Abundance of unimodal maps with dense critical orbit and prefixed critica orbit

For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invariant measure for almost all parameters in E; it is also proved that there is a dense set in E such that the critical orbit of the corresponding map enter the reversed fixed point.     

ABUNDANCE OF UNIMODAL MAPS WITH DENSE CRITICAL ORBIT AND PREFIXED CRITICAL ORBIT

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An existence theory for nonlinear elasticity that allows for cavitation

In this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body. Such cavities have been observed in experiments on elastomers, and a mathematical theory for radially symmetric cavities has been developed by Ball. Here the full three-dimensional problem is considered and an additional, physically motivated, energy term that is proportional to the area of the boundary of the deformed body is included. The minimizers lie in a subclass of those maps in W ^{1, p} , 2<p<3, that are one-to-one almost everywhere and preserve orientation. Roughly speaking, this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body. Such maps are shown to be much more regular than expected. In particular, some ideas of verák are used to show that each map in this subclass has a representative which is continuous outside a set of Hausdorff dimension 3 — p and that this representative also satisfies Lusin's condition (N), i.e., it maps Lebesgue null sets onto such sets. It is also shown that the distributional Jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to Lebesgue measure and (at most) a countable number of Dirac measures.     

The Representation Theorem for Linear, Isotropic Tensor Functions Revisited

Two recent notes [1–2] express ideas that can be combined in order to get an elementary proof of the representation theorem of the title, a proof stressing the geometrical action of 4-tensors in subspaces. Thus we examine direct consequences, for linear maps C of Lin into Lin, of the notion of invariance (we recall this concept and we recall other standard notations in the first section). Suppose we classify the invariant (under Orth) subspaces of Lin, that an invariant linear C maps invariant subspaces into invariant subspaces and that Lin = Skw Sph Dev expresses the unique decomposition of Lin into a direct sum of three-, one- and five-dimensional invariant subspaces. Then, if we assume that C is also invertible, the structure of C comes from the knowledge of the invariant linear maps in Skw and Dev. As they must be, in each case, multiples of the identity, CE=C(E _{_3}+E _{_1}+ E _{_5})=_{_3} E _{_3} +_{_1} E _{_1}+_{_5} E _{_5} for (nonzero) constants _{_3}, _{_1} and _{_5}. The noninvertible case results from considering C+I.     

Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.     

The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise

In the present paper,the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise.By using a perturbation method,the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases,in which different forms of the matrix B,that is included in the noise excitation term,are assumed and then,as a result,all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed.Via Monte-Carlo simulation,we find that the analytical expressions of the invariant measures meet well the numerical ones.And furthermore,the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process.Finally,for the three cases of singular boundaries for one-dimensional phase diffusion process,analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.     

A pseudo-stable structure in a completely invertible bouncer system

It is shown that a pseudo-stable structure of non-asymptotic convergence may exist in a completely invertible bouncing ball model. Visualization of the pattern of H-ranks helps to identify this structure. It appears that this structure is similar to the stable manifold of non-invertible nonlinear maps which govern the non-asymptotic convergence to unstable periodic orbits. But this convergence to the unstable repeller of the bouncing ball problem is only temporary since non-asymptotic convergence cannot exist in completely invertible maps. This nonlinear effect is exploited for temporary stabilization of unstable periodic orbits in completely reversible nonlinear maps.     

Pesin’s Formula for Random Dynamical Systems on \mathbf {R}^d

Pesin’s formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on $\mathbf {R}^d$ which have an invariant probability measure absolutely continuous to the Lebesgue measure on $\mathbf {R}^d$ . Finally we will show that a broad class of stochastic flows on $\mathbf {R}^{d}$ of a Kunita type satisfies Pesin’s formula.     

On the existence of low-period orbits in <Emphasis Type="Italic">n</Emphasis>-dimensional piecewise linear discontinuous maps

Discontinuous maps occur in many practical systems, and yet bifurcation phenomena in such maps is quite poorly understood. In this paper, we report some important results that help in analyzing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. In this paper, we extend that line of work for maps with discontinuity to obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map. This work was supported in part by the BRNS, Department of Atomic Energy (DAE), Government of India under project no. 2003/37/11/BRNS.     

磨损图研究的发展现状与趋势

磨损研究作为摩擦学领域的一个重要分支日益受到广泛重视，并且已经积螺了大量的实验数据，建立了不少磨损模型和以此为基础的磨损计算公式，但是，这些大都是在某些特定条件下将某一磨损机理孤立起来进行研究的结果，故其实际应用具有很大的局限性，在本世纪８０年代初问世的磨损图研究能够有效地改变这种状况。已有的研究成果表明，从一幅简单的磨损图上，不仅可以看出磨损的分类，以及每种磨损形式的特征及其控制因素和类型，而且     

The Conley Index Over the Circle

We study the Conley index over a base in the case when the base is the circle. Such an index arises in a natural way when the considered flow admits a Poincaré section. In that case the fiberwise pointed spaces over the circle generated by index pairs are semibundles, i.e., admit a special structure similar to locally trivial bundles. We define a homotopy invariant of semibundles, the monodromy class. We use the monodromy class to prove that the Conley index of the Poincaré map may be expressed in terms of the Conley index over the circle.     

城市地质图的特点·编图原则·方法

论述了城市地质图的主要特点、编图原则和方法, 其主要内容概括如下: (1)编制城市地质图的重要意义首先在于为不断完善和适应城市建设和发展的需要, 更好地使地质工作为城市建设总体规划和政府决策服务, 同时为具体工程设计和施工提供基础资料。其次为合理地开发利用城市土地资源、保护城市地质环境提供科学依据。(2)城市地质图是一个具有以专业地质规划目标与工程地质要求、城市发展总体效益评估为内容的系列图件, 是一个分别具有适应于地质专业人员、城市规划设计工程师和政府决策管理官员需要的多功能特点的系统性图件。(3)城市地质图编制原则, 第一在于客观性; 第二为评价性; 第三为预测性; 第四为实用性。前三项是前提和基础, 后者为目的。力求做到内容准确、明晰可读、易懂, 紧紧围绕城市规划设计和政府决策对地质技术的要求进行。(4)城市地质图包容了基础地质图类、分析评价图类、效益决策评估类三个等级的功能各异的图件, 在制图方法和程式上体现了图件的等级—内容—性质—用途的统一、技术性与管理决策的统一。     

碰撞振动系统的一类余维二分岔及T2环面分岔

建立了三自由度碰撞振动系统的动力学模型及其周期运动的Poincaré映射,当Jacobi矩阵存在两对共轭复特征值同时在单位圆上时,通过中心流形-范式方法将六维映射转变为四维范式映射.理论分析了这种余维二分岔问题,给出了局部动力学行为的两参数开折.证明系统在一定的参数组合下,存在稳定的Hopf分岔和T2环面分岔.数值计算验证了理论结果.     

Normal form for librational curves of the standard map

We present a method for constructing analytic expansions approximating librational invariant curves in the case of the so-called generalized standard map. After some preliminary changes of variables, we apply a direct Birkhoff normal form to a fixed order. The resulting system describes homotopically trivial invariant curves close to a periodic orbit. We investigate the stability of the librational curves applying a numerical method developed by J. Greene.

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Sommario Presentiamo un metodo per la costruzione di espansioni analitiche, adatte a descrivere le curve invarianti di librazione nel caso della standard map generalizzata. Dopo aver effettuato alcuni cambiamenti preliminari di variabili, calcoliamo la forma normale di Birkhoff ad un ordine fissato. Il sistema risultante descrive curve invarianti attorno ad orbite periodiche. Studiamo la stabilità di tali curve di librazione tramite un metodo numerico sviluppato da J. Greene.

     

Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations

A vector bundle morphism of a vector bundle with strongly ordered Banach spaces as fibers is studied. It is assumed that the fiber maps of this morphism are compact and strongly positive. The existence of two complementary, dimension-one and codimension-one, continuous subbundles invariant under the morphism is established. Each fiber of the first bundle is spanned by a positive vector (that is, a nonzero vector lying in the order cone), while the fibers of the other bundle do not contain a positive vector. Moreover, the ratio between the norms of the components (given by the splitting of the bundle) of iterated images of any vector in the bundle approaches zero exponentially (if the positive component is in the denominator). This is an extension of the Krein-Rutman theorem which deals with one compact strongly positive map only. The existence of invariant bundles with the above properties appears to be very useful in the investigation of asymptotic behavior of trajectories of strongly monotone discrete-time dynamical systems, as demonstrated by Poláik and Tereák (Arch. Ration. Math. Anal. 116, 339–360, 1991) and Hess and Poláik (preprint). The present paper also contains some new results on typical asymptotic behavior in scalar periodic parabolic equations.     

Extremal exponents of random dynamical systems do not vanish

A family of random diffeomorphisms on a manifoldM is said to be a random dynamical system or RDS if it has the so-called cocycle property. The multiplicative ergodic theorem assignsd (=dimM) Lyapunov exponents to every invariant measure of the system. Take the maximum of the leading exponents associated with the various invariant measures. The resulting number is said to be the maximal exponent of the system. The minimal exponent is defined in a similar fashion. It is shown that the minimal exponent of an RDS on a compact manifold is negative, provided not all invariant measures are determined by the future of. A similar statement relates the maximal exponent with the past of. We proceed by introducing Markov systems and Markov measures. This notion covers flows of stochastic differential equations as well as products of random diffeomorphisms in Markovian dependence, in particular, products of iid diffeomorphisms. Markov measures are characterized by the fact that they are functionals of the past. Consequently, if there exists a non-Markovian invariant measure, then the maximal exponent does not vanish. Typically, Markov systems do have non-Markovian invariant measures. Finally, for linear systems we recover results of Ledrappier. In particular, these results provide another proof of Furstenberg's theorem on the positivity of the leading exponent of a product of iid unimodular matrices.     

Lebesgue measure for creep

The Lebesgue strain measure for creep has been developed by taking the Lebesgue integral of the function of a weighted function instead of the power of a weighted function as was taken in Seth's strain measure. The problem of a spherical shell under internal pressure is considered and it is shown that the results obtained by using Seth's concept of measure can be derived from the more general analysis presented herein.     

Bifurcation analysis of a two-DoF mechanical system subject to digital position control. Part I: theoretical investigation

This paper analyzes the double Neimark–Sacker bifurcation occurring in a two-DoF system, subject to PD digital position control. In the model the control force is considered piecewise constant. Introducing a nonlinearity related to the saturation of the control force, the bifurcations occurring in the system are analyzed. The system is generally losing stability through Neimark–Sacker bifurcations, with relatively simple dynamics. However, the interaction of two different Neimark–Sacker bifurcations steers the system to much more complicated behavior. Our analysis is carried out using the method proposed by Kuznetsov and Meijer. It consists of reducing the dynamics of the nonlinear map to its local center manifold, eliminating the non-internally resonant nonlinear terms and transforming the nonlinear map to an amplitude map, that describes the local dynamics of the system. The analysis of this amplitude map allows us to define regions, in the space of the control gains, with a close interaction of the two bifurcations, which generates unstable quasiperiodic motion on a 3-torus, coexisting with two stable 2-torus quasiperiodic motions. Other regions in the space of the control gains show the coexistence of 2-torus quasiperiodic solutions, one stable and the other unstable. All the results described in this work are analytical and obtained in closed form, numerical simulations illustrate and confirm the analytical results.     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, a rather general model is obtained by assuming that the real noise is an output of a linear filter system-a zeromean stationary Gaussian diffusion process which satisfies detailed balance condition. By means of the asymptotic analysis approach given by L. Arnold and the expression of the eigenvalue spectrum of Fokker-Planck operator, the asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are obtained. Foundation item: the National Natural Science Foundation of China (19602016)     

Chaos and Entropy for Interval Maps

In this paper, various chaotic properties and their relationships for interval maps are discussed. It is shown that the proximal relation is an equivalence relation for any zero entropy interval map. The structure of the set of f-nonseparable pairs is well demonstrated and so is its relationship to Li-Yorke chaos. For a zero entropy interval map, it is shown that a pair is a sequence entropy pair if and only if it is f-nonseparable. Moreover, some equivalent conditions of positive entropy which relate to the number “3” are obtained. It is shown that for an interval map if it is topological null, then the pattern entropy of every open cover is of polynomial order, answering a question by Huang and Ye when the space is the closed unit interval.