A. R. Humphries

Runge-Kutta methods for dissipative and gradient dynamical systems

The numerical approximation of dissipative initial value problems by fixed time-stepping Runge–Kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared. A general class of ordinary differential equations, for which dissipativity is induced through an inner product, is studied throughout. This class arises naturally in many finite dimensional applications (such as the Lorenz equations) and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics. It is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step-size h > 0. The numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor A_h exists; upper-semicontinuity of A_h at h = 0 is established, which shows that, for h small, every point on the numerical attractor is close to a point on the true global attractor A. Under the additional assumption that the problem is globally Lipschitz, it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of A_h at h = 0 is again established. For gradient systems with globally Lipschitz vector fields it is shown that any Runge–Kutta method preserves the gradient structure for h sufficiently small. For general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small. Convergence of the numerical attractor is studied and, for a dissipative gradient system with hyperbolic equilibria, lower semicontinuity at h = 0 is established. Thus, for such a system, A_h converges to A in the Hausdorff metric as h → 0.

Model problems in numerical stability theory for initial value problems

In the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics. Whilst this has resulted in a coherent and self-contained body of knowledge, it has not thoroughly addressed the problems of real interest in applications. Recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics. This on-going work is unified and possible directions for future work are outlined. In particular, striking similarities between this new developing stability theory and the classical non-linear stability theory are emphasised. The classical theories of

Computation of Mixed Type Functional Differential Boundary Value Problems

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The essential stability of local error control for dynamical systems

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Neutrophil dynamics during concurrent chemotherapy and G-CSF administration: Mathematical modelling guides dose optimisation to minimise neutropenia.

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Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

, and algebraic stability for Runge-Kutta methods are briefly reviewed, and it is emphasised that the classes of equations to which these theories apply - linear decay and contractive problems - only admit trivial dynamics. Four other categories of equations - gradient, dissipative, conservative and Hamiltonian systems - are considered. Relationships and differences between the possible dynamics in each category, which range from multiple competing equilibria to fully chaotic solutions, are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications. Runge-Kutta schemes which preserve the dynamical structure of the underlying problem are sought, and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given. Algebraic stability, in particular, is seen to play a central role. The effects of error control are considered, and multi-step methods are discussed briefly. Finally, various open problems are described.

Synchronization in ecological systems by weak dispersal coupling with time delay

We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the model- ing of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such prob- lems including discrete Nagumo equations, an Ising model, and Frenkel-Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals. 1. Introduction. Nonlinear spatially discrete diffusion equations occur as models in many areas of science and engineering. When the underlying mathematical models contain differ- ence terms or delays as well as derivative terms, the resulting differential-difference equations present challenging analytical and computational problems. We demonstrate how functional differential boundary value problems with advances and delays arise from such models and describe a general approach for the numerical computation of solutions. Solutions are approx- imated for several such problems, and the numerical issues arising in their computation are

Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

Although most adaptive software for initial value problems is designed with an accuracy requirement—control of the local error—it is frequently observed that stability is imparted by the adaptation. This relationship between local error control and numerical stability is given a firm theoretical underpinning.The dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations: dissipative, contractive, and gradient systems. Dissipative dynamical systems are characterised by having a bounded absorbing set

Deterministic and random dynamical systems: theory and numerics

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A Mathematical Model of Granulopoiesis Incorporating the Negative Feedback Dynamics and Kinetics of G-CSF/Neutrophil Binding and Internalization

which all trajectories eventually enter and remain inside. The exponentially contractive problems studied have a unique, globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set