Covering relations and Lyapunov condition for topological conjugacy

In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrix A, we show that any small C⁰ perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to σ_A, the two-sided subshift of finite type induced by A. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any small C¹ perturbed systems is topological conjugate to σ_A. We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate.     

A survey of the automorphism groups of some fulleroids

An automorphism of the graph X is a bijection on which preserves the edge set. By a (x,6) fullerene graph, we mean a fulleroid graph composed of x-gonal and hexagonal faces which satisfies in the Euler's theorem. The aim of this paper is to report the automorphism group of some fulleroid graphs such as fullerene graphs. We also determine the size of automorphism group of some fullerenes.     

Node-pair reliability of network systems with small distances between adjacent nodes

A new method for computing the node-pair reliability of network systems modeled by random graphs with nodes arranged in sequence is presented. It is based on a recursive algorithm using the “sliding window” technique, the window being composed of several consecutive nodes. In a single step, the connectivity probabilities for all nodes included in the window are found. Subsequently, the window is moved one node forward. This process is repeated until, in the last step, the window reaches the terminal node. The connectivity probabilities found at that point are used to compute the node-pair reliability of the network system considered. The algorithm is designed especially for graphs with small distances between adjacent nodes, where the distance between two nodes is defined as the absolute value of the difference between the nodes’ numbers. The maximal distance between any two adjacent nodes is denoted by Γ(G), where G symbolizes a random graph. If Γ(G)=2 then the method can be applied for directed as well as undirected graphs whose nodes and edges are subject to failure. This is important in view of the fact that many algorithms computing network reliability are designed for graphs with failure-prone edges and reliable nodes. If Γ(G)=3 then the method's applicability is limited to undirected graphs with reliable nodes. The main asset of the presented algorithms is their low numerical complexity—O(n), where n denotes the number of nodes.     

Co-citation,bibliographic coupling and a characterization of lattice citation networks

In this article we study directed, acyclic graphs. We introduce the head and tail order relations and study some of their properties. Recalling the notions of generalized bibliographic coupling and generalized co-citation, and introducing a new property, called the l - property, we come to a characterization of lattices. As document citation networks are concrete realizations of directed acyclic graphs all our results are directly applicable to citation analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

Thermal stability of superconducting magnets: static criteria

To analyse the stability of a superconductor, a static analysis is often employed. The static solution of the energy equation describing the superconductor forms the boundary between a stable and an unstable region of operation. We solved the static energy equation analytically for the constant property case, and numerically for the variable property case. The results are gathered in graphs, from which the critical energy as a function of j and αj² can be found. In the variable property case we introduced θ_s as an additional variable to characterize the conductor. In a second paper we will analyse the dynamic behaviour of a conductor and compare it with the criteria introduced in this paper.     

State splitting,strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras

We prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz–Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behaviour of the stable isomorphism between the associated stable Cuntz–Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Markov shifts is described in terms of some equivalence relation of the Cuntz–Krieger algebras with canonical masas and the gauge actions without stabilization.     

Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations

In planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects. In this article, we propose a graph-aided method to solve this problem for two-level experiments. First, we choose the defining relations for a 2 ^n–k design according to a goodness criterion such as the minimum aberration criterion. Then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two-factor interactions for the given defining relations. In each graph a vertex represents a factor and an edge represents the interaction between the two factors. For the experiment planner, the job is simple: Draw a graph representing the specified interactions and compare it with the list of graphs obtained previously. Our approach is a substantial improvement over Taguchi's linear graphs.     

The concept of bounded mean motion for toral homeomorphisms

A conservative irrational pseudo-rotation of the two-torus is semi-conjugate to the irrational rotation if and only if it has the property of bounded mean motion [T. Jäger, Linearisation of conservative toral homeomorphisms, Invent. Math., published online 9 December 2008, DOI: 10.1007/s00222-088-0171-5]. (Here ‘irrational pseudo-rotation’ means a toral homeomorphism with a unique and totally irrational rotation vector). The aim of this note is to explore this concept further. For instance, we provide an example which shows that the preceding statement does not hold in the non-conservative case. Further, we collect a number of observations concerning the case where the bounded mean motion property fails. In particular, we show that a non-wandering irrational pseudo-rotation of the two-torus with unbounded mean motion has sensitive dependence on initial conditions.     

Explicit bounds for separation between Oseledets subspaces

We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and satisfy two properties (SVG) and (FI). The singular value gap (SVG) property says that two successive singular values of the cocycle at some index d admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest d-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension d and a slow space of codimension d. We compute an explicit constant lower bound on the angle between these two spaces using solely the constants defining the properties (SVG) and (FI). We extend the results obtained by Bochi and Gourmelon in the finite-dimensional case for bijective operators and the results obtained by Blumenthal and Morris in the infinite dimensional case for injective norm-continuous cocycles, in the direction that the operators are not required to be globally injective, that no dynamical system is involved and no compactness of the underlying system or smoothness of the cocycle is required. Moreover we give quantitative estimates of the angle between the fast and slow spaces that are new even in the case of finite-dimensional bijective operators in Hilbert spaces.     

Ergodic measure-expansive diffeomorphisms

In this paper, we consider the set of diffeomorphisms which are measure-expansive for any ergodic measure, and study the set from the viewpoint of geometric theory of dynamical systems. It is proved that (1) there exists a non-empty C¹-open set of robustly non-hyperbolic and transitive diffeomorphisms such that each element of the set is measure-expansive for any ergodic measure, and that (2) C¹-generically, a diffeomorphism is measure-expansive for any generic ergodic measure.     

STEINER PROBLEMS IN GRAPHS: ALGORITHMS AND APPLICATIONS

Consider an undirected graph G = (V, E) with positive edge weights and a nonempty set S ? V, where Vand E are the sets of vertices and edges of G respectively. The Steiner problem in graphs is that of finding a subgraph of G which spans S and is of minimum total edge weight. In this paper we survey solution procedures for this problem. As the associated decision problem is NP-Complete we place special emphasis on heuristic methods of solution. We also examine special cases, related problems, and applications. The paper concludes with ideas for the development of new, efficient heuristics.     

Fast matrix exponent for deterministic or random excitations

The solution of ż=Az is z(t)=exp(At)z₀=E_tz₀, z₀=z(0). Since z(2t)=E_2tz₀=Ez₀, z(4t)=E_4tz₀=Ez₀, etc., one function evaluation can double the time step. For an n‐degree‐of‐freedoms system, A is a 2n matrix of the nth‐order mass, damping and stiffness matrices M, C and K. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean‐square response P to a white noise random force with intensity W(t) is governed by the Lyapunov differential equation: Ṗ=AP+PA^T+W. The solution of the homogeneous Lyapunov equation is P(t)=exp(At) P₀ exp(A^Tt), P₀=P(0). One function evaluation can also double the time step. If W(t) is given as piecewise polynomials, the mean‐square response can also be obtained analytically. In fact, exp(At) consists of the impulsive‐ and step‐response functions and requires no special treatment. The method is extended further to coloured noise. In particular, for a linear system initially at rest under white noise excitation, the classical non‐stationary response is resulted immediately without integration. The method is further extended to modulated noise excitations. The method gives analytical mean‐square response matrices for lightly damped or heavily damped systems without using modal expansion. No integration over the frequency is required for the mean‐square response. Four examples are given. The first one shows that the method include the result of Caughy and Stumpf as a particular case. The second one deals with non‐white excitation. The third finds the transient stress intensity factor of a gun barrel and the fourth finds the means‐square response matrix of a simply supported beam by finite element method. Copyright © 2001 John Wiley & Sons, Ltd.     

Two simple stabilization criteria for interconnected systems

Abstract

The stabilization of an interconnected system is considered. Based on the Lyapunov stability theorem, two simple stabilization criteria for interconnected systems are derived using different Lyapunov functions. These main results are expressed by the measurement of the system matrices directly, without solving the Lyapunov equations. The two stabilization criteria are also compared.     

On bending strength of superhard composite materials based on WC–Co hardmetals

We present analytical algorithms for computing the ultimate bending strength of superhard composite materials based on WC–Co hardmetals. The study is performed for fine-grained materials (where mean particle size of the dispersed superhard phase d _C and that of carbide grains d _WC are of the same order of magnitude) and coarse-grained materials (with d _C ≫ d _WC ). The strength of the composite is assumed to be governed by the strength of its hardmetal matrix. The stressed state of the matrix is assessed through volume-average microstresses for fine-grained materials and interface-average stresses for coarse-grained composites. The calculated results presented in the form of tables and graphs have been analyzed. The strength has been found to decrease drastically with increasing particle size of the superhard phase and its concentration in the composite.     

Identification of parameters with different orders of magnitude in chaotic systems

The synchronization and parameter identification of six unknown parameters in a chaotic neuron model, which one parameter (about 0.006) is 3 orders of magnitude smaller than the others (about 1–5), is investigated by using Lyapunov stability theory and adaptive synchronization in detail. Two gain coefficients (δ₁,?δ₂) are introduced into the Lyapunov function to obtain certain optimized controllers and parameter observers. A selectable amplification factor k ₀ is presented using scale conversion and it is used to improve the accuracy of parameter estimation with the smallest order. The parameter space for gain coefficient (δ) versus amplification factor k ₀, and the parameter space δ₁ versus δ₂ at certain fixed amplification factor k ₀ are calculated numerically. It is found that the selection values of optimized gain coefficients and amplification factor are critical to estimate the six unknown parameters, particularly for the smallest unknown parameters with an order 0.001. The extensive numerical results show that it is more effective to estimate the smallest unknown parameter r when the two gain coefficients δ₁ and δ₂ are given the same value and a higher amplification factor k ₀ is used. It could be useful to estimate the unknown parameters with large deviation of order magnitude, such as a single chaotic Josephson junction coupled to a Resonant tank and other chaotic systems with potential application [Z.Y. Wang, H.Y. Liao, and S.P. Zhou, Study of the DC biased Josephson junction coupled to a Resonant tank, Acta. Phys. Sin. 50(10) (2001), pp. 1996–2000 (in Chinese)].     

Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing

We study bifurcations of invariant graphs in skew-product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with one-dimensional concave fibre maps under multiplicative forcing when the forcing is scaled by a parameter r = e ^?t . For a range of parameters two invariant graphs (a trivial and a non-trivial one) coexist, and we use thermodynamic formalism to characterize the parameter dependence of the Hausdorff and packing dimension of the set of points where both graphs coincide. As a corollary we characterize the parameter dependence of the dimension of the global attractor : Hausdorff and packing dimension have a common value , and there is a critical parameter γ^? _c determined by the SRB measure of T ^?1 such that for t?γ^? _c and is strictly decreasing for t∈[γ^? _c , γ_max ).     

Hamiltonian stability in open surfaces with simple singularities

We extend to open surfaces with simple singularities the theorem proposed by Xavier Jarque and Zbigniew Nitecki for Hamiltonian flows in the plane that are structurally stable among Hamiltonian flows. We describe the Hamiltonian dynamics on {x² + y² = z²} and {xy = 0} by presenting characterization theorems for Hamiltonian stability and some natural consequences. The stability of planar Hamiltonian flows with an invariant line is also studied.     

Computational structural and material stability analysis in finite electro‐elasto‐statics of electro‐active materials

Dielectric materials like electro‐active polymers (EAPs) exhibit coupled electro‐mechanical behavior at large strains. They respond by a deformation to an applied electrical field and are used in advanced industrial environments as sensors and actuators, for example, in robotics, biomimetics and smart structures. In field‐activated or electronic EAPs, the electric activation is driven by Coulomb‐type electrostatic forces, resulting in Maxwell stresses. These materials are able to provide finite actuation strains, which can even be improved by optimizing their composite microstructure. However, EAPs suffer from different types of instabilities. This concerns global structural instabilities, such as buckling and wrinkling of EAP devices, as well as local material instabilities, such as limit‐points and bifurcation‐points in the constitutive response, which induce snap‐through and fine scale localization of local states. In this work, we outline variational‐based definitions for structural and material stability, and design algorithms for accompanying stability checks in typical finite element computations. The formulation starts from stability criteria for a canonical energy minimization principle of electro‐elasto‐statics, and then shifts them over to representations related to an enthalpy‐based saddle point principle that is considered as the most convenient setting for numerical implementation. Here, global structural stability is analyzed based on a perturbation of the total electro‐mechanical energy, and related to statements of positive definiteness of incremental finite element tangent arrays. We base the local material stability on an incremental quasi‐convexity condition of the electro‐mechanical energy, inducing the positive definiteness of both the incremental electro‐mechanical moduli as well as a generalized acoustic tensor. It is shown that the incremental arrays to be analyzed in the stability criteria appear within the enthalpy‐based setting in a distinct diagonal form, with pure mechanical and pure electrical partitions. Applications of accompanying stability analyses in finite element computations are demonstrated by means of representative model problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Normal Lyapunov exponents and asymptotically stable attractors

Suppose f is a C ^1+α map and leaves a lower-dimensional compact attractor A. In this article, we show that if for every f-ergodic probability measure supported on A, the normal Lyapunov exponents are negative, then this attractor could be a high-dimensional attractor. Moreover, we prove that the supremum of the normal Lyapunov exponents on the set of all ergodic measures can be achieved.