Notion Of Hyperbolicity Research Articles

Exponential splittings of products of matrices and accurately computing singular values of long products

Accurately computing the singular values of long products of matrices is important for estimating Lyapunov exponents: λ i= lim n→∞(1/n) logσ i(A n⋯A 1) . Algorithms for computing singular values of products, in fact, compute the singular values of a perturbed product (A n+E n)⋯(A 1+E 1) . The question is how small are the relative errors of the singular values of the product with respect to these factorwise perturbations. In general, the relative errors in the singular values can be quite large. However, if the product has an exponential splitting, then the error in the singular values is O(n 2 max iκ 2(A i)∥E i∥ F) , uniformly in n. The exponential splitting property is not directly comparable with the notion of hyperbolicity in dynamical systems, but is similar in philosophy.

STABILITY VERSUS HYPERBOLICITY IN DYNAMICAL AND ITERATED FUNCTION SYSTEMS

In this paper we investigate a certain notion of stability, for one function or for iterated function systems, and discuss why this notion can be a good extension and complement to the notion of hyperbolicity. This last notion is very well-known in the literature and plays an important role in the investigation of the dynamical behavior of a system. The main result is that although some classical sets of functions like the stable Lipschitz functions are conjugate to hyperbolic functions there exist continuous stable functions which are not conjugate to hyperbolic functions. A sufficient condition for not being conjugate to a hyperbolic function is given.

Linearization of vector fields on the orbit space of the action of a compact Lie group

In this paper we present results which relate the linearization of a G-equivariant vector field at an equilibrium to the linearization of the vector field obtained by projection on the orbit space of the group action. This allows us to propose a notion of hyperbolicity for an equilibrium of the projected vector field which is compatible with the orbit space structure. It provides an alternative to the notion of normal hyperbolicity for relative equilibria (flow invariant group orbits) developed by Field [9, 10].

A class of fully nonlinear 2×2 systems of partial differential equations

This paper is a study of certain fully nonlinear 2x2 systems of partial differential equations in one space variable and time. The nonlinearity contains a term proportional to {vert_bar}{partial_derivative}U/{partial_derivative}x{vert_bar} where U - U(x,t) {element_of} {Re}{sup 2} is the unknown function and {vert_bar}.{vert_bar} is the Euclidean norm on {Re}{sup 2}; i.e., a term homogeneous of degree 1 in {partial_derivative}U/{partial_derivative}x and singular at the origin. Such equations are motivated by hypoplasticity. The paper introduces a notion of hyperbolicity for such equations and, in the hyperbolic case, proves existence of solutions for two initial value problems admitting similarity solutions: the Riemann problem and the scale-invariant problem. Uniqueness is addressed in a companion paper. 10 refs., 10 figs.

Hyperbolic Convolution Operators

Hyperbolic differential operators with constant coefficients introduced and studied systematically by Gårding (4), were characterized by the existence of the fundamental solution with some cone condition, according to Hörmander (6). Recently Ehrenpreis, extending the notion of hyperbolicity due to Gårding, has defined hyperbolic operators for distributions with compact support in the convolution sense. Under the hypothesis that the operator is invertible as a distribution, he has established a theorem analogous to the theorem of Hörmander mentioned above (3).