Tangent Map Research Articles

Fractal dimension estimate for invariant set in complete Riemannian manifold

Abstract In this paper, we give a simple upper bounds for the fractal dimensions of a forward invariant set and a negatively invariant set of a C1-diffeomorphism of the complete Riemannian manifold with non-negative Ricci curvature. These results generalize the corresponding result of C. Robinson [Dynamics systems: stability symbolic dynamics and chaos. 2nd ed. Boca Raton, London, New York, Washington: CRC Press; 1999] in R n , and weakens the condition to the singular values of the tangent map in [Boichenko VA, Franz A, Leonov GA, Reitmann V. Hausdorff and fractal dimension estimates for invariant sets of non-injective maps. Z Anal Anw 1998;17:207–23]. Finally, as application, we give the upper bound of the fractal dimension of an invariant set of the flow on Riemannian manifold.

Modeling initial strain distribution in soft tissues with application to arteries

A general theory for computing and identifying the stress field in a residually stressed tissue is presented in this paper. The theory is based on the assumption that a stress free state is obtained by letting each point deform independently of its adjacent points. This local unloading represents an initial strain, and can be described by a tangent map. When experimental data is at hand in a specific situation, the initial strain field may be identified by stating a nonlinear minimization problem where this data is fitted to its corresponding model response. To illustrate the potential of such a method for identifying initial strain fields, the application to an in vivo pressure-radius measurement for a human aorta is presented. The result shows that the initial strain is inconsistent with the strain obtained with the opening-angle-method. This indicates that the opening-angle-method has a too restrictive residual strain parameterization, in this case.

Conditions for the linear stability of libration points and effects of drag on the linear stability of triangular libration points in the restricted three-body problem

A number of criteria for linear stability of libration points in the perturbed restricted three-body problem are presented. The criteria involve only the coefficients of the characteristic equation of the tangent map of the libration points and can be easily applied. With these criteria the effect of drag on the linear stability of the triangular libration points in the classical restricted three-body problem is investigated. Some of Murray et al.'s results are improved.

A criterion for the linear stability of theequilibrium points in the perturbed restricted three-body problem and its application in Robe's problem

A criterion for the linear stability of the equilibrium points in the perturbed restricted three-body problem is given. This criterion is related only to the coefficients of the characteristic equation of the tangent map of an equilibrium point, and this is convenient to use. With this criterion, we have discussed the linear stability of the equilibrium points in the Robe problem under the perturbation of a drag force, derived the linearly stable region of the equilibrium point in the perturbed Robe's problem with the drag given by Hallen et al., and improved as well the results obtained by Giordano et al.

Lower Bound for Energies of Harmonic Tangent Unit-Vector Fields on Convex Polyhedra

We derive a lower bound for energies of harmonic maps of convex polyhedra in $ \R^3 $ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^\infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy.

Differential Invariants of Immersions of Manifolds with Metric Fields

The problem of finding all r-th order differential invariants of immersions of manifolds with metric fields, with values in a left (G^1_m x G^1_n)-manifold is formulated. For obtaining the basis of higher order differential invariants the orbit reduction method is used. As a new result it appears that r-th order differential invariants depending on an immersion f:M->N of manifolds M and N and on metric fields on them can be factorized through metrics, curvature tensors and their covariant derivatives up to the order (r-2), and covariant differentials of the tangent mapping Tf up to the order r. The concept of a covariant differential Tf is also introduced in this paper. The obtained results are geometrically interpreted as well.

Upper and lower Hausdorff dimension estimates for invariant sets of weak k‐1‐endomorphisms

AbstractFor a special class of non‐injective maps called the weak k‐1‐endomorphisms on Riemannian manifolds upper and lower bounds for Hausdorff dimensions of invariant sets are given in terms of the singular values of the tangent maps, which generalize Franz's corresponding results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Infinitesimal Liouville Distributions for Teichmüller Space

We consider an arbitrary Riemann surface $X$, possibly of infinite hyperbolic area. The Liouville measure of the hyperbolic metric defines a measure on the space $G(\tilde{X})$ of geodesics of the universal covering $\tilde{X}$ of $X$. As we vary the Riemann surface structure, this gives an embedding from the Teichmuller space of $X$ into the Frechet space of Holder distributions on $G(\tilde{X})$. We show that the embedding is continuously differentiable. In particular, we obtain an explicit integral representation of the tangent map.

A framework for multiplicative elastoplasticity with kinematic hardening coupled to anisotropic damage

The objective of this contribution is the formulation and algorithmic treatment of a phenomenological framework to capture anisotropic geometrically nonlinear inelasticity. We consider in particular the coupling of viscoplasticity with anisotropic continuum damage whereby both, proportional and kinematic hardening are taken into account. As a main advantage of the proposed formulation standard continuum damage models with respect to a fictitious isotropic configuration can be adopted and conveniently extended to anisotropic continuum damage. The key assumption is based on the introduction of a damage tangent map that acts as an affine pre-deformation. Conceptually speaking, we deal with an Euclidian space with respect to a non-constant metric. The evolution of this field is directly related to the degradation of the material and allows the modeling of specific classes of elastic anisotropy. In analogy to the damage mapping we introduce an internal variable that determines a back-stress tensor via a hyperelastic format and therefore enables the incorporation of plastic anisotropy. Several numerical examples underline the applicability of the proposed finite strain framework.

Fractal dimension estimates for invariant sets of non-injective maps

For a special class of non-injective maps on Riemannian manifolds an upper bound for the fractal dimension of invariant set in terms of singular values of the tangent map and degree of non-injectivity is given.

LYAPUNOV EXPONENTS FOR BILINEAR SYSTEMS: APPLICATION TO THE BUCK CONVERTER

Using the QR decomposition of the tangent map, we explicitly write the differential equations satisfied by the Lyapunov exponents of a two-dimensional bilinear system, and specialize them to the buck converter in continuous current mode. These equations are solved numerically for an arbitrary value of the bifurcation parameter, and analytically for periodic orbits, recovering for the later results already known from the computation of Floquet exponents.

A dynamical analysis of the HD 37124 planetary system

In this paper we study the stability of the HD 37124 planetary system. Using initial conditions found by Butler et al. (2002) we estimate the dynamical bounds on orbital parameters that provide stable (quasi-periodic) motions of the system. The stability analysis has been performed with the help of the MEGNO technique. This method, invented by Cincotta & Simo (2000), makes it possible to distinguish efficiently between chaotic and regular dynamics of a dynamical system. The MEGNO analysis helps us to confirm the result of Butler et al. (2002) who found that the critical factor for system stability is the eccentricity of the outer planet. This parameter is not well constrained by the current set of observations. For coplanar configurations, the limiting value of the eccentricity, providing quasi-periodic motion of the system, is roughly equal to 0.55. This value is typically slightly smaller when the system inclination increases (and companions' masses grow) but for i � 25 ◦ it can be as low as � 0.25. The system is located in a wide stable zone in the plane of eccentricities. The dynamics are insensitive to the initial phases of the planets. If the eccentricity of the outer planet is close to the current fit value � 0.4, then the system is regular over wide ranges of the relative inclination of the planets. We analyze whether a telluric planet can survive orbitally in the habitable zone of the system. This zone is covered by the investigated region of semi-major axes between 0.6 AU and 2.4 AU. The dynamics in this zone strongly depend on the eccentricity of the outer companion. For moderately low values of this parameter, ec � 0.1-0.2, the habitable zone is a dynamical analogue of the Asteroid belt in the Solar system, and the habitable zone of the 47 UMa system. Moreover, in this instance, relatively wide stable areas exist. For increasing eccentricity of the outer planet the stable zones shrink rapidly, and if the parameter is larger than 0.4 they practically disappear. We describe an alternative integration of the MEGNO indicator in the planetary N-body problem, based on the symplectic leapfrog scheme and the tangent map by Mikkola & Innanen (1999).

An atomistic-based finite deformation membrane for single layer crystalline films

A general methodology to develop hyper-elastic membrane models applicable to crystalline films one-atom thick is presented. In this method, an extension of the Born rule based on the exponential map is proposed. The exponential map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a tangent map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential map is locally approximated. The effectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed configurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by finite elements.

Energy quantization for harmonic maps

In this paper we establish the higher-dimensional energy bubbling results for harmonic maps to spheres. We have shown in particular that the energy density of concentrations has to be the sum of energies of harmonic maps from the standard 2-dimensional spheres. The result also applies to the structure of tangent maps of stationary harmonic maps at either a singularity or infinity.

Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach.

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long-range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for alpha>d whereas it decays as an inverse power law of N for alpha<d. A recent theoretical calculation based on a geometrization of the dynamics is consistent with these numerical results. Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

Abstract The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.

Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map

For the area preserving maps, the linearized tangent map determines the stability of the fixed point. When the trace of the tangent map is less than −2, the fixed point is inversion hyperbolic, thus the subsequent points of mapping alternate across the destabilized fixed point. That is to say, the fixed point undergoes periodic doubling bifurcation. While for the trace of the tangent map is larger than +2, the fixed point undergoes the hyperbolic bifurcation without reflection. Here, the processes of the hyperbolic bifurcation without reflection in the standard map have been examined in terms of the higher order symmetry in the momentum inversion. It is shown that the higher order symmetry lines approach asymptotically to the separatrix of the hyperbolic fixed point, and the existing symmetry lines cannot determine the structure of the periodic islands born after the hyperbolic bifurcation without reflection.

Upper and Lower Hausdorff Dimension Estimates for Invariant Sets ofk — 1 — Endomorphisms

For a special class of non–injective maps on Riemannian manifolds upper and lower bounds for the Hausdorff dimension of invariant sets are given in terms of the singular values of the tangent map. The upper estimation is based on a theorem by Douady and Oesterlé and its generalization to Riemannian manifolds by Noack and Reitmann, but additionally information about the noninjectivity is used. The lower estimation can be reached by modifying a method, derived by Shereshevskij for geometric constructions on the real line (also described by Barreira, for similar constructions in general metric spaces. The upper and lower dimension estimates for k — 1 — endomorphisms can for instance be applied to Julia sets of quadratic maps on the complex plane.

On the theory and computation of anisotropic damage at large strains

The goal of this contribution is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of an isotropic, fictitious and undamaged configuration a Finger—type damage metric tensor in terms of the fictitious linear tangent map is introduced. Referring to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration the free Helmholtz energy can be formulated within the standard reference configuration and treated in the spirit of standard dissipative materials. Thereby, the introduced damage potential substantially affects the anisotropic nature of the damage formulation. Finally, some numerical examples demonstrate the applicability of the proposed framework.

On the Comparison of Two Strategies to Formulate Orthotropic Hyperelasticity

The main goal of this work is to clarify the relation between two strategies to formulate constitutive equations for orthotropic materials at large strains. On the one hand, the classical approach is based on the incorporation of structural tensors into the free energy function via an enriched set of invariants. On the other hand, a fictitious isotropic configuration is introduced which renders an anisotropic, undeformed reference configuration via an appropriate linear tangent map. This formulation results in a reduced (with respect to the more general setting based on structural tensors) but nevertheless physically motivated set of invariants which are related to the invariants defined by structural tensors. As a main conceptual advantage standard isotropic constitutive equations can be applied and moreover, due to the reduced set of physically motivated invariants, the numerical treatment within a finite element setting becomes manageable.