Invariant curves for numerical methods
Author:
H. T. Doan
Journal:
Quart. Appl. Math. 43 (1985), 385-393
MSC:
Primary 65L05; Secondary 58F99
DOI:
https://doi.org/10.1090/qam/814235
MathSciNet review:
814235
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The problem of finding periodic orbits of dynamical systems numerically is considered. It is shown that if a convergent, strongly stable, multi-step method is employed then under some suitable conditions, there exist invariant curves. The result also shows that the rates of convergence toward the invariant curves are roughly the same for different methods and different step sizes.
- M. Braun and J. Hershenov, Periodic solutions of finite difference equations, Quart. Appl. Math. 35 (1977/78), no. 1, 139–147. MR 510330, DOI https://doi.org/10.1090/S0033-569X-1977-0510330-3
- Hai Thanh Doan, INVARIANT CURVES FOR NUMERICAL METHODS AND THE HOPF BIFURCATION, ProQuest LLC, Ann Arbor, MI, 1982. Thesis (Ph.D.)–Michigan State University. MR 2632586
- K. Georg, Numerical integration of the Davidenko equation, Numerical solution of nonlinear equations (Bremen, 1980) Lecture Notes in Math., vol. 878, Springer, Berlin-New York, 1981, pp. 128–161. MR 644329
- K. P. Hadeler, Effective computation of periodic orbits and bifurcation diagrams in delay equations, Numer. Math. 34 (1980), no. 4, 457–467. MR 577410, DOI https://doi.org/10.1007/BF01403681
- Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR 0419901
- Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
O. E. Lansford, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens, Nonlinear Problems in the Physical Sciences and Biology, Springer Lecture Notes, vol. 322, 1972
- Ira Bruce Schwartz, Estimating regions of existence of unstable periodic orbits using computer-based techniques, SIAM J. Numer. Anal. 20 (1983), no. 1, 106–120. MR 687371, DOI https://doi.org/10.1137/0720008
M. Braun and J. Hershenov, Periodic solutions of finite difference equations, Quart. Appl. Math. 35, 139–147 (1977)
H. T. Doan, Invariant curves for numerical methods and the Hopf bifurcation, Ph.D. dissertation, Michigan State University, 1982
K. Georg, Numerical integration of the equation, Numerical Solutions of Nonlinear Equations, Springer Lecture Notes, vol. 878, 1981
K. P. Hadeler, Computation of periodic orbits and bifurcation diagrams, Numer. Math. 34, 439–455 (1980)
J. K. Hale, Ordinary differential equations, Wiley-Interscience, New York, 1969
P. Henrici, Discrete variable methods in ordinary differential equations, John Wiley and Sons, New York, 1962
O. E. Lansford, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens, Nonlinear Problems in the Physical Sciences and Biology, Springer Lecture Notes, vol. 322, 1972
I. B. Schwartz, Existence of unstable periodic orbits, SIAM J. Numer. Anal. 20, 107–120 (1983)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
65L05,
58F99
Retrieve articles in all journals
with MSC:
65L05,
58F99
Additional Information
Article copyright:
© Copyright 1985
American Mathematical Society
