NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS(Ⅱ)──PART 1

ＮＯＴＥＳＯＮＡＳＴＵＤＹＯＦＶＥＣＴＯＲＢＵＮＤＬＥＤＹＮＡＭＩＣＡＬＳＹＳＴＥＭＳ（Ⅱ）──ＰＡＲＴ１ＬｉａｏＳｈａｎｔａｏ（廖山涛）Ａｂｓｔｒａｃｔ：Ｔｈｅｓｔｕｄｙｏｆｌｉｎｅａｒａｎｄｇｌｏｂａｌ．ｐｒｏｐｅｒｔｉｅｓｏｆｌｉｎｅａｒｄｙｎ...     

Generic diffeomorphisms with measure-expansive homoclinic classes

We prove that for C¹ generic diffeomorphisms, every measure-expansive locally maximal homoclinic class is hyperbolic.     

Singular hyperbolicity of star flows on surfaces

In this paper, we prove that every star flow on the closed surface has finitely many chain recurrent classes. Furthermore, it is singular hyperbolic if every non-trivial singular chain component is a graph. As a consequence, every star flow on the 2-sphere or the projective plane is singular hyperbolic.     

New approach to differential equations with countable impulses

This paper provides a new approach to study the solutions of a class of generalized Jacobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n + 1.A new class of generalized differential operators is defined.We investigate the kernel of the corresponding maximal operators by applying operator theory.It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic,in which there are n dimension solutions with exponential...     

Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

In this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface.     

Hyperbolicity of C~1 -star invariant sets for C~1 -class dynamical systems

In this paper, we simply prove, in the framework of Liao, the hyperbolicity of C 1 -star invariant sets of C 1 -class differential systems on a closed manifold of dimension≤ 4, without using the C 1 -connecting lemma and even the ergodic closing lemma.     

A Grobman-Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators Am, we establish a version of the Grobman-Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(n), determined by a sequence ρ(n). For all sufficiently small Lipschitz perturbations Am+fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators Am. We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ(n)=n, but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ(n)=logn.     

Persistence of nonhyperbolic measures for <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-diffeomorphisms

In the space of diffeomorphisms of an arbitrary closed manifold of dimension ≥ 3, we construct an open set such that each difteomorphism in this set has an invariant ergodic measure with respect to which one of its Lyapunov exponents is zero. These difteomorphisins are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with the circle as the fiber. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products.     

Instabilities in the Chapman-Enskog Expansion and Hyperbolic Burnett Equations

It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnett equations (the next step after the Navier-Stokes equations). Roughly speaking, the reason is that the solutions of higher equations of hydrodynamics (Burnett's, etc.) become unstable with respect to short-wave perturbations. This problem was recently attacked by several authors who proposed different ways to deal with it. We present in this paper one of possible alternatives. First we deduce a criterion for hyperbolicity of Burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases. Then we discuss in more detail the problem of truncation of the Chapman-Enskog expansion and show that the way of truncation is not unique. The general idea of changes of coordinates (based on analogy with the theory of dynamical systems) leads finally to nonlinear Hyperbolic Burnett Equations (HBEs) without using any information beyond the classical Burnett equations. It is proved that HBEs satisfy the linearized H-theorem. The linear version of the problem is studied in more detail, the complete Chapman-Enskog expansion is given for the linear case. A simplified proof of the Slemrod identity for Burnett coefficients is also given.     

Extended thermodynamics—a theory of symmetric hyperbolic field equations

Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation, provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday