Numerical integration of the Davidenko equation
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  predictor — corrector method and is a modification of Davidenko's  equation. In order to numerically trace c(s), this modified equation may be integrated by some standard IVP - code .
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 . 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 . Instead of using a local perturbation  in the sense of Jürgens - Peitgen - Saupe , a technique more closely related to Sard's theorem  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.
A homotopy method for solving a difficult fixed point test problem .
A bifurcation problem for highly symmetric periodic solutions of a differential delay equation [18,28].
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 . The numerical calculations were performed on a Hewlett Packard 85 and are illustrated by the standard plots which have a rather coarse grid.
KeywordsBifurcation Point Functional Differential Equation Differential Delay Equation Continuation Method Solution Curve
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- ABRAHAM,R. and ROBBIN,J.: Transversal mappings and flows. W.A.Benjamin (1967).Google Scholar
- ALEXANDER,J., KELLOGG,R.B., LI,T.Y. and YORKE,J.A.: Piecewise smooth continuation. Preprint, University of Maryland (1979).Google Scholar
- ANGELSTORF,N.: Global branching and multiplicity results for periodic solutions of functional differential equations. In: Functional Differential Equations and Approximation of Fixed Points, H.O.Peitgen, H.O.Walther (eds), Springer Lecture Notes in Math. 730 (1979) 32–45.Google Scholar
- BEN-ISRAEL,A. and GREVILLE,T.N.E.: Generalized inverses: theory and applications. Wiley-Interscience publ. (1974).Google Scholar
- BRANIN,JR.,F.H. and HOO,S.K.: A method for finding multiple extrema of a function of n variables. Numerical Methods for Nonlinear Optimization, F.Lootsma, ed., Academic Press (1972) 231–237.Google Scholar
- BROYDEN,C.G.: Quasi-Newton, or modification methods. Numerical Solution of Systems of Nonlinear Equations, G.Byrne and C.Hall (eds), Academic Press (1973) 241–280.Google Scholar
- GEORG,K.: On tracing an implicitly defined curve by Quasi — Newton steps and calculating bifurcation by local perturbation. To appear in: SIAM Journal of Scientific and Statistical Computing.Google Scholar
- GEORG,K.: Zur numerischen Lösung nichtlinearer Gleichungssysteme mit simplizialen und kontinuierlichen Methoden. Unfinished manuscript.Google Scholar
- JÜRGENS,H., PEITGEN,H.-O. and SAUPE,D.: Topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems. in: Proceedings Symposium on Analysis and Computation of Fixed Points, S.M.Robinson (ed.), Academic Press, 1980, 139–181.Google Scholar
- JÜRGENS, H. and SAUPE, D.: Methoden der simplizialen Topologie zur numerischen Behandlung von nichtlinearen Eigenwert-und Verzweigungsproblemen. Diplomarbeit, Bremen (1979).Google Scholar
- KEARFOTT,R.B.: A derivative-free arc continuation method and a bifurcation technique. Preprint.Google Scholar
- KELLER,H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. Applications of Bifurcation Theory, P.H.Rabinowitz (ed.), Academic Press (1977) 359–384.Google Scholar
- KELLER,H.B.: Global homotopies and Newton methods. Numerical Analysis, Academic Press (1978) 73–94.Google Scholar
- KRASNOSEL'SKII,M.A.: Topological methods in the theory of nonlinear integral equations. Pergamon Press (1964).Google Scholar
- MILNOR,J.W.: Topology from the differentiable viewpoint. University Press of Virginia (1969).Google Scholar
- PEITGEN,H.-O. and PRÜFER,M.: The Leray — Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems. In: Proceedings Functional Differential Equations and Approximation of Fixed Points, H.O.Peitgen and H.O.Walther (eds), Springer Lecture Notes in Math. 730 (1979) 326–409.Google Scholar
- POTTHOFF,M.: Diplom thesis, in preparation.Google Scholar
- PRÜFER,M.: Calculating global bifurcation. In: Continuation Methods, H.J.Wacker (ed.) Academic Press (1978) 187–213.Google Scholar
- PRÜFER,M.: Simpliziale Topologie und globale Verzweigung. Dissertation, Bonn (1978).Google Scholar
- RHEINBOLDT,W.C.: Methods for solving systems of nonlinear equations. Regional conference series in applied mathematics 14, SIAM (1974).Google Scholar
- SCHMIDT,C.: Approximating differential equations that describe homotopy paths. Preprint No 7931, Univ. of Santa Clara (1979).Google Scholar
- SCHWETLICK,H.: Numerische Lösung nichtlinearer Gleichungen. VEB Deutscher Verlag der Wissenschaften (1979).Google Scholar
- SHAMPINE,L.F. and GORDON,M.K.: Computer solution of ordinary differential equations: The initial value problem. ordinary differential equations: The initial value problem. W.H.Freeman and Company (1975).Google Scholar
- TANABE,K.: A geometric method in nonlinear programming. Preprint STAN-CS-77-643, Stanford University (1977).Google Scholar
- WACKER,H.J.: A summary of the developments on imbedding methods. Continuation Methods, H.J.Wacker (ed), Academic Press (1978) 1–35.Google Scholar