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## Annales Polonici Mathematici

2000 | 73 | 1 | 37-57
Tytuł artykułu

### Topological conjugacy of cascades generated by gradient flows on the two-dimensional sphere

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This article presents a theorem about the topological conjugacy of a gradient dynamical system with a constant time step and the cascade generated by its Euler method. It is shown that on the two-dimensional sphere S² the gradient dynamical flow is, under some natural assumptions, correctly reproduced by the Euler method for a sufficiently small time step. This means that the time-map of the induced dynamical system is globally topologically conjugate to the discrete dynamical system obtained via the Euler method.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
37-57
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-02-18
poprawiono
1999-12-08
Twórcy
autor
• Institute of Computer Science, Jagiellonian University, Nawojki 11, 30-072 Kraków, Poland
Bibliografia
• [AD] F. Alouges and A. Debussche, On the qualitative behaviour of the orbits of a parabolic partial differential equation and its discretization in the neighbourhood of a hyperbolic fixed point, Numer. Funct. Anal. Optim. 12 (1991), 253-269.
• [Bey1] W. J. Beyn, On invariant closed curves for one-step methods, Numer. Math. 51 (1987), 103-122.
• [Bey2] W. J. Beyn, On the numerical approximation of phase portraits near stationary points, SIAM J. Numer. Anal. 24 (1987), 1095-1113.
• [BL] W. J. Beyn and J. Lorenz, Center manifolds of dynamical systems under discretizations, Numer. Funct. Anal. Optim. 9 (1987), 381-414.
• [Bie] A. Bielecki, Gradient dynamical systems and learning process of layer artificial neural networks, PhD thesis, Faculty of Mathematics and Physics, Jagiellonian Univ., 1998 (in Polish).
• [Fec1] M. Fečkan, Asymptotic behaviour of stable manifolds, Proc. Amer. Math. Soc. 111 (1991), 585-593.
• [Fec2] M. Fečkan, Discretization in the method of averaging, ibid. 113 (1991), 1105-1113.
• [Fec3] M. Fečkan, The relation between a flow and its discretization, Math. Slovaca 42 (1992), 123-127.
• [Gar1] B. Garay, Discretization and some qualitative properties of ordinary differential equations about equilibria, Acta Math. Univ. Comenian. 62 (1993), 245-275.
• [Gar2] B. Garay, Discretization and Morse-Smale dynamical systems on planar discs, ibid. 63 (1994), 25-38.
• [Gar3] B. Garay, On structural stability of ordinary differential equations with respect to discretization methods, J. Numer. Math. 4 (1996), 449-479.
• [Gar4] B. Garay, On $C^j$-closeness between the solution flow and its numerical approximation, J. Differential Equations Appl. 2 (1996), 67-86.
• [Gar5] B. Garay, The discretized flow on domains of attraction: a structural stability result, IMA J. Numer. Anal. 18 (1998), 77-90.
• [KL] P. E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and their one-step discretizations, SIAM J. Numer. Anal. 23 (1986), 986-996.
• [Kru] A. Krupowicz, Numerical Methods for Boundary Value Problems of Ordinary Differential Equations, PWN, Warszawa, 1986 (in Polish).
• [Li] M. C. Li, Structural stability of Morse-Smale gradient-like flows under discretization, SIAM J. Math. Anal. 28 (1997), 381-388.
• [Man] R. Ma né, A proof of $C^1$ stability conjecture, Publ. Math. IHES 66 (1988).
• [MR] M. Mrozek and K. P. Rybakowski, Discretized ordinary differential equations and the Conley index, J. Dynam. Differential Equations 4 (1992), 57-63.
• [PM] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer, New York, 1982.
• [Rob] J. Robbin, A structural stability theorem, Ann. of Math. 94 (1971), 447-493.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv73z1p37bwm
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