Abstract
For nearly a Century now the concept of a manifold has played a fundamental role throughout mathematics. In the global view of modern mechanics, pioneered by H. Poincare, the phase-space of a dynamical system constitutes a differentiable manifold. Accordingly, manifolds have become essential tools wherever a global study of nonlinear phenomena is undertaken. A Ii st of such areas would be extensive. It would include, for example, bi-furcation theory and the study of stability and chaos in dynamical systems, gravitational studies and other work involving modern field theories in physics, as well as many other problems concerning nonlinear operator equations.
This work was in part supported by the National Science Foundation under Grant MCS-78-05299.
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References
E. Abraham and J. Marsden, Foundations of Mechanics, Second Edition, The Benjamin/Cummings Publ. Co., London 1978.
S. Agmon, A. Douglis and L. Nirenberg, Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions I, Comm. Pure Appl. Math. 12, 1959, 623–727.
E. Allgower and K. Georg, Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations, SIAM Review 22, 1980, 28–85.
M. Bier, O. A. Palusinski, R. A. Mosher and D. A. Saville, Electro-phoresis: Mathematical Modeling and Computer Simulation, Science, Vol. 219, 18 March 1983, No. 4590, 1281–1286.
M. A. Crisfield, A Fast Incremental/Iterative Solution Procedure that Handies “Snap Through”, Comp, and Structures 13, 1981, 55–62.
J. P. Fink and W. C. Rheinboldt, On the Discretization Error of Para-metrized Nonlinear Equations, SIAM J. Num. Anal. 20, 1983, 732–746.
J. P. Fink and W. C. Rheinboldt, Solution Manifolds and Submanifolds of Parametrized Equations and Their Discretization Error, Univ. of Pittsburgh, Inst. f. Comp. Math, and Appl., Tech. Report ICMA-83–59, 1983.
W. Fleming, Functions of Several Variables, Second Edition, Springer Verlag, New York, 1977.
C. W. Gear, Simultaneous Numerical Solution of Differential-Algebraic Equations, IEEE Trans, on Circuit Theory, CT-18, 1971, 89–95.
C. W. Gear and L. R. Petzold, ODE Methods for the Solution of Differential-Algebraic Systems, Univ. of Illinois at Urbana-Champaign, Dept. of Comp. Science, Tech. Report 82–1103.
A. C. Hindmarsh, ODE Solvers for Use with the Method of Lines, in “Adv. in Computer Methods for Partial Diff. Eqn. IV”, ed. by R. Vichnevetsky and R. S. Stepleman, IMACS, New Brunswick, NJ 1981, 312–316.
H. B. Keller, Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, in “Applications of Bifurcation Theory”, ed. by P. Rabinowitz, Academic Press, New York, NY 1977, 359–384.
H. B. Keller, Global Homotopies and Newton Methods, in “Recent Advances in Numerical Analysis”, ed. by C. deBoor, G. H. Golub, Academic Press, New York, NY 1978, 73–94.
S. Lang, Introduction to Differentiable Manifolds, Wiley and Sons, New York 1962.
H. D. Mittelmann and H. Weber, A Bibliography on Numerical Methods for Bifurcation Problems, Univ. Dortmund, Angew. Mathem., Bericht 56, 1981.
R. W. Newcomb, The Semistate Description of Nonlinear Time-Variable Circuits, IEEE Trans, on Circuits and Systems, CAS-28, 1981, 62–71.
L. R. Petzold, A Description of DASSL: A Differential-Algebraic System Solver, in “Proc. IMACS World Congress 1982”, to appear.
L. R. Petzold, Differential-Algebraic Equations are not ODE’s, SIAM J. on Scientific and Statist. Computing 3, 1982, 367–384.
T. Poston and I. Stewart, Catastrophe Theory and its Applications, Pitman Publ. Ltd., London 1978.
W. C. Rheinboldt, Solution Fields of Nonlinear Equations and Continuation Methods, SIAM J. Num. Anal. 17, 1980, 221–237.
W. C. Rheinboldt, Differential-Algebraic Systems as Differential Equations on Manifolds, Univ. of Pittsburgh, Inst. f. Comp. Math, and Appl., Tech. Report ICMA-83–55, 1983.
W. C. Rheinboldt and J. V. Burkardt, A Locally Parametrized Continuation Process, ACM Trans, on Math. Software 9, 1983, 215–235.
W. C. Rheinboldt and J. V. Burkardt, Algorithm 596: A Program for a Locally Parametrized Continuation Process, ACM Trans, on Math. Software 9, 1983, 236–241.
M. Schechter, Principles of Functional Analysis, Academic Press, NY 1971.
L. F. Shampine and R. C. Allen, Jr., Numerical Computing: An Introduction, W. B. Saunders Co., Philadelphia, PA 1973.
M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. I, Second Edition, Publish or Perish, Inc., Berkeley, CA 1979.
H. J. Wacker (editor), Continuation Methods, Academic Press, New York, NY 1978.
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Rheinboldt, W.C. (1984). On Some Methods for the Computational Analysis of Manifolds. In: Küpper, T., Mittelmann, H.D., Weber, H. (eds) Numerical Methods for Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 70. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6256-1_28
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