On the local classification of smooth maps induced by Newton's method

Journal of Differential Equations

Volumn 217, Issue 2, 2005, Pages 377-392

  Riaza, Ricardo ^a  

^a UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 26844517730     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2005.06.009     Document Type: Article

References (41)

1. H. Amann Ordinary Differential Equations 1990 Walter de Gruyter Berlin
2. V.I. Arnol'd Singularities of smooth mappings in: V.I. Arnol'd (Eds) Singularity Theory. Selected Papers 1981 Cambridge University Press Cambridge 3-45
3. V.I. Arnol'd Geometrical Methods in the Theory of Ordinary Differential Equations 1988 Springer Berlin
4. V.I. Arnol'd Dynamical Systems VI: Singularity Theory I 1993 Springer Berlin
5. V.I. Arnol'd S.M. Gusein-Zade A.N. Varchenko Singularities of Differentiable Maps vol. I 1985 Birkhauser Basel
6. R.E. Beardmore R. Laister The flow of a DAE near a singular equilibrium SIAM J. Matrix Anal. Appl. 24 2002 106-120
7. R.E. Beardmore R. Laister A. Peplow Trajectories of a DAE near a pseudo-equilibrium Nonlinearity 17 2004 253-279
8. K.E. Brenan S.L. Campbell L.R. Petzold Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations 1996 SIAM Philadelphia, PA
9. J.W. Bruce P.J. Giblin Curves and Singularities second ed. 1992 Cambridge University Press Cambridge
10. L.O. Chua H. Oka Normal forms for constrained nonlinear differential equations, Part I theory IEEE Trans. Circuits Systems 35 1988 881-901
11. M. Golubitsky V. Guillemin Stable Mappings and Their Singularities 1973 Springer Berlin
12. M. Golubitsky D.G. Schaeffer Singularities and Groups in Bifurcation Theory I 1985 Springer Berlin
13. A.O. Griewank On solving nonlinear equations with simple singularities or nearly singular solutions SIAM Rev. 27 1985 537-572
14. E. Hairer G. Wanner Solving Ordinary Differential Equations II Stiff and Differential-Algebraic Problems 1996 Springer Berlin
15. P. Hartman Ordinary Differential Equations second ed. 2002 SIAM Philadelphia, PA
16. M.W. Hirsch Differential Topology 1976 Springer Berlin
17. V. Janovsky V. Seige Qualitative analysis of Newton's flow SIAM J. Numer. Anal. 33 1996 2068-2097
18. H.B. Keller Geometrically isolated nonisolated solutions and their approximation SIAM J. Numer. Anal. 18 1981 822-838
19. J. Llibre J. Sotomayor Structural stability of constrained polynomial systems Bull. London Math. Soc. 30 1998 589-595
20. J. Llibre, J. Sotomayor, M. Zhitomirskii, Impasse bifurcations of constrained systems, in: A. Galves, et al. (Eds.), Differential Equations and Dynamical Systems, Fields Institute of Communication, vol. 31, 2002, pp. 235-255.
21. J. Martinet Singularities of Smooth Functions and Maps Lecture Note Series vol. 58 1982 London Mathematical Society, Cambridge University Press Cambridge
22. J.N. Mather Stability of C ∞ mappings. III: Finitely determined map germs Publ. Math. Inst. Hautes Etud. Sci. 35 1968 127-156
23. M. Medved Normal forms of implicit and observed implicit differential equations Riv. Mat. Pura Appl. 10 1991 95-107
24. J. Milnor Morse Theory 1963 Princeton University Press Princeton, NJ
25. M. Morse Relations between the critical points of a real function of n independent variables Trans. AMS 27 1925 345-396
26. P.J. Olver Equivalence, Invariants, and Symmetry 1995 Cambridge University Press Cambridge
27. L. Perko Differential Equations and Dynamical Systems 2001 Springer Berlin
28. P.J. Rabier Implicit differential equations near a singular point J. Math. Anal. Appl. 144 1989 425-449
29. P.J. Rabier W.C. Rheinboldt A geometric treatment of implicit differential-algebraic equations J. Differential Equations 109 1994 110-146
30. P.J. Rabier W.C. Rheinboldt On the computation of impasse points of quasi-linear differential-algebraic equations Math. Comput. 62 1994 133-154
31. P.J. Rabier W.C. Rheinboldt Theoretical and numerical analysis of differential-algebraic equations Handbook of Numerical Analysis vol. VIII 2002 North-Holland/Elsevier Amsterdam 183-540
32. G. Reiszig H. Boche On singularities of autonomous implicit ordinary differential equations IEEE Trans. Circuits Systems, Part I 50 2003 922-931
33. R. Riaza Preliminary normal forms for a class of singular equilibria in implicit ODEs in: R. Agarwal D. O'Regan (Eds) Nonlinear Analysis and Applications 2003 Kluwer Dordrecht 831-850
34. R. Riaza Attraction domains of degenerate singular equilibria in quasi-linear ODEs SIAM J. Math. Anal. 36 2004 678-690
35. R. Riaza P.J. Zufiria Stability of singular equilibria in quasilinear implicit differential equations J. Differential Equations 171 2001 24-53
36. R. Riaza P.J. Zufiria Discretization of implicit ODEs for singular root-finding problems J. Comput. Appl. Math. 140 2002 695-712
37. J. Sotomayor Structurally stable differential systems of the form A(x) x′ = F(x) Differential Equations Dynamic Systems 5 1997 415-422
38. J. Sotomayor M. Zhitomirskii Impasse singularities of differential systems of the form A(x)x′ = F(x) J. Differential Equations 169 2001 567-587
39. V. Venkatasubramanian, A taxonomy of the dynamics of large differential algebraic systems such as the power system, Ph.D. Thesis, Washington University, 1992.
40. H. Whitney On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane Ann. of Math. 62 1955 374-410
41. M. Zhitomirskii Local normal forms for constrained systems on 2-manifolds Bol. Soc. Bras. Mat. 24 1993 211-232