanical constitutive theories, and important future trends. This revised version was published online in July 2006 with corrections to the Cover Date.     

L2 Well-Posedness of Planar Div-Curl Systems

Criteria for the existence and uniqueness of solutions of div-curl boundary value problems on bounded planar regions with nice boundaries are developed. The boundary conditions to be treated include prescribed normal component of the field, tangential component of the field and disjoint combinations of these conditions. Under natural assumptions on the data, when either tangential or normal components are given on the whole boundary, weak (finite-energy) solutions exist if and only if a compatibility condition holds. If the region is simply connected this solution is unique. When the region is multiply connected, there is a finite-dimensional family of solutions. The dimension of the solution space is the Betti number of the region. The problem is well-posed with a unique solution when certain line integrals are further prescribed. L ² estimates of the solutions are given. If mixed tangential, and normal, components of the field are specified on different parts of the boundary, no compatibility condition is required for solvability. In general, though, there is considerable non-uniqueness of solutions. Well-posedness is recovered by specifying certain line integrals. L ² estimates of the solutions are given. The dimensionality of the solution space depends on the topology of the boundary data. These results depend on certain weighted orthogonal decompositions of L ² vector fields on the region which are related to classical Hodge-Weyl decomposition results.     

Planar jet flow of a radiating gas

Propagation of a laminar jet of a thermodynamically equilibrious gray gas is examined in the boundary layer theory approximation. The one-dimensional radiative energy transport is accounted for in the P ₁ approximation of the spherical harmonic method. Numerical solution of the problem is made under additional simplifying assumptions for various values of the radiation parameters to illustrate the radiation effect. The method and the computational scheme used are applicable to the study of complex jet flows of a radiating gas.The author thanks Yu. P. Lun'kin for his assistance in posing the problem and for his continued interest in the study.     

Stability of Periodic Clusters in Globally Coupled Maps

The phenomenon of partial synchronization, or clustering, in a system of globally coupled C ¹-smooth maps is analyzed. We prove the stability of equally populated K-clustered states with n-periodic temporal dynamics, referred to as P _n C _K-states. For this purpose, we first obtain formulas giving a relation between longitudinal and transverse multipliers of the in-cluster periodic orbits, and then, using these formulas, we find exact parameter intervals for transverse stability. We conclude that, typically, for symmetric P _n C _K-states, in-cluster stability implies transverse stability. Moreover, transverse stability can take place even if in-cluster dynamics are unstable.     

Planar dynamics of a dimer on a wave

Nonlinear Dynamics - We study the 2D dynamics of a rigid dimer, a dumbbell-shaped extended body, on an elastic surface carrying a harmonic traveling wave. The impact of the dimer with the surface...