Rua Murray

Computing invariant measures for expanding circle maps

Let f be a sufficiently expanding circle map. We prove that a certain Markov approximation scheme based on a partition of into equal intervals produces a probability measure whose total variation norm distance from the exact absolutely continuous invariant measure is bounded by ; C is a constant depending only on the map f.

Information on Biotic Interactions Improves Transferability of Distribution Models

Predicting changes in species’ distributions is a crucial problem in ecology, with leading methods relying on information about species’ putative climatic requirements. Empirical support for this approach relies on our ability to use observations of a species’ distribution in one region to predict its range in other regions (model transferability). On the basis of this observation, ecologists have hypothesized that climate is the strongest determinant of species’ distributions at large spatial scales. However, it is difficult to reconcile this claim with the pervasive effects of biotic interactions. Here, we resolve this apparent paradox by demonstrating how biotic interactions can affect species’ range margins yet still be compatible with model transferability. We also identify situations where small changes in species’ interactions dramatically shift range margins.

Dynamical Probing of the Mechanisms Underlying Calcium Oscillations

AbstractOscillations in the concentration of free intracellular calcium ([Ca2+]) play an important role in many cell types. Thus, understanding the mechanisms underlying Ca2+ oscillations is of significant scientific import. There are two basic classes of mechanism that cause these oscillations: (1) positive and negative feedback from calcium to the inositol trisphosphate (IP3) receptor, and (2) positive and negative feedback from calcium to IP3 metabolism. These two classes can be distinguished experimentally by their different responses to pulses of IP3. In general most cells will have both types of mechanism present simultaneously. We show that, when Ca2+ oscillations are driven by these two mechanisms at the same time, one mechanism is dominant. As the strength of each mechanism is varied, the response of the cell exhibits a threshold phenomenon, being governed either by one mechanism or the other, with no ambiguity in the response to a pulse of IP3. We interpret these results, and other responses to IP3 pulses, in terms of a fast-slow time scale analysis of the calcium dynamics, where calcium transport across the cell membrane occurs on a slow time scale.

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate

Optimal partition choice for invariant measure approximation for one-dimensional maps

L^1

Ulam's Method for Lasota--Yorke Maps with Holes

convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulams method and provide a formal explanation for the numerical behaviour of Ulams method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.

Maximum Entropy Estimates for Risk-Neutral Probability Measures with Non-Strictly-Convex Data

The problem of finding absolutely continuous invariant measures (ACIMs) for a dynamical system can be formulated as a fixed point problem for a Markov operator (the Perron–Frobenius operator). This is an infinite-dimensional problem. Ulams method replaces the Perron–Frobenius operator by a sequence of finite rank approximations whose fixed points are relatively easy to compute numerically. This paper concerns the optimal choice of Ulam approximations for one-dimensional maps; an adaptive partition selection is used to tailor the approximations to the structure of the invariant measure. The main idea is to select a partition which equally distributes the square root of the derivative of the invariant density amongst the bins of the partition. The results are illustrated for the logistic map where the ACIMs may have inverse square root singularities in their density functions. O(log n/n) convergence rates can be expected, whereas a non-adaptive algorithm yields O(n−1/2) at best. Studying the convergence of the adaptive algorithm allows an estimate to be made of the measure of the Jakobson parameter set (those logistic maps which admit an ACIM).

Duality and the Computation of Approximate Invariant Densities for Nonsingular Transformations

Ulams method is a rigorous numerical scheme for approximating invariant densities of dynamical systems. The phase space is partitioned into a grid of connected sets, and a set-to-set transition matrix is computed from the dynamics; an approximate invariant density is read off as the leading left eigenvector of this matrix. When a hole in phase space is introduced, one instead searches for conditional invariant densities and their associated escape rates. For Lasota-Yorke maps with holes we prove that a simple adaptation of the standard Ulam scheme provides convergent sequences of escape rates (from the leading eigenvalue), conditional invariant densities (from the corresponding left eigenvector), and quasi-conformal measures (from the corresponding right eigenvector). We also immediately obtain a convergent sequence for the invariant measure supported on the survivor set. Our approach allows us to consider relatively large holes. We illustrate the approach with several families of examples, including a class of Lorenz-like maps.

Dynamical conditions for convergence of a maximum entropy method for Frobenius-Perron operator equations

This article investigates use of the Principle of Maximum Entropy for approximation of the risk-neutral probability density on the price of a financial asset as inferred from market prices on associated options. The usual strict convexity assumption on the market-price to strike-price function is relaxed, provided one is willing to accept a partially supported risk-neutral density. This provides a natural and useful extension of the standard theory. We present a rigorous analysis of the related optimization problem via convex duality and constraint qualification on both bounded and unbounded price domains. The relevance of this work for applications is in explaining precisely the consequences of any gap between convexity and strict convexity in the price function. The computational feasibility of the method and analytic consequences arising from non-strictly-convex price functions are illustrated with a numerical example.

The dimensional reduction method for identification of parameters that trade-off due to similar model roles

We investigate a class of optimization problems which arise in the approximation of invariant densities for a nonsingular, measurable transformation