Structural stability on basins for numerical methods

Li, Ming-Chia

doi:10.1090/S0002-9939-99-04591-8

Structural stability on basins for numerical methods
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Proc. Amer. Math. Soc. 127 (1999), 289-295 Request permission

Abstract:

In this paper, we show that a flow $\Phi$ 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

Kendall E. Atkinson, An introduction to numerical analysis, 2nd ed., John Wiley & Sons, Inc., New York, 1989. MR 1007135
W.-J. Beyn, On the numerical approximation of phase portraits near stationary points, SIAM J. Numer. Anal. 24 (1987), no. 5, 1095–1113. MR 909067, DOI 10.1137/0724072
M. Braun and J. Hershenov, Periodic solutions of finite difference equations, Quart. Appl. Math. 35 (1977/78), no. 1, 139–147. MR 510330, DOI 10.1090/S0033-569X-1977-0510330-3
Timo Eirola, Invariant curves of one-step methods, BIT 28 (1988), no. 1, 113–122. MR 928439, DOI 10.1007/BF01934699
Michal Fe kan, The relation between a flow and its discretization, Math. Slovaca 42 (1992), no. 1, 123–127. MR 1159497
B. M. Garay, Discretization and some qualitative properties of ordinary differential equations about equilibria, Acta Math. Univ. Comenian. (N.S.) 62 (1993), no. 2, 249–275. MR 1270512
—, Discretization and normal hyperbolicity, Z. Angew. Math. Mech. 74 (1994), T662–T663.
—, The discretized flow on domains of attraction: a structural stability result, (1997), IMA J. Numer. Anal., to appear.
Barnabas M. Garay, On structural stability of ordinary differential equations with respect to discretization methods, Numer. Math. 72 (1996), no. 4, 449–479. MR 1376108, DOI 10.1007/s002110050177
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal. 31 (1994), no. 5, 1452–1485. MR 1293524, DOI 10.1137/0731075
P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Anal. 23 (1986), no. 5, 986–995. MR 859010, DOI 10.1137/0723066
M.-C. Li, Structural stability of Morse-Smale gradient-like flows under discretizations, SIAM J. Mathematical Analysis 28 (1997), 381–388.
Charles Pugh and Michael Shub, $C^r$ stability of periodic solutions and solution schemes, Appl. Math. Lett. 1 (1988), no. 3, 281–285. MR 963700, DOI 10.1016/0893-9659(88)90093-6
Clark Robinson, The geometry of the structural stability proof using unstable disks, Bol. Soc. Brasil. Mat. 6 (1975), no. 2, 129–144. MR 494284, DOI 10.1007/BF02584780
R. Clark Robinson, Structural stability of $C^{1}$ flows, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 262–277. MR 0650640
Clark Robinson, Structural stability of $C^{1}$ diffeomorphisms, J. Differential Equations 22 (1976), no. 1, 28–73. MR 474411, DOI 10.1016/0022-0396(76)90004-8
Clark Robinson, Dynamical systems, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. Stability, symbolic dynamics, and chaos. MR 1396532
Michael Shub, Some remarks on dynamical systems and numerical analysis, Dynamical systems and partial differential equations (Caracas, 1984) Univ. Simon Bolivar, Caracas, 1986, pp. 69–91. MR 882014