Skip to main content
SpringerLink
Log in

Perturbation of Systems with Nilpotent Real Part

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

  • 253 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

Complex and Real Jordan Canonical Forms

Nilpotent Perturbation and Formal Normal Forms of Vector Fields and Maps Near a Fixed Point

Loss of Gevrey Regularity in Siegel Domains in the Presence of Jordan Blocks

First-Order Singular Partial Differential Equations

Normal Forms for Real Commuting Vector Fields with Linear Parts Admitting Nontrivial Jordan Blocks

Analytic Maps near a Fixed Point in the Presence of Jordan Blocks

Weakly Hyperbolic Systems and Nilpotent Perturbations

Bibliography

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

Perturbation :

Typically, one starts with an “initial” system S 0, which is usually simple and/or well understood. We perturb the system by adding a (small) perturbation R so that the new object becomes \({S_0+R}\). In our context the typical examples for S 0 will be systems of linear ordinary differential equations with constant coefficients in \({{\mathbb{R}}^n}\) or the associated linear vector fields.

Nilpotent linear transformation:

Let \({A\colon {\mathbb{K}}^n \mapsto {\mathbb{K}}^n}\) be a linear map, where \({{\mathbb{K}}={\mathbb{R}}}\) or \({{\mathbb{K}}={\mathbb{C}}}\). We call A nilpotent if there exists a positive integer r such that the rth iteration A r become the zero map, in short \({A^r= 0}\).

Gevrey spaces :

Let Ω be an open domain in \({{\mathbb{R}}^n}\) and let \({\sigma \geq 1}\). The Gevrey space \({G^\sigma (\Omega)}\) stands for the set of all functions \({f\in C^\infty(\Omega)}\) such that for every compact subset \({K\subset\subset \Omega}\) one can find \({C=C_{K,f} > 0}\) such that

$$ \sup_{x\in K} |\partial^\alpha_x f(x)|\leq C^{|\alpha|+1} \alpha!^\sigma $$
(1)

for all \( \alpha =(\alpha_1, \ldots, \alpha_n)\in {\mathbb{Z}}_+^n \), \( \alpha! = \alpha_1! \ldots \alpha_n! \), \( |\alpha|:=\alpha_1+\ldots + \alpha_n \). If \( \sigma =1 \) we recapture the space of real analytic functions in Ω while the scale \( G^\sigma (\Omega) \), \( \sigma > 1 \), serves as an intermediate space between the real analytic functions and the set of all \( C^\infty \) functions in Ω. By the Stirling formula one may replace \( \alpha!^\sigma \) by \({|\alpha|!^\sigma}\), \( |\alpha|^{\sigma |\alpha|}\) or \( \Gamma (\sigma |\alpha|) \), where \( \Gamma(z) \) stands for the Euler Gamma function cf. the book of Rodino [41] for more details on the Gevrey spaces.

One associates also Gevrey index to formal power series, namely, given a (formal) power series

$$ f(x)= \sum_\alpha f_\alpha x^\alpha $$

this is in the formal Gevrey space \({G^\sigma_f({\mathbb{\mathbb{K}}}^n)}\) if there exist \({C > 0}\) and \({R > 0}\) such that

$$ |f_\alpha| \leq C^{|\alpha|+1} |\alpha|!^{\sigma-1}$$
(2)

for all \({\alpha \in {\mathbb Z}^n_+}\).

In fact, one can find in the literature another definition of the formal Gevrey spaces \({G^\tau_f}\) of index τ, namely replacing \({\sigma-1}\) by τ (see e. g. Ramis [40]).

Bibliography

  1. Abate M (2000) Diagonalization of nondiagonalizable discrete holomorphic dynamical systems. Amer J Math 122:757–781

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold VI (1971) Matrices depending on parameters. Uspekhi Mat Nauk 26:101–114 (in Russian); Russ Math Surv 26:29–43

    Google Scholar 

  3. Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New York

    Book  MATH  Google 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 1–155

    Google Scholar 

  5. Bambusi D, Cicogna G, Gaeta G, Marmo G (1998) Normal forms, symmetry and linearization of dynamical systems. J Phys A 31:5065–5082

    Article  MathSciNet  MATH  Google Scholar 

  6. Belitskii GR (1978) Equivalence and normal forms of germs of smooth mappings. Uspekhi Mat Nauk 33:95–155, 263 (in Russian); Russ Math Surv 33:107–177

    MathSciNet  Google Scholar 

  7. Belitskii GR (1979) Normal forms, invariants, and local mappings. Naukova Dumka, Kiev (in Russian)

    Google Scholar 

  8. Bove A, Nishitani T (2003) Necessary conditions for hyperbolic systems. II. Jpn J Math (NS) 29:357–388

    MathSciNet  MATH  Google 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) 26:199–239

    Google Scholar 

  10. Bruno AD, Walcher S (1994) Symmetries and convergence of normalizing transformations. J Math Anal Appl 183:571–576

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen KT (1965) Diffeomorphisms: \({C^\infty}\)-realizations of formal properties. Amer J Math 87:140–157

    Article  MathSciNet  MATH  Google Scholar 

  12. Cicogna G, Gaeta G (1999) Symmetry and perturbation theory in nonlinear dynamics. Lecture Notes in Physics. New Series m: Monographs, vol 57. Springer, Berlin

    Google 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 70:95–111

    Article  MathSciNet  MATH  Google Scholar 

  14. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. McGraw‐Hill, New York

    MATH  Google Scholar 

  15. Craig W (1987) Nonstrictly hyperbolic nonlinear systems. Math Ann 277:213–232

    Article  MathSciNet  MATH  Google 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. IMA Vol Math Appl, vol 19. Springer, New York, pp 82–106

    Google 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):1–27

    MathSciNet  Google Scholar 

  18. Dumortier F, Roussarie R (1980) Smooth linearization of germs of R 2‑actions and holomorphic vector fields. Ann Inst Fourier Grenoble 30:31–64

    Article  MathSciNet  MATH  Google Scholar 

  19. Gaeta G, Walcher S (2005) Dimension increase and splitting for Poincaré‐Dulac normal forms. J Nonlinear Math Phys 12(1):327–342

    Article  MathSciNet  Google Scholar 

  20. Gaeta G, Walcher S (2006) Embedding and splitting ordinary differential equations in normal form. J Differ Equ 224:98–119

    Article  MathSciNet  MATH  Google Scholar 

  21. Gantmacher FR (1959) The theory of matrices, vols 1, 2. Chelsea, New York

    Google 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 Roy Sci Liège 71:169–187

    MathSciNet  MATH  Google 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 Sci. Publ., River Edge, pp 106–115

    Google Scholar 

  24. Gramchev T, Tolis E (2006) Solvability of systems of singular partial differential equations in function spaces. Integral Transform Spec Funct 17:231–237

    Article  MathSciNet  MATH  Google Scholar 

  25. Gramchev T, Walcher S (2005) Normal forms of maps: formal and algebraic aspects. Acta Appl Math 85:123–146

    Article  MathSciNet  Google Scholar 

  26. 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 Sci Publ, River Edge, pp 203–217

    Google Scholar 

  27. Hasselblatt B, Katok A (2003) A first course in dynamics: with a panorama of recent developments. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  28. 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 Sci Publ, Singapore, pp 138–184

    Google Scholar 

  29. Hibino M (1999) Divergence property of formal solutions for first order linear partial differential equations. Publ Res Inst Math Sci 35:893–919

    Article  MathSciNet  MATH  Google Scholar 

  30. Hibino M (2003) Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients. J Math Sci Univ Tokyo 10:279–309

    MathSciNet  MATH  Google Scholar 

  31. Hibino M (2006) Formal Gevrey theory for singular first order quasi–linear partial differential equations. Publ Res Inst Math Sci 42:933–985

    Article  MathSciNet  MATH  Google Scholar 

  32. Il'yashenko Y (1979) Divergence of series reducing an analytic differential equation to linear form at a singular point. Funct Anal Appl 13:227–229

    MathSciNet  Google Scholar 

  33. Iooss G, Lombardi E (2005) Polynomial normal forms with exponentially small remainder for vector fields. J Differ Equ 212:1–61

    Article  MathSciNet  MATH  Google Scholar 

  34. Kajitani K (1979) Cauchy problem for non–strictly hyperbolic systems. Publ Res Inst Math 15:519–550

    Article  MathSciNet  MATH  Google Scholar 

  35. Katok A, Katok S (1995) Higher cohomology for Abelian groups of toral automorphisms. Ergod Theory Dyn Syst 15:569–592

    Article  MathSciNet  MATH  Google Scholar 

  36. Murdock J (2002) On the structure of nilpotent normal form modules. J Differ Equ 180:198–237

    Article  MathSciNet  MATH  Google Scholar 

  37. Murdock J, Sanders JA (2007) A new transvectant algorithm for nilpotent normal forms. J Differ Equ 238:234–256

    Article  MathSciNet  MATH  Google Scholar 

  38. Petkov VM (1979) Microlocal forms for hyperbolic systems. Math Nachr 93:117–131

    Article  MathSciNet  MATH  Google Scholar 

  39. Pérez Marco R (2001) Total convergence or small divergence in small divisors. Commun Math Phys 223:451–464

    Google Scholar 

  40. Ramis J-P (1984) Théorèmes d'indices Gevrey pour les équations différentielles ordinaires. Mem Amer Math Soc 48:296

    MathSciNet  Google Scholar 

  41. Rodino L (1993) Linear partial differential operators in Gevrey spaces. World Science, Singapore

    Book  MATH  Google Scholar 

  42. Sanders JA (2005) Normal form in filtered Lie algebra representations. Acta Appl Math 87:165–189

    Article  MathSciNet  MATH  Google Scholar 

  43. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, 2nd edn. Applied Mathematical Sciences, vol 59. Springer, New York

    Google Scholar 

  44. Siegel CL (1942) Iteration of analytic functions. Ann Math 43:607–614

    Article  MATH  Google Scholar 

  45. Sternberg S (1958) The structure of local homeomorphisms. II, III. Amer J Math 80:623–632, 81:578–604

    Google Scholar 

  46. Stolovitch L (2000) Singular complete integrability. Publ Math IHES 91:134–210

    Google Scholar 

  47. Stolovitch L (2005) Normalisation holomorphe d'algèbres de type Cartan de champs de vecteurs holomorphes singuliers. Ann Math 161:589–612

    Article  MathSciNet  MATH  Google Scholar 

  48. Taylor M (1981) Pseudodifferential operators. Princeton Mathematical Series, vol 34. Princeton University Press, Princeton

    Google Scholar 

  49. Thiffeault J-L, Morison PJ (2000) Classification and Casimir invariants of Lie–Poisson brackets. Physica D 136:205–244

    Article  MathSciNet  MATH  Google Scholar 

  50. Vaillant J (1999) Invariants des systèmes d'opérateurs différentiels et sommes formelles asymptotiques. Jpn J Math (NS) 25:1–153

    MathSciNet  MATH  Google Scholar 

  51. Yamahara H (2000) Cauchy problem for hyperbolic systems in Gevrey class. A note on Gevrey indices. Ann Fac Sci Toulouse Math 19:147–160

    Article  MathSciNet  Google Scholar 

  52. 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 231:3–88

    MathSciNet  Google Scholar 

  53. 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 287–294

    Google Scholar 

  54. 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) 58:263–297

    Article  MathSciNet  MATH  Google Scholar 

  55. Zung NT (2002) Convergence versus integrability in Poincaré‐Dulac normal form. Math Res Lett 9:217–228

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Cagliari, Cagliari, Italy

    Todor Gramchev

Authors
  1. Todor Gramchev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Gramchev, T. (2012). Perturbation of Systems with Nilpotent Real Part. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_78

Download citation

Publish with us

Policies and ethics

Search

Navigation