Warwick Tucker

Uppsala University | UU · Department of Mathematics

Many scientific computing applications demand massive numerical computations on parallel architectures such as Graphics Processing Units (GPUs). Usually, either floating-point single or double precision arithmetic is used. Higher precision is generally not available in hardware, and software extended precision libraries are much slower and rarely s...

We describe a modern approach to parameter estimation, based on set-valued computations combined with a branch and bound step. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. In addition, we show that the method can be accelerated by set-valued constraint propagation, which...

We consider the family of destabilized Kuramoto–Sivashinsky equations in one spatial dimension for α, ν ≥ 0 and . For certain parameter values, shock-like stationary solutions have been numerically observed. In this work we verify the existence of several such solutions using the framework of self-consistent bounds and validated numerics.

By performing a systematic study of the Hénon map, we find low-period sinks for parameter values extremely close to the classical ones. This raises the question whether or not the well-known Hénon attractor-the attractor of the Hénon map existing for the classical parameter values-is a strange attractor, or simply a stable periodic orbit. Using res...

An extensive search for stable periodic orbits (sinks) for the Hénon map in a small neighborhood of the classical parameter values is carried out. Several parameter values which generate a sink are found and verified by rigorous numerical computations. Each found parameter value is extended to a larger region of existence using a simplex continuati...

Today, GPUs represent an important hardware development platform for many problems in dynamical systems, where massive parallel computations are needed. Beside that, many numerical studies of chaotic dynamical systems require a computing precision higher than common oating point (FP) formats. One such application is locating invariant sets for chao...

The question of coexisting attractors for the Hénon map is studied numerically by performing an exhaustive search in the parameter space. As a result, several parameter values for which more than two attractors coexist are found. Using tools from interval analysis, we show rigorously that the attractors exist. In the case of periodic orbits, we ver...

A regular paving is a finite succession of bisections that partition a root box x in ℝd into sub-boxes using a binary tree-based data structure. The sequence of splits that generate such a partition is given by the sub-boxes associated with the nodes of the tree. The leaf boxes, i.e., the sub-boxes associated with the leaf nodes, form a partition o...

The problem of existence of stable periodic orbits (sinks) for the Hénon map in a neighborhood of classical parameter values is studied numerically. Several parameter values which sustain a sink are found. It is shown rigorously that the sinks exist. Regions of existence in the parameter space of the sinks are located using the continuation method.

We develop techniques for the verification of the Chebyshev property of Abelian integrals. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [F. Dumort...

In this paper, we analyze the stability properties of a system of ordinary differential equations describing the thermodynamic limit of a microscopic and stochastic model for file sharing in a peer-to-peer network introduced by Kesidis et al. We show, under certain assumptions, that this BitTorrent-like system has a unique locally attracting equili...

A regular paving is a finite succession of bisections that partition a root box x in ℝd into sub-boxes using a binary tree-based data structure. We extend regular pavings to mapped regular pavings which map sub-boxes in a regular paving of x to elements in some set Y. Arithmetic operations defined on Y can be extended point-wise over x and carried...

We propose a new method for optimal experimental design of population pharmacometric experiments based on global search methods using interval analysis; all variables and parameters are represented as intervals rather than real numbers. The evaluation of a specific design is based on multiple simulations and parameter estimations. The method requir...

This textbook provides a comprehensive introduction to the theory and practice of validated numerics, an emerging new field that combines the strengths of scientific computing and pure mathematics. In numerous fields ranging from pharmaceutics and engineering to weather prediction and robotics, fast and precise computations are essential. Based on...

We establish the existence and local uniqueness of traveling wave solutions to the one–dimensional Euler equations with artificial viscosity. The equations are expressed as a fixed–point problem, which is solved by a computer–assisted method based on Yamamoto's application of the Banach fixed–point theorem.

In this paper, we present a computer–assisted method that establishes the existence and local uniqueness of a stationary solution to the viscous Burgers' equation. The problem formulation involves a left boundary condition and one integral boundary condition, which is a variation of a previous approach.

The limit cycle bifurcations of a Z2 equivariant planar Hamiltonian vector field of degree 7 under Z2 equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cy...

The limit cycle bifurcation of a Z2 equivariant quintic planar Hamiltonian vector field under Z2 equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quint...

We establish a lower bound on the measure of the set of stable parameters a for the quadratic map Qa(x)=ax(1−x). For these parameters, we prove that Qa either has a single stable periodic orbit or a period-doubling bifurcation. From this result, we also obtain a non-trivial upper bound on the set of stochastic parameters for Qa.

We consider a hyper-elliptic Hamiltonian of degree five, chosen from a generic set of parameters, and study what configurations of limit cycles can bifurcate from the corresponding differential system under quartic perturbations. Perturbations of Lienard type are considered separately. Several different configurations with seven (four) limit cycles...

It is shown that the problem of existence of periodic orbits can be studied rigorously by means of a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate initial positions of periodic points and interval operators are used to prove the existence of periodic orbits in a neighborhood of the computer...

In this chapter, we will give a brief introduction to some aspects of chaos theory. This task is by no means easy: despite more than four decades of intense research in this area, there is still no general agreement as to what the word chaos should really mean. In fact, there appears to exist almost a continuum of definitions of a chaotic system, a...

Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to...

We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighborhood of the saddle point. The normal form is suitable for computations aimed at enclosing the fl...

The existence of short periodic orbits for the Lorenz system is studied rigorously. We describe a method for finding all short cycles embedded in the chaotic attractor. We use the method of close returns to find initial points for the Newton operator, combined with interval tools for proving the existence of periodic orbits in a neighborhood of a p...

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian...

We present a method that–given a data set, a finitely parametrized system of ordinary differential equations (ODEs), and a search space of parameters–discards portions of the search space that are inconsistent with the model ODE and data. The method is completely rigorous as it is based on validated integration of the vector field. As a consequence...

The existence of short periodic orbits for the Lorenz system is studied rigorously. We describe a method for finding all short cycles embedded in a chaotic singular attractor (i.e. an attractor containing an equilibrium). The method uses an interval operator for proving the existence of periodic orbits in regions where it can be evaluated, and boun...

We present a method to find all zeros of an analytic function in a rectangular domain. The approach is based on finding guaranteed enclosures rather than approx-imations of the zeros. Well–isolated simple zeros are determined fast and with high accuracy. Clusters of zeros can in many cases be distinguished from multiple zeros by applying the argume...

We present a fast hybrid method designed to enclose all zeros of an analytic function on a triangulated domain. The method consists of three parts: first the zeros are isolated (up to some resolution) using a combination of winding number computations and bisections; in the sec-ond step we approximate the location of each zero using a floating poin...

As modern molecular biology moves towards the analysis of biological systems as opposed to their individual components, the need for appropriate mathematical and computational techniques for understanding the dynamics and structure of such systems is becoming more pressing. For example, the modeling of biochemical systems using ordinary differentia...

Biochemical systems are commonly modelled by systems of ordinary differential equations (ODEs). A particular class of such models called S-systems have recently gained popularity in biochemical system modelling. The parameters of an S-system are usually estimated from time-course profiles. However, finding these estimates is a difficult computation...

In this paper we prove that the L2 spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in R2 is within 10−2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radiu...

In recent years, the modeling and simulation of biochemical networks has attracted increasing attention. Such networks are commonly modeled by systems of ordinary differential equations, a special class of which are known as S-systems. These systems are specifically designed to mimic kinetic reactions, and are sufficiently general to model genetic...

Recently, there has been growing interest in the modelling and simulation of biological systems. Such systems are often modelled in terms of coupled ordinary differential equations that involve parameters whose (often unknown) values correspond to certain fundamental properties of the system. For example, in metabolic modelling, concentrations of m...

The aim of this paper is to give a very brief introduction to the emerg-ing area of validated numerics. This is a rapidly growing field of research faced with the challenge of interfacing computer science and pure mathematics. Most validated numerics is based on interval analysis, which allows its users to account for both rounding and discretizati...

The aim of this paper is to introduce a technique for describing trajectories of systems of ordinary differential equations (ODEs) passing near saddle-fixed points. In contrast to classical linearization techniques, the methods of this paper allow for perturbations of the underlying vector fields. This robustness is vital when modelling systems con...

The goal of this paper is to produce a series of counterexamples for the L p spectral radius conjecture, 1<p<∞, for double-layer potential operators associated to a distinguished class of elliptic systems in polygonal domains in ℝ 2 . More specifically the class under discussion is that of second-order elliptic systems in two dimensions whose coeff...

We present a numerical method particularly suited for computing Poincaré maps for systems of ordinary differential equations. The method is a generalization of a stopping procedure described by Hénon [Physica D 5 (1982) 412], and it applies to a wide family of systems.

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. Thi...

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic. We illustrate the presented method by computing solution sets for two explicit systems.

We investigate a one-parameter family of interval maps arising in the study of the geometric Lorenz flow for non-classical parameter values. Our conclusion is that for all parameters in a set of positive Lebesgue measure, the map has a positive Lyapunov exponent. Furthermore, this set of parameters has a density point which plays an important dynam...

Starting from the classical Baker's map, we define a family of "Tired Baker's" maps. It is proved that for this family, uniform expansion is not a sufficient condition for topological transitivity. As an application, we show that some of the one-dimensional Poincar'e maps of the geometric Lorenz model are not good models of the real flow. We conclu...

We prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. The proof is based on a combination of normal form theory and rigorous numerical computations....

Inspired by numerical solutions of the Lorenz equations, we model the Poincar'e map of the flow by a one-parameter map of the unit interval. For a certain region in the parameter space of the Lorenz equations, we show that the corresponding one-dimensional map is chaotic, imposing only minimal conditions on its derivative. Perturbing the map, we ge...

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. Thi...