Kenneth J. Palmer

Exponential dichotomies and transversal homoclinic points

Let F be a diffeomorphism on a two-dimensional manifold. Smale [24] shows that if F has a transversal homoclinic point there is a Cantor-like set near it on which some iterate of F is invariant and isomorphic to the Bernoulli shift on a finite number of symbols. Examples arise as period maps of periodic systems of differential equations. Melnikov [ 171, Chow, Hale and Mallet-Paret [5] and Holmes [13, 141 have considered periodic perturbations of two-dimensional autonomous systems of the form

A generalization of Hartman's linearization theorem☆

If x is in Rn we denote its norm by 1 x / and if A is an n x n matrix we denote its operator norm by 1 A 1 . (Although we are concerned only with Rn here, our theorem is true with only minor changes when Rn is replaced by an arbitrary Banach space.) Suppose that A(t) is a matrix function defined and continuous for all t on the real line R. Then we say that the linear differential equation, x’ = A(t) x, (1)

Smooth convergence in the binomial model

In this article, we consider a general class of binomial models with an additional parameter λ. We show that in the case of a European call option the binomial price converges to the Black–Scholes price at the rate 1/n and, more importantly, give a formula for the coefficient of 1/n in the expansion of the error. This enables us, by making special choices for λ, to prove that convergence is smooth in Tian’s flexible binomial model and also in a new center binomial model which we propose.

Exponential dichotomy, integral separation and diagonalizability of linear systems of ordinary differential equations

Abstract This paper is concerned with linear systems of ordinary differential equations. A criterion for integral separation in terms of exponential dichotomy is given. As corollaries we obtain the roughness theorem for integral separation and the new result that an upper triangular system on a half-line is integrally separated if and only if the system corresponding to its diagonal is. We then show that a diagonal system on a half-line is integrally separated if and only if a certain perturbed system is diagonalizable. Using this result we are able to deduce that the interior of the set of diagonalizable systems on a half-line is the set of systems with integral separation and that the interior for the whole line is the set of systems which are integrally separated on both half-lines.

A shadowing lemma with applications to semilinear parabolic equations

The property of hyperbolic sets that is embodied in the Shadowing Lemma is of great importance in the theory of dynamical systems. In this paper a new proof of the lemma is presented, which applies not only to the usual case of a diffeomorphism in finite-dimensional space but also to a sequence of possibly noninvertible maps in a Banach space. The approach is via Newton’s method, the main step being the verification that a certain linear operator is invertible. At the end of the paper an application to parabolic evolution equations is given.

On the numerical computation of orbits of dynamical systems: the higher dimensional case

A finite time version of the shadowing theorem is used to develop a procedure to determine the accuracy of numerically computed orbits of one-dimensional maps. The procedure works forward. After any given number of iterates, we can decide whether our theorem applies and, if it does, we can estimate how far the computed orbit is from a true orbit.

Chaos in almost periodic systems

AbstractLet

A Shadowing Theorem for ordinary differential equations

\dot x = f(t,x)