Algorithms for computing normally hyperbolic invariant manifolds

An effcient algorithm is developed for the numerical computation of normally hyperbolic invariant manifolds, based on the graph transform and Newton's method. It fits in the perturbation theory of discrete dynamical systems and therefore allows application to the setting of continuation. A convergence proof is included. The scope of application is not restricted to hyperbolic attractors, but extends to normally hyperbolic manifolds of saddle type. It also computes stable and unstable manifolds. The method is robust and needs only little specification of the dynamics, which makes it applicable to e.g. Poincaré maps. Its performance is illustrated on examples in 2D and 3D, where a numerical discussion is included.

他臓器癌の現病歴あるいは既往歴が大腸腫瘍の危険因子となるか？大腸内視鏡検査を行った患者を対象にした解析結果

We present an algorithm for computing one-dimensional stable and unstable manifolds of saddle periodic orbits in a, Poincaré section. The computation is set up as a, boundary value problem by restricting both end points of orbit segments to the section. Starting from the periodic orbit itself, we use collocation routines from AUTO to continue the solutions of the boundary value problem such that one end point of the orbit segment varies along a part of the manifold that was already computed. In this way, the other end point of the orbit segment traces out a new piece of the manifold. As opposed to standard methods that use shooting to compute the Poincaré map as the kth return map, our approach defines the Poincaré map as the solution of a boundary value problem. This enables us to compute global manifolds through points where the flow is tangent to the section - a situation that is typically encountered unless one is dealing with a periodically forced system. Another major advantage of our approach is that it deals effectively with the problem of extreme sensitivity of the Poincaré map to its argument, which is a typical feature in the important class of slow-fast systems. We illustrate and test our algorithm by computing stable and unstable manifolds for three examples: the forced Van der Pol oscillator, a model of a semiconductor laser with optical injection, and a slow-fast chemical oscillator. All examples are accompanied by animations demonstrating how the manifolds grow during the computation. © 2005 Society for Industrial and Applied Mathematics

De overgang van primair naar voortgezet onderwijs in internationaal perspectief:Een systematische overzichtsstudie van onderwijstransities in relatie tot kenmerken van verschillende Europese onderwijsstelsels.

NRO-OPRO project 405-15-72

The Lorenz manifold as a collection of geodesic level sets

We demonstrate a method to compute a two-dimensional global stable or unstable manifold of a vector field as a sequence of approximate geodesic level sets. Specifically, we compute the Lorenz manifold-the two-dimensional stable manifold of the origin of the well-known Lorenz equations-which has emerged as a test example for manifold computations. The information given by the geodesic level sets can be used to visualize and understand the geometry of the Lorenz manifold, and one such visualization can be seen as the cover illustration

Interactions between a locally separating stable manifold and a bursting periodic orbit

Multi-spike bursting of the membrane potential is understood to be a key mechanism for cell signalling in neurons. During the active phase of a burst, the voltage potential across the cell membrane exhibits a series of spikes. This is followed by a silent (recovery) phase during which there is relatively little change in the potential. Mathematical models of this behaviour are frequently based on Hodgkin–Huxley formalism; the dynamics of the voltage is expressed in terms of ionic currents that lead to a system of ordinary differential equations in which some variables (voltage, in particular) are fast and others are slow. The bursting patterns observed in such slow-fast models are often explained in terms of transitions between different coexisting attracting states associated with the so-called fast subsystem, for which the slow variables are viewed as parameters. In particular, the threshold that determines when the voltage starts to burst is identified with the basin boundary between two attractors associated with the active and silent phases. In reality, however, the bursting threshold is a more complicated object. Numerical methods recently developed by the authors approximate the bursting threshold as a locally separating stable manifold of the full slow-fast system. Here, we use these numerical techniques to investigate how a bursting periodic orbit interacts with this stable manifold. We focus on a Morris–Lecar model, which is three dimensional with one slow and two fast variables, as a representative example. We show how the locally separating stable manifold organises the number of spikes in a bursting periodic orbit, and illustrate its role in a spike-adding transition as a parameter is varied

SUPPLY RESPONSE IN THE NORTHEASTERN FRESH TOMATO MARKET: COINTEGRATION AND ERROR CORRECTION ANALYSIS

This paper reexamines supply response in the Northeastern fresh tomato market during the 1949-94 period by employing cointegration and error correction technique. It tests whether there has been a long-run equilibrium relationship between Northeastern production and a set of price and nonprice factors that influence it. Findings suggest that wage rate, imports from competing regions, and urban pressure have had significant negative impacts on regional production. The negative relationship between price and production may have resulted from the strong negative effects exerted by the nonprice factors