74 research outputs found
Numerical computation and continuation of invariant manifolds connecting fixed points
No full textFriedman, Mark J.; Doedel, Eusebius J.. (1989). Numerical computation and continuation of invariant manifolds connecting fixed points. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/5110
Computational methods for global analysis of homoclinic and hetero-clinic orbits: a case study
No full textFriedman, Mark J.; Doedel, Eusebius J.. (1991). Computational methods for global analysis of homoclinic and hetero-clinic orbits: a case study. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/1619
Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations
No full textBeyn W-J, Doedel E. Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations. SIAM Journal on Scientific and Statistical Computing. 1981;2(1):107-120.A large class of consistent and unconditionally stable discretizations of nonlinear boundary value problems is defined. The number of solutions to the discretizations is compared to the number of solutions of the continuous problem. We state conditions under which these numbers must agree for all sufficiently small mesh sizes. Various examples, including bifurcation problems, illustrate our theoretical results
AUTO94P: An Experimental Parallel Version of AUTO
No full textA detailed description is given of the parallel algorithms used in AUTO94P, an experimental parallel version of the software AUTO for the numerical bifurcation analysis of systems of ordinary differential equations. Timing results and user instructions for the Intel Delta are included. The sequential version of the software, AUTO94, is fully described in [8]. For a related tutorial paper see [5, 6]. 1 Introduction In this report we give a detailed presentation of the parallel algorithms used in AUTO94P, an experimental version of the software package AUTO for the bifurcation analysis of systems of ordinary differential equations. The latest version of the standard sequential software, AUTO94, is described in [8], where user instructions and many illustrative examples are given. A description of the numerical algorithms used in AUTO, as well as related algorithms, can be found in [5, 6]. To obtain a copy of AUTO94 or for information on AUTO94P send email to [email protected]. We..
The Construction of Finite Difference Approximations to Ordinary Differential Equations
No full textFinite difference approximations of the form Σ^(si)_(i=-rj)d_(j,i)u_(j+i)=Σ^(mj)_(i=1) e_(j,if)(z_(j,i)) for the numerical solution of linear nth order ordinary differential equations are analyzed. The order of these approximations is shown to be at least r_j + s_j + m_j - n, and higher for certain special choices of the points Z_(j,i). Similar approximations to initial or boundary conditions are also considered and the stability of the resulting schemes is investigated
Nonlinear Numerics
No full textThe objectives and some basic methods of numerical bifurcation analysis are described. Several computational examples are used to illustrate the power as well as the limitations of these techniques. Future directions of algorithmic and software development are also discussed. Contents 1 Introduction 4 2 Continuation 4 2.1 Regular solution points : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Simple singular points : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Linear algebra : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3 Numerical Bifurcation Analysis of ODEs 8 3.1 Boundary value problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3.2 Periodic solutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.3 Connecting orbits : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.4 Discretization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :..
'Ex J. Nieremberg'
No full textList of plants extracted from Johannes Eusebius Nieremberg's (1595–1658) work: Nieremberg, J. E. 1635. "Historia naturae, maxime peregrinae, libris XVI."Antwerp. Numbers refer to the page numbers in Nieremberg's work.
Author and date unknown
Numerical continuation methods for dynamical systems: path following and boundary value problems
No full textPath following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel''s 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits
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