Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Perturbation of Systems with Nilpotent Real Part

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_395-4

Definition of the Subject

The main goal of this entry is to dwell upon the influence of the presence (explicit and/or hidden) of nontrivial real nilpotent perturbations appearing in problems in dynamical systems, partial differential equations, and mathematical physics. Under the term nilpotent perturbation, we will mean, broadly speaking, a classical linear algebra typesetting: we start with an object (vector field or map near a fixed point, first-order singular partial differential equations, system of evolution partial differential equations) whose “linear part” A is semisimple (diagonalizable) and we add a (small) nilpotent part N. The problems of interest might be summarized as follows: are the “relevant properties” (in suitable functional framework) of the initial “object” stable under the perturbation N? If instabilities occur, to classify, if possible, the novel features of the perturbed systems.

Broadly speaking, the cases when the instabilities occur are rare; they form some...

Keywords

Normal Form Jordan Block Small Divisor Convergent Power Series Normal Form Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

References

  1. Abate M (2000) Diagonalization of nondiagonalizable discrete holomorphic dynamical systems. Am J Math 122:757–781CrossRefMATHMathSciNetGoogle Scholar
  2. Arnold VI (1971) Matrices depending on parameters. Uspekhi Mat Nauk 26:101–114; Russ Math Surv 26:29–43 (in Russian)Google Scholar
  3. Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New YorkCrossRefMATHGoogle Scholar
  4. Arnold VI, Ilyashenko YU (1988) In: Anosov DV, Arnold VI (eds) Encyclopedia of Math Sci, vol 1. Dynamical systems I. Springer, New York, pp 1–155Google Scholar
  5. Bambusi D, Cicogna G, Gaeta G, Marmo G (1998) Normal forms, symmetry and linearization of dynamical systems. J Phys A 31:5065–5082ADSCrossRefMATHMathSciNetGoogle Scholar
  6. Belitskii GR (1978) Equivalence and normal forms of germs of smooth mappings. Uspekhi Mat Nauk 33:95–155, 263; Russ Math Surv 33:107–177 (in Russian)MATHMathSciNetGoogle Scholar
  7. Belitskii GR (1979) Normal forms, invariants, and local mappings. Naukova Dumka, Kiev (in Russian)MATHGoogle Scholar
  8. Bove A, Nishitani T (2003) Necessary conditions for hyperbolic systems. II. Jpn J Math (NS) 29:357–388MATHMathSciNetGoogle Scholar
  9. Bruno AD (1971) The analytic form of differential equations. Tr Mosk Mat O-va 25:119–262; (1972) 26:199–239 (in Russian); See also (1971) Trans Mosc Math Soc 25:131–288; (1972) 26:199–239Google Scholar
  10. Bruno AD, Walcher S (1994) Symmetries and convergence of normalizing transformations. J Math Anal Appl 183:571–576CrossRefMATHMathSciNetGoogle Scholar
  11. Chen KT (1965) Diffeomorphisms: C -realizations of formal properties. Am J Math 87:140–157CrossRefMATHGoogle Scholar
  12. Cicogna G, Gaeta G (1999) Symmetry and perturbation theory in nonlinear dynamics, vol 57, Lecture notes in physics. New series M: monographs. Springer, BerlinMATHGoogle Scholar
  13. Cicogna G, Walcher S (2002) Convergence of normal form transformations: the role of symmetries. (English summary) Symmetry and perturbation theory. Acta Appl Math 70:95–111CrossRefMATHMathSciNetGoogle Scholar
  14. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. McGraw-Hill, New YorkMATHGoogle Scholar
  15. Craig W (1987) Nonstrictly hyperbolic nonlinear systems. Math Ann 277:213–232CrossRefMATHMathSciNetGoogle Scholar
  16. Cushman R, Sanders JA (1990) A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part. In: Stanton D (ed) Invariant theory and tableaux. Springer, New York, pp 82–106, IMA vol Math Appl, vol 19Google Scholar
  17. DeLatte D, Gramchev T (2002) Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena. Math Phys Electron J 8(2):1–27MathSciNetGoogle Scholar
  18. Dumortier F, Roussarie R (1980) Smooth linearization of germs of R 2-actions and holomorphic vector fields. Ann Inst Fourier Grenoble 30:31–64CrossRefMATHMathSciNetGoogle Scholar
  19. Gaeta G, Walcher S (2005) Dimension increase and splitting for Poincaré-Dulac normal forms. J Nonlinear Math Phys 12(1):327–342ADSCrossRefMathSciNetGoogle Scholar
  20. Gaeta G, Walcher S (2006) Embedding and splitting ordinary differential equations in normal form. J Differ Equ 224:98–119ADSCrossRefMATHMathSciNetGoogle Scholar
  21. Gantmacher FR (1959) The theory of matrices, vols 1, 2. Chelsea, New YorkGoogle Scholar
  22. Ghedamsi M, Gourdin D, Mechab M, Takeuchi J (2002) Équations et systèmes du type de Schrödinger à racines caractéristiques de multiplicité deux. Bull Soc R Sci Liège 71:169–187MATHMathSciNetGoogle Scholar
  23. Gramchev T (2002) On the linearization of holomorphic vector fields in the Siegel Domain with linear parts having nontrivial Jordan blocks. In: Abenda S, Gaeta G, Walcher S (eds) Symmetry and perturbation theory, Cala Gonone, 16–22 May 2002. World Scientific Publication, River Edge, pp 106–115CrossRefGoogle Scholar
  24. Gramchev T, Orrú N (2011) Cauchy problem for a class of nondiagonalizable hyperbolic systems. Dynamical systems, differential equations and applications. 8th AIMS Conference. Discrete Contin Dyn Syst I(Suppl):533–542Google Scholar
  25. Gramchev T, Ruzhansky M (2013) Cauchy problem for some 2 × 2 hyperolic systems of pseudo-differential equations with nondiagonalisable principal part. In: Cicognani M, Colombini F, Del Santo D (eds) Studies in space phase analysis with applications to PDEs, progress in nonlinear differential equations and their applications, vol 84, Chapter 7., pp 129–146CrossRefGoogle Scholar
  26. Gramchev T, Tolis E (2006) Solvability of systems of singular partial differential equations in function spaces. Integr Transform Spec Funct 17:231–237CrossRefMATHMathSciNetGoogle Scholar
  27. Gramchev T, Walcher S (2005) Normal forms of maps: formal and algebraic aspects. Acta Appl Math 85:123–146CrossRefMathSciNetGoogle Scholar
  28. Gramchev T, Yoshino M (2007) Normal forms for commuting vector fields near a common fixed point. In: Gaeta G, Vitolo R, Walcher S (eds) Symmetry and perturbation theory, Oltranto, 2–9 June 2007. World Scientific Publication, River Edge, pp 203–217Google Scholar
  29. Hasselblatt B, Katok A (2003) A first course in dynamics: with a panorama of recent developments. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  30. Herman M (1987) Recent results and some open questions on Siegel’s linearization theorem of germs of complex analytic diffeomorphisms of C n near a fixed point. VIIIth international congress on mathematical physics, Marseille 1986. World Scientific Publication, Singapore, pp 138–184Google Scholar
  31. Hibino M (1999) Divergence property of formal solutions for first order linear partial differential equations. Publ Res Inst Math Sci 35:893–919CrossRefMATHMathSciNetGoogle Scholar
  32. Hibino M (2003) Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients. J Math Sci Univ Tokyo 10:279–309MATHMathSciNetGoogle Scholar
  33. Hibino M (2006) Formal Gevrey theory for singular first order quasi-linear partial differential equations. Publ Res Inst Math Sci 42:933–985CrossRefMATHMathSciNetGoogle Scholar
  34. Il’yashenko Y (1979) Divergence of series reducing an analytic differential equation to linear form at a singular point. Funct Anal Appl 13:227–229MathSciNetGoogle Scholar
  35. Iooss G, Lombardi E (2005) Polynomial normal forms with exponentially small remainder for vector fields. J Differ Equ 212:1–61ADSCrossRefMATHMathSciNetGoogle Scholar
  36. Kajitani K (1979) Cauchy problem for non-strictly hyperbolic systems. Publ Res Inst Math 15:519–550CrossRefMATHMathSciNetGoogle Scholar
  37. Katok A, Katok S (1995) Higher cohomology for Abelian groups of toral automorphisms. Ergod Theory Dyn Syst 15:569–592CrossRefMATHMathSciNetGoogle Scholar
  38. Murdock J (2002) On the structure of nilpotent normal form modules. J Differ Equ 180:198–237ADSCrossRefMATHMathSciNetGoogle Scholar
  39. Murdock J, Sanders JA (2007) A new transvectant algorithm for nilpotent normal forms. J Differ Equ 238:234–256ADSCrossRefMATHMathSciNetGoogle Scholar
  40. Pérez Marco R (2001) Total convergence or small divergence in small divisors. Commun Math Phys 223:451–464ADSCrossRefMATHGoogle Scholar
  41. Petkov VM (1979) Microlocal forms for hyperbolic systems. Math Nachr 93:117–131CrossRefMATHMathSciNetGoogle Scholar
  42. Raissy J (2012) Holomorphic linearization of commuting germs of holomorphic maps. J Geom Anal. doi:10.1007/s12220-012-9316-2, Online firstGoogle Scholar
  43. Ramis J-P (1984) Théorèmes d’indices Gevrey pour les équations différentielles ordinaires. Mem Am Math Soc 48:296MathSciNetGoogle Scholar
  44. Rodino L (1993) Linear partial differential operators in Gevrey spaces. World Science, SingaporeCrossRefMATHGoogle Scholar
  45. Sanders JA (2005) Normal form in filtered Lie algebra representations. Acta Appl Math 87:165–189CrossRefMATHMathSciNetGoogle Scholar
  46. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, vol 59, 2nd edn, Applied mathematical sciences. Springer, New YorkMATHGoogle Scholar
  47. Siegel CL (1942) Iteration of analytic functions. Ann Math 43:607–614CrossRefMATHGoogle Scholar
  48. Sternberg S (1958) The structure of local homeomorphisms. II, III. Am J Math 80:623–632; 81:578–604CrossRefMATHMathSciNetGoogle Scholar
  49. Stolovitch L (2000) Singular complete integrability. Publ Math IHES 91:134–210CrossRefGoogle Scholar
  50. Stolovitch L (2005) Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers. Ann Math 161:589–612CrossRefMATHMathSciNetGoogle Scholar
  51. Taylor M (1981) Pseudodifferential operators, vol 34, Princeton mathematical series. Princeton University Press, PrincetonMATHGoogle Scholar
  52. Thiffeault J-L, Morison PJ (2000) Classification and Casimir invariants of Lie-Poisson brackets. Phys D 136:205–244CrossRefMATHMathSciNetGoogle Scholar
  53. Vaillant J (1999) Invariants des systèmes d’opérateurs différentiels et sommes formelles asymptotiques. Jpn J Math (NS) 25:1–153MATHMathSciNetGoogle Scholar
  54. Yamahara H (2000) Cauchy problem for hyperbolic systems in Gevrey class. A note on Gevrey indices. Ann Fac Sci Toulouse Math 19:147–160CrossRefMathSciNetGoogle Scholar
  55. Yoccoz J-C (1995) A remark on Siegel’s theorem for nondiagonalizable linear part. Manuscript, 1978; See also Théorème de Siegel, nombres de Bruno e polynômes quadratic. Astérisque 231:3–88Google Scholar
  56. Yoshino M (1999) Simultaneous normal forms of commuting maps and vector fields. In: Degasperis A, Gaeta G (eds) Symmetry and perturbation theory SPT 98, Rome, 16–22 December 1998. World Scientific, Singapore, pp 287–294Google Scholar
  57. Yoshino M, Gramchev T (2008) Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks. Ann Inst Fourier (Grenoble) 58:263–297CrossRefMATHMathSciNetGoogle Scholar
  58. Zung NT (2002) Convergence versus integrability in Poincaré-Dulac normal form. Math Res Lett 9:217–228CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di CagliariCagliariItaly