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Numerical integration of the Davidenko equation

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Numerical Solution of Nonlinear Equations

Part of the book series: Lecture Notes in Mathematics ((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.

Partially supported by the Deutsche Forschungsgemeinschaft through SFB 72, Bonn

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Wegelerstraße 6, D-5300, Bonn

    K. Georg

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Eugene L. Allgower Klaus Glashoff Heinz-Otto Peitgen

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© 1981 Springer-Verlag

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Georg, K. (1981). Numerical integration of the Davidenko equation. In: Allgower, E.L., Glashoff, K., Peitgen, HO. (eds) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol 878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090680

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  • DOI: https://doi.org/10.1007/BFb0090680

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  • Print ISBN: 978-3-540-10871-9

  • Online ISBN: 978-3-540-38781-7

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