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Numerical integration of the Davidenko equation

  • K. Georg
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 878)

Abstract

Given a solution curve c(s) in the kernel H−1(0) of a smooth map H : IRN+1 → IRN, we consider a differential equation such that c(s) is an asymptotically stable solution. The equation may be viewed as a continuous version of Haselgrove's [17] predictor — corrector method and is a modification of Davidenko's [12] equation. In order to numerically trace c(s), this modified equation may be integrated by some standard IVP - code [40].

A curve — tracing algorithm is then discussed which makes one predictor step along the kernel of the Jacobian DH and one subsequent corrector (Newton) step perpendicular to this kernel. Instead of using the exact Jacobian, we update an approximate Jacobian in the sense of Broyden [9]. The algorithm differs somewhat from the recently described methods [15,20] in that we emphasize on "safe" curve — following. A simple and robust step — size control is given which may be improved in particular for less "nasty" problems.

Finally, it is discussed how such derivative — free curve — tracing methods may be used to deal with bifurcation points caused by an index jump in the sense of Crandall — Rabinowitz [11]. Instead of using a local perturbation [15] in the sense of Jürgens - Peitgen - Saupe [18], a technique more closely related to Sard's theorem [37] is proposed. This had the advantage that sparseness of DH is not destroyed near a bifurcation point, and hence the given method may be applied to large eigenvalue problems arising from discretizations of differential equations.

The following numerical examples are discussed:
  1. 1.

    A homotopy method for solving a difficult fixed point test problem [49].

     
  2. 2.

    A bifurcation problem for highly symmetric periodic solutions of a differential delay equation [18,28].

     
  3. 3.

    A secondary bifurcation problem for periodic solutions of a differential delay equation, where a highly symmetric solution bifurcates into a solution with less symmetries [18,28].

     

Some ideas are only roughly sketched and will be appropriately discussed elsewhere [16]. The numerical calculations were performed on a Hewlett Packard 85 and are illustrated by the standard plots which have a rather coarse grid.

Keywords

Bifurcation Point Functional Differential Equation Differential Delay Equation Continuation Method Solution Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • K. Georg
    • 1
  1. 1.Institut für Angewandte MathematikUniversität BonnBonn

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