Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Structural stability on basins
for numerical methods

Author: Ming-Chia Li
Journal: Proc. Amer. Math. Soc. 127 (1999), 289-295
MSC (1991): Primary 58F10, 58F12, 65L20, 34D30, 34D45
MathSciNet review: 1469420
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that a flow $\varphi $ with a hyperbolic compact attracting set is structurally stable on the basin of attraction with respect to numerical methods. The result is a generalized version of earlier results by Garay, Li, Pugh, and Shub. The proof relies heavily on the usual invariant manifold theory elaborated by Hirsch, Pugh, and Shub (1977), and by Robinson (1976).

References [Enhancements On Off] (What's this?)

  • 1. K. E. Atkinson, An Introduction to Numerical Analysis, second ed., John Wiley & Sons, New York, 1989. MR 90m:65001
  • 2. W. J. Beyn, On the numerical approximation of phase portraits near stationary points, SIAM J. Numer. Anal. 24 (1987), no. 5, 1095-1113. MR 88j:65143
  • 3. M. Braun and J. Hershenov, Periodic solutions of finite difference equations, Quarterly of Applied Math. 35 (1977), 139-147. MR 58:23212
  • 4. T. Eirola, Invariant curves of one-step methods, BIT (1988), no. 28, 113-122. MR 89g:65109
  • 5. M. Fe\v{c}kan, The relation between a flow and its discretization, Math. Slovaca 42 (1992), 123-127. MR 93f:34071
  • 6. B. M. Garay, Discretization and some qualitative properties of ordinary differential equations about equilibria, Acta Math. Univ. Commeniznae 62 (1993), 249-275. MR 95i:34079
  • 7. -, Discretization and normal hyperbolicity, Z. Angew. Math. Mech. 74 (1994), T662-T663.
  • 8. -, The discretized flow on domains of attraction: a structural stability result, (1997), IMA J. Numer. Anal., to appear.
  • 9. -, On structural stability of ordinary differential equations with respect to discretization methods, Numer. Math. 72 (1996), 449-479. MR 97d:34007
  • 10. M. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Math, vol. 583, Springer-Verlag, New York, 1977. MR 58:18595
  • 11. A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal. 31 (1994), 1452-1485. MR 95m:65116
  • 12. P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and their one-step discretizations, SIAM J. Numer. Anal. 23 (1986), 986-995. MR 87k:34074
  • 13. M.-C. Li, Structural stability of Morse-Smale gradient-like flows under discretizations, SIAM J. Mathematical Analysis 28 (1997), 381-388. CMP 97:08
  • 14. C. Pugh and M. Shub, $C^r$ stability of periodic solutions and solution schemes, Appl. Math. Letters 1 (1988), 281-285. MR 89m:58117
  • 15. C. Robinson, The geometry of the structural stability proof using unstable disks, Bol. Soc. Brasil. Math. 6 (1975), 129-144. MR 58:13190
  • 16. -, Structural stability of $C^1$ flows, Lectures Notes in Math., vol. 468, Springer-Verlag, Berlin, 1975, pp. 262-277. MR 58:31251
  • 17. -, Structural stability of $C^1$ diffeomorphisms, J. Differential Equations 22 (1976), 28-73. MR 57:14051
  • 18. -, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1995. MR 97e:58064
  • 19. M. Shub, Some remarks on dynamical systems and numerical analysis, Dynamical Systems and Partial Differential Equations, Proceeding of the VII Elam, Equinoccio, Universidad Simon Bolivar, Caracas, 1986, pp. 69-91. MR 88j:58065

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58F10, 58F12, 65L20, 34D30, 34D45

Retrieve articles in all journals with MSC (1991): 58F10, 58F12, 65L20, 34D30, 34D45

Additional Information

Ming-Chia Li
Affiliation: Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan

Keywords: Structural stability, dynamical systems, hyperbolic attracting set, basin of attraction, numerical method, Euler's method
Received by editor(s): January 28, 1997
Received by editor(s) in revised form: May 6, 1997
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society