Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system using a technique combining the generalized Lyapunov function theory and optimization. For the Lorenz–Haken system, we derive a four-dimensional ellipsoidal ultimate bound and positively invariant set. Furthermore, the two-dimensional parabolic ultimate bound with respect to x–z is established. Finally, numerical results to estimate the ultimate bound are also presented for verification. The results obtained in this paper are important and useful in control, synchronization of hyperchaos and their applications.     

Generic points for dynamical systems with average shadowing

We study the Besicovitch pseudometric \(D_B\) for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to \(D_B\). It is proved that the weak specification property implies the average asymptotic shadowing property and the latter property does not imply the former one nor the almost specification property. Furthermore an example of a proximal system with the average shadowing property is constructed. It is proved that to every invariant measure \(\mu \) of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for \(\mu \). A simple consequence of the theory presented is that every invariant measure has a generic point in a system with the asymptotic average shadowing property.     

Quasi-Armendariz rings relative to a monoid

For a monoid M, we introduce M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings, and investigate their properties. The M-quasi-Armendariz condition is a Morita invariant property. The class of M-quasi-Armendariz rings is closed under some kinds of upper triangular matrix rings. Every semiprime ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M. Moreover, we study the relationship between the quasi-Baer property of a ring R and those of the monoid ring R[M]. Every quasi-Baer ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M.     

Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. This paper has investigated the ultimate bound and positively invariant set of a permanent magnet synchronous motor system. We combine the Lyapunov stability theory with the comparison principle method. For this system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set for all the positive values of its parameters σ, γ. In addition, the two-dimensional bound with respect to x ? y are established. Then, it is the two-dimensional estimation about x ? z. Finally, the result is applied to the study of completely chaos synchronization. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme. At the same time, one numerical example illustrating a localization of a chaotic attractor is presented as well. Numerical simulation is consistent with the results of theoretical calculation.     

Roth's theorems and decomposition of modules

It is shown that there is a connection between Roth's theorems on similarity and equivalence of block-triangular matrices and decomposition of modules. The module property is that if $\text{M?}\text{N⊕M}\text{N}$, then N is a summand of M. This holds for any commutative ring if M is finitely presented. New proofs of Roth's theorems are given for commutative rings. Some results are established in the noncommutative case.     

Coherent measures and the existence of smooth Lyapunov 1-forms for flows

Let a smooth vector field V on a smooth closed manifold M be given and let Z⊂M be an isolated invariant set for the flow of V. In this situation, we give a necessary and sufficient condition for the existence of a Lyapunov 1-form for (V,Z) in terms of the relative asymptotic cycles associated with certain invariant measures of the flow.     

Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

On the Conley index in the invariant manifolds

Let ? be a flow on a manifold M and assume that N⊂M is an invariant manifold. The aim of this note is to compare the Conley indices of an isolated invariant set S⊂N with respect to the flow ? and the flow ? restricted to N.     

Convergence in measure with respect to a matrix-valued measure and some matrix completion problems

For a matrix-valued measure M we introduce a notion of convergence in measure M, which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of M. Let S be a subset of the set of matrices of given size. It is easy to see that the set of S-valued measurable functions is closed under convergence in measure with respect to a matrix-valued measure if and only if S is a ρ-closed set, i.e. if and only if SP is closed for any orthoprojector P. We discuss the behaviour of ρ-closed sets under operations of linear algebra and the ρ-closedness of particular classes of matrices.     

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II₁ factors and Mn(C)) and symmetric gauge norms on L^∞[0,1] and Cn. As the first application, we obtain that the class of unitarily invariant norms on a type II₁ factor coincides with the class of symmetric gauge norms on L^∞[0,1] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286-300] on unitarily invariant norms on Mn(C). As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760-766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp-theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative Lp-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M₂(C)) and some extreme points of N(Mn(C)) (n?3). For a type II₁ factor M, we prove that if t (0?t?1) is a rational number then the Ky Fan tth norm is an extreme point of N(M).     

Divergence theorems in path space

We obtain divergence theorems on the solution space of an elliptic stochastic differential equation defined on a smooth compact finite-dimensional manifold M. The resulting divergences are expressed in terms of the Ricci curvature of M with respect to a natural metric on M induced by the stochastic differential equation. The proofs of the main theorems are based on the lifting method of Malliavin together with a fundamental idea of Driver.     

DIFFEOMORPHISMS WITH VARIOUS C1 STABLE PROPERTIES

Let M be a smooth compact manifold and Λ be a compact invariant set.In this article,we prove that,for every robustly transitive set Λ,f|Λ satisfies a C1-genericstable shadowable property (resp.,C1-gene...     

Local index theory over foliation groupoids

We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P→M along with a G-invariant fiberwise Dirac-type operator D on P. The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory group, with a closed graded trace on a certain noncommutative de Rham algebra Ω^*B associated to G. The proof is by means of superconnections in the framework of noncommutative geometry.     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

Book review of Israel Gohberg,Seymour Goldberg,Marinus A. Kaashoek,Basic Classes of Linear Operators,Birkhäuser Verlag,Basel–Boston–Berlin 2003, 423 p. ISBN 3-7643-6930-2

In projective space three notions of convexity (weak convexity, strong convexity, p-convexity) are regarded systematically. Since these notions are defined only by incidence relations, there can be introduced dual notions. We consider relations.between the introduced notions and the most essential properties of convex sets. To all assertions. can be formulated dual assertions, too. The most important theorems given by Fenchel can be generalised. The property of a point set (a set of hyperplanes) to be strongly convex or p-convex, respectively, is invariant with respect to correlations.     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

Generalized closed sets with respect to an ideal in bitopological spaces

An ideal on a set X is a nonempty collection of subsets of X with heredity property which is also closed under finite unions. The concept of generalized closed sets in bitopological spaces was introduced by Sundaram. In this paper, we introduce and study the concept of generalized closed sets with respect to an ideal in an ideal bitopological space.     

Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization

In this paper, we investigate the ultimate bound and positively invariant set for a new chaotic system via the generalized Lyapunov function theory. For this system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set. In addition, the two-dimensional bound with respect to x-z and y-z are established. Finally, the result is applied to the study of completely chaos synchronization, an exact threshold is given with the system parameters. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

The degree of multivalued maps from manifolds to spheres

In this paper, for closed connected oriented manifolds M and N of the same dimension, we study the degree of a triple (??, p, q), where p is a Vietoris map from a compact space ?? to M and q is a continuous map from ?? to N. In particular, we have Borsuk?CUlam-type degree theorems on manifolds with involutions.