Tangent Map Research Articles

Generic inflationary and noninflationary behavior in toy-cosmology

Cosmological solutions of a toy homogeneous isotropic universe filled with a superfluid Bose condensate described by a complex scalar field (with relativistic barotropic fluid interpretation) are studied. The eigenvalues of the tangent map for the resulting Hamiltonian system are used to classify the phase space regions and to understand the typical toy-universe evolution. After a transient, inflation is obtained in the hyperbolic (real eigenvalues) region. This new independent eigenvalue-based inflationary criterion is shown to be compatible and complementary to the standard slow roll-over conditions. For the later evolution of the toy-universe, a family of adiabatic trajectories oscillating about a conventional cosmology filled with a relativistic fluid is obtained once the system falls into the elliptic (imaginary eigenvalue) regions. The corresponding thermodynamic functions are computed.

Hausdorff dimension estimates fornon-injective maps using the cardinality of the pre-image sets

Upper bounds for the Hausdorff dimension of compact invariant sets ofC1-maps on smooth Riemannian manifolds are given in terms of thesingular values of the tangent map and the multiplicity function of themap, describing the number of pre-images of a certain point in a given set.For non-injective maps this improves previous results using only thesingular values.

Convergence of the Eckmann and Ruelle algorithm for the estimation of Liapunov exponents

We analyse the convergence conditions of the Eckmann and Ruelle algorithm (ERA) used to estimate the Liapunov exponents, for the tangent map, of an ergodic measure, invariant under a smooth dynamical system. We find sufficient conditions for this convergence that are related to those ensuring the convergence to the tangent map of the best linear $L^p$-fittings of the action of a mapping $f$ on small balls. Under such conditions, we show how to use ERA to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of ERA for the computation of Liapunov exponents in smooth manifolds, which allows us to avoid the problem of detecting the spurious exponents.We prove, for a Borel measurable dynamics $f$, the existence of Liapunov exponents for the function $S_r(x)$, mapping each point $x$ to the matrix of the best linear $L^p$-fitting of the action of $f$ on the closed ball of radius $r$ centred at $x$, and we show how to use ERA to get reliable estimates of the Liapunov exponents of $S_r$. We also propose a test for checking the differentiability of an empirically observed dynamics.

Basic Properties of the Projective Product with Application to Products of Column-Allowable Nonnegative Matrices

We use basic properties of the projective product to obtain exponential bounds for the Lipschitz constant associated with the projective product of column-allowable nonnegative matrices. We obtain similar bounds for the associated linear tangent maps.

The Blow-up Locus of Heat Flows for Harmonic Maps

Let M and N be two compact Riemannian manifolds. Let u k (x, t) be a sequence of strong stationary weak heat flows from M×R + to N with bounded energies. Assume that u k→u weakly in H 1, 2(M×R +, N) and that Σt is the blow-up set for a fixed t > 0. In this paper we first prove Σt is an H m−2-rectifiable set for almost all t∈R +. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σt is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪t<0Σt× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.

Lp(μ)-Estimation of Tangent Maps

We analyze the conditions under which the best Lp(μ)-linear fittings of the action of a mapping f on small balls give reliable estimates of the tangent map Df. We show that there is an inverse relationship between the conditions on the regularity, in terms of local densities, of the measure μ and the smoothness of the mapping f which are required to ensure the goodness of the estimates. The above results can be applied to the estimation of tangent maps in two empirical settings: from finite samples of a given probability distribution on Rn and from finite orbits of smooth dynamical systems. As an application of the results of this paper we obtain sufficient conditions on the measure μ to ensure the convergence of the Eckmann–Ruelle algorithm for computing the Liapunov exponents of smooth dynamical systems.

Symplectic Tangent map for Planetary Motions

Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in general.

Positive Semicharacters of Lie Semigroups

We study positive semicharacters of generating Lie subsemigroup $$S$$ of a connected Lie group $$G$$ . These semicharacters are important for positive representations of $$S$$ in Hilbert space and for completely monotonic functions in $$S$$ . We describe the tangent map for a positive semicharacter and then obtain a necessary and sufficient condition for nontriviality of the wedge $$S_1^*$$ consisting of all bounded positive semicharacters of $$S$$ . In particular $$S_1^*$$ is nontrivial for a solvable simply connected $$G$$ and invariant $$S$$ without nontrivial subgroups, but it is trivial for a semisimple $$G$$ .

Secondary characteristic classes for the isotropic Grassmannian

0. Introduction. Let M be a 2n-dimensional manifold and ω be a 2-form on M . The pair (M,ω) is called a symplectic manifold if ω is closed (i.e. dω = 0) and nondegenerate, [Wei]. A submanifold L of M is called Lagrangian if the restriction of the symplectic form to the tangent space TyL vanishes for each y ∈ L and TyL = (TyL) = {v ∈ TyM : ω(v, u) = 0 for every u ∈ TyL}. A submanifold I of M is called isotropic if for every y ∈ I a skew-orthogonal complement (with respect to ω) of the tangent space to I contains this tangent space, i.e. (TyI) ⊇ TyI. A submanifold C2n−k is said to be coisotropic if for every y ∈ C holds (TyC) ⊆ TyC. Note that the dimension of an isotropic submanifold is less than or equal to n, and the dimension of a coisotropic one is greater than or equal to n. A Lagrangian submanifold is an isotropic submanifold of maximal dimension or a coisotropic one of minimal dimension. These submanifolds play a significant role in symplectic geometry and they appear naturally in many branches of physics ([GS], [Ja1], [Ja2]). A fundamental example of a symplectic manifold is the real cotangent bundle T ∗Rn endowed with the canonical symplectic structure ω = dΘ, where 1-form Θ is the Liouville form on T ∗Rn. It is defined by Θ(u) = 〈TπRn(u), τT∗Rn(u)〉, where u ∈ T (T ∗Rn), the mapping TπRn is the tangent mapping of πRn : T ∗Rn −→ R and τT∗Rn : T (T ∗Rn) −→ T ∗Rn is the tangent bundle projection. If (p, x) are coordinates of the bundle T ∗Rn, i.e. x1, . . . , xn are coordinates of the base and p1, . . . , pn are coordinates of the fibre, then Θ and ω have the local Darboux form (see [Wei])

A framework for geometrically nonlinear continuum damage mechanics

The objective of this work is the derivation of a framework for geometrically nonlinear continuum damage mechanics which allows for the description of damage by second order tensors. To this end, so called fictitious undamaged or rather microscopic configurations are considered which are related to the macroscopic configurations by linear tangent maps which allow for the interpretation as damage deformation gradients. The corresponding Finger tensors defined in the macroscopic configuration are then understood as damaged metrics for measuring the free energy for given strains. Thereby, the underlying motivation is provided by the hypothesis of strain energy equivalence between microscopic and macroscopic configurations. Based on the standard dissipative material approach the constitutive framework is completed by stresses, damage stresses, a damage condition and the associated evolution laws for the damage metric and the internal hardening variable. The relationship of the present damage metric based theory to the classical understanding of damage as an area reduction is highlighted. Finally, a simple prototype model problem is presented which allows for an efficient and accurate algorithmic treatment. Numerical examples demonstrate the applicability of the proposed framework.

Increasing the Dynamic Aperture of Accelerator Lattices

A method for reducing chaotic dynamics in four-dimensional symplectic maps is presented. The method is to reduce certain chaos measures related to the invariants of the tangent map of fixed points. This method is applied to circular accelerators, where chaotic dynamics determines the dynamic aperture, the region of long-time stable orbits. A factor of 3--4 increase in the phase-space volume of confined trajectories is obtained.

Compatibility of large deformations in nonlinear shell theory

In this paper, we propose to establish compatibility conditions satisfied by the membrane strain tensor Δ and the curvature strain tensor K of a shell midsurface. To this end, we use the polar decomposition RU of the linear tangent map of a mapping function φ. First, we characterize the tangential part of the derivative of R by a new variable A. Then the compatibility equations are decomposed in a natural way into a first-order differential system. The use of A greatly simplifies the differential calculus and gives an intrinsic and explicit formulation for compatibility conditions in nonlinear shell theory.

Analysis of flexible multibody systems with spatial beams using mixed variational principles

A general finite element formulation is presented for dynamic analysis of spatial elastic beams, for small strains, in a multi-body configuration. The tangent maps associated to the finite rotation vector are used to compute the tangent matrices used to integrate implicitly the equations of motion in descriptor form. A corotational method and a mixed variational method are used to compute the tangent stiffness matrix. The tangent constraint matrices are obtained using consistent linearization of the constraint equations. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include: dynamic analysis of flexible beam structures and multi-flexible body systems with open and closed kinematic loops. A comparison with the previous results in the literature shows a very good performance in terms of time integration step and number of elements used. © 1998 John Wiley & Sons, Ltd.

Analysis of flexible multibody systems with spatial beams using mixed variational principles

A general finite element formulation is presented for dynamic analysis of spatial elastic beams, for small strains, in a multi-body configuration. The tangent maps associated to the finite rotation vector are used to compute the tangent matrices used to integrate implicitly the equations of motion in descriptor form. A corotational method and a mixed variational method are used to compute the tangent stiffness matrix. The tangent constraint matrices are obtained using consistent linearization of the constraint equations. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include: dynamic analysis of flexible beam structures and multi-flexible body systems with open and closed kinematic loops. A comparison with the previous results in the literature shows a very good performance in terms of time integration step and number of elements used. © 1998 John Wiley & Sons, Ltd.

Hausdorff dimension estimates for invariant sets with an equivariant tangent bundle splitting

Upper bounds for the Hausdorff dimension of compact and invariant sets of diffeomorphisms are given using a singular value function of the tangent map and the topological entropy under the assumption, that there exists an equivariant splitting of the tangent bundle. This improves previous results for compact uniformly hyperbolic sets of diffeomorphisms satisfying an additional pinching condition. Furthermore, it is shown that the results can be extended to a special class of noninjective maps.

Characteristic Spectrum for Differential Systems

The characteristic spectrum of the tangent map of a differential system is determined by using qualitative functions and ergodic measures over the bundle of orthonormaln-frames. The characteristic spectrum of the parallelotope cocycle of a differential system is determined by using a qualitative function and ergodic measures over the state manifold.

Lyapunov Exponents without Rescaling and Reorthogonalization

We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or $M \~{M}$ where M is the tangent map. This method uses a minimal set of variables, does not require renormalization or reorthogonalization, can be used to efficiently compute partial Lyapunov spectra, and does not break down when the Lyapunov spectrum is degenerate.

Hausdorff and Fractal Dimension Estimates for Invariant Sets of Non-Injective Maps

In this paper we are concerned with upper bounds for the Hausdorff and fractal dimensions of negatively invariant sets of maps on Riemannian manifolds. We consider a special class of non-injective maps, for which we introduce a factor describing the "degree of non-injectivity". This factor can be included in the Hausdorif dimension estimates of Douady-Oesterlé type [2, 7, 10] and in fractal dimension estimates [5, 13, 15] in order to weaken the condition to the singular values of the tangent map. In a number of cases we get better upper dimension estimates.

Stochasticity in the Josephson map

The Josephson map describes the nonlinear dynamics of systems characterized by the standard map with a uniform external bias superposed. The intricate structures of the phase-space portrait of the Josephson map are examined here on the basis of the associated tangent map. A numerical investigation of stochastic diffusion in the Josephson map is compared with the renormalized diffusion coefficient calculated using the characteristic function. The global stochasticity of the Josephson map occurs at far smaller values of the stochastic parameter than is the case of the standard map.

Primal and mixed variational principles for dynamics of spatial beams

A general formulation is presented for static and dynamic analysis of spatial elastic beams capable of undergoing finite rotations and small strains. The tangent maps associated to the finite rotation vector are used to compute the tangent iteration matrices used to integrate implicitly the equations of motion in descriptor form. A total Lagrangian primal corotational method and an updated Lagrangian mixed variational method are proposed to compute the tangent stiffness matrix. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include static (pre- and postbuckling) and dynamic analysis of flexible beams structures. The new finite elements show a very good performance, in terms of fewer number of elements used and accuracy during the simulation, both for static and dynamic problems.