Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Nonlinear Dynamics, Symmetry and Perturbation Theory in

  • Giuseppe Gaeta
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_361-3

Definition of the Subject

Given a differential equation or system of differential equations Δ with independent variables ξaΞRq and dependent variables xaMRp, a symmetry of Δ is an invertible transformation of the extended phase space \( \tilde{M}=\varXi \times M \)

Keywords

Vector Field Perturbation Theory Normal Form Symmetry Reduction Approximate Symmetry 
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.

Bibliography

  1. Abenda S, Gaeta G, Walcher S (eds) (2003) Symmetry and perturbation theory – SPT2002. In: Proceedings of Cala Gonone workshop, 19–26 May 2002. World Scientific, SingaporeGoogle Scholar
  2. Abud M, Sartori G (1983) The geometry of spontaneous symmetry breaking. Ann Phys 150:307–372ADSzbMATHMathSciNetGoogle Scholar
  3. Aleekseevskij DV, Vinogradov AM, Lychagin VV (1991) Basic ideas and concepts of differential geometry. In: Gamkrelidze RV (ed) Geometry I. Encyclopaedia of mathematical sciences, vol 28. Springer, BerlinGoogle Scholar
  4. Arnal D, Ben Ammar M, Pinczon G (1984) The Poincaré-Dulac theorem for nonlinear representations of nilpotent Lie algebras. Lett Math Phys 8:467–476ADSzbMATHMathSciNetGoogle Scholar
  5. Arnold VI (1974) Equations differentielles ordinaires, 2nd edn. MIR, Moscow, 1990zbMATHGoogle Scholar
  6. Arnold V (1976) Les méthodes mathématiques de la mecanique classique. MIR, MoscowzbMATHGoogle Scholar
  7. Arnold V (1980) Chapitres supplementaires de la théorie des equations differentielles ordinaires. MIR, MoscowzbMATHGoogle Scholar
  8. Arnold VI (1983a, 1989) Mathematical methods of classical mechanics. Springer, BerlinGoogle Scholar
  9. Arnold VI (1983b) Geometrical methods in the theory of ordinary differential equations. Springer, BerlinzbMATHGoogle Scholar
  10. Arnold VI (1992) Ordinary differential equations. Springer, BerlinGoogle Scholar
  11. Arnold VI, Il’yashenko YS (1988) Ordinary differential equations. In: Anosov DV, Arnold VI (eds) Dynamical systems I. Encyclopaedia of mathematical sciences, vol 1. Springer, Berlin, pp 1–148Google Scholar
  12. Arnold VI, Kozlov VV, Neishtadt AI (1993) Mathematical aspects of classical and celestial mechanics. In: Arnold VI (ed) Dynamical systems III. Encyclopaedia of mathematical sciences, vol 2, 2nd edn. Springer, Berlin, pp 1–291Google Scholar
  13. Baider A (1989) Unique normal form for vector fields and Hamiltonians. J Diff Eqs 78:33–52ADSzbMATHMathSciNetGoogle Scholar
  14. Baider A, Churchill RC (1988) Uniqueness and non-uniqueness of normal forms for vector fields. Proc R Soc Edinburgh A 108:27–33zbMATHMathSciNetGoogle Scholar
  15. Baider A, Sanders J (1992) Further reduction of the Takens-Bogdanov normal form. J Diff Eqs 99:205–244ADSzbMATHMathSciNetGoogle Scholar
  16. Bakri T, Nabergoj R, Tondl A, Verhulst F (2004) Parametric excitation in non-linear dynamics. Int J Nonlinear Mech 39:311–329zbMATHMathSciNetGoogle Scholar
  17. Bambusi D, Gaeta G (eds) (1997) Symmetry and perturbation theory. In: Proceedings of Torino workshop, ISI, Dec 1996. GNFM-CNR, RomaGoogle Scholar
  18. Bambusi D, Gaeta G (2002) On persistence of invariant tori and a theorem by Nekhoroshev. Math Phys El J 8:1–13MathSciNetGoogle Scholar
  19. Bambusi D, Cicogna G, Gaeta G, Marmo G (1998) Normal forms, symmetry, and linearization of dynamical systems. J Phys A Math Gen 31:5065–5082ADSzbMATHMathSciNetGoogle Scholar
  20. Bambusi D, Gaeta G, Cadoni M (eds) (2001) Symmetry and perturbation theory – SPT2001. In: Proceedings of the international conference SPT2001, Cala Gonone, 6–13 May 2001. World Scientific, SingaporeGoogle Scholar
  21. Bargmann V (1961) On a Hilbert space of analytic functions and an associated integral transform. Comm Pure Appl Math 14:187–214zbMATHMathSciNetGoogle Scholar
  22. Baumann G (2000) Symmetry analysis of differential equations with Mathematica. Springer, New YorkzbMATHGoogle Scholar
  23. Belitskii GR (1978) Equivalence and normal forms of germs of smooth mappings. Russ Math Surveys 33(1):107–177ADSzbMATHMathSciNetGoogle Scholar
  24. Belitskii GR (1981) Normal forms relative to the filtering action of a group. Trans Moscow Math Soc 40(2):1–39Google Scholar
  25. Belitskii GR (1987) Smooth equivalence of germs of vector fields with a single eigenvalue or a pair of purely imaginary eigenvalues. Funct Anal Appl 20:253–259Google Scholar
  26. Belitskii GR (2002) C -Normal forms of local vector fields. Acta Appl Math 70:23–41zbMATHMathSciNetGoogle Scholar
  27. Belmonte C, Boccaletti D, Pucacco G (2006) Stability of axial orbits in galactic potentials. Cell Mech Dyn Astr 95:101–116ADSzbMATHMathSciNetGoogle Scholar
  28. Benettin G, Galgani L, Giorgilli A (1984) A proof of the Kolmogorov theorem on invariant tori using canonical transformations defined by the Lie method. Nuovo Cimento B 79:201–223ADSMathSciNetGoogle Scholar
  29. Bluman GW, Anco SC (2002) Symmetry and integration methods for differential equations. Springer, BerlinzbMATHGoogle Scholar
  30. Bluman GW, Kumei S (1989) Symmetries and differential equations. Springer, BerlinzbMATHGoogle Scholar
  31. Bogoliubov NN, Mitropolsky VA (1961) Asymptotic methods in the theory of nonlinear oscillations. Hindustan, New DelhiGoogle Scholar
  32. Bogoliubov NN, Mitropolsky VA (1962) Méthodes asymptotiques dans la théorie des oscillations non-linéaires. Gauthier-Villars, ParisGoogle Scholar
  33. Broer HW (1979) Bifurcations of singularities in volume preserving vector fields. Ph.D. thesis, GroningenGoogle Scholar
  34. Broer HW (1981) Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case. In: Rand DA, Young LS (eds) Dynamical systems and turbulence. Lecture notes in mathematics, vol 898. Springer, BerlinGoogle Scholar
  35. Broer HW, Takens F (1989) Formally symmetric normal forms and genericity. Dyn Rep 2:39–59MathSciNetGoogle Scholar
  36. Bryuno AD (1971a) Analytical form of differential equations I. Trans Moscow Math Soc 25:131–288Google Scholar
  37. Bryuno AD (1971b) Analytical form of differential equations II. Trans Moscow Math Soc 26:199–239Google Scholar
  38. Bryuno AD (1988) The normal form of a Hamiltonian system. Russ Math Sur 43(1):25–66zbMATHMathSciNetGoogle Scholar
  39. Bryuno AD (1989) Local methods in the theory of differential equations. Springer, BerlinGoogle Scholar
  40. Bryuno AD, Walcher S (1994) Symmetries and convergence of normalizing transformations. J Math Anal Appl 183:571–576zbMATHMathSciNetGoogle Scholar
  41. Cantwell BJ (2002) Introduction to symmetry analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  42. Carinena JF, Grabowski J, Marmo G (2000) Lie-Scheffers systems: a geometric approach. Bibliopolis, NapoliGoogle Scholar
  43. Chen G, Della Dora J (2000) Further reductions of normal forms for dynamical systems. J Diff Eqs 166:79–106ADSzbMATHMathSciNetGoogle Scholar
  44. Chern SS, Chen WH, Lam KS (1999) Lectures on differential geometry. World Scientific, SingaporezbMATHGoogle Scholar
  45. Chossat P (2002) The reduction of equivariant dynamics to the orbit space for compact group actions. Acta Appl Math 70:71–94zbMATHMathSciNetGoogle Scholar
  46. Chossat P, Lauterbach R (1999) Methods in equivariant bifurcations and dynamical systems with applications. World Scientific, SingaporeGoogle Scholar
  47. Chow SN, Hale JK (1982) Methods of bifurcation theory. Springer, BerlinzbMATHGoogle Scholar
  48. Chow SN, Li C, Wang D (1994) Normal forms and bifurcations of planar vector fields. Cambridge University Press, CambridgeGoogle Scholar
  49. Chua LO, Kokubu H (1988) Normal forms for nonlinear vector fields part I: theory. IEEE Trans Circ Syst 35:863–880zbMATHMathSciNetGoogle Scholar
  50. Chua LO, Kokubu H (1989) Normal forms for nonlinear vector fields part II: applications. IEEE Trans Circ Syst 36:851–870MathSciNetGoogle Scholar
  51. Churchill RC, Kummer M, Rod DL (1983) On averaging, reduction and symmetry in Hamiltonian systems. J Diff Eqs 49:359–414ADSzbMATHMathSciNetGoogle Scholar
  52. Cicogna G, Gaeta G (1994a) Normal forms and nonlinear symmetries. J Phys A 27:7115–7124ADSzbMATHMathSciNetGoogle Scholar
  53. Cicogna G, Gaeta G (1994b) Poincaré normal forms and Lie point symmetries. J Phys A 27:461–476ADSzbMATHMathSciNetGoogle Scholar
  54. Cicogna G, Gaeta G (1994c) Symmetry invariance and center manifolds in dynamical systems. Nuovo Cim B 109:59–76ADSMathSciNetGoogle Scholar
  55. Cicogna G, Gaeta G (1999) Symmetry and perturbation theory in nonlinear dynamics. Springer, BerlinzbMATHGoogle Scholar
  56. Cicogna G, Walcher S (2002) Convergence of normal form transformations: the role of symmetries. Acta Appl Math 70:95–111zbMATHMathSciNetGoogle Scholar
  57. Courant R, Hilbert D (1962) Methods of mathematical physics. Wiley, New York; (1989)zbMATHGoogle Scholar
  58. Crawford JD (1991) Introduction to bifurcation theory. Rev Mod Phys 63:991–1037ADSMathSciNetGoogle Scholar
  59. Crawford JD, Knobloch E (1991) Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu Rev Fluid Mech 23:341–387ADSMathSciNetGoogle Scholar
  60. Cushman R, Sanders JA (1986) Nilpotent normal forms and representation theory of sl2,R. In: Golubitsky M, Guckenheimer J (eds) Multi-parameter bifurcation theory. Contemporary mathematics, vol 56. AMS, ProvidenceGoogle Scholar
  61. de Zeeuw T, Merritt D (1983) Stellar orbits in a triaxial galaxy I Orbits in the plane of rotation. Astrophys J 267:571–595ADSMathSciNetGoogle Scholar
  62. Degasperis A, Gaeta G (eds) (1999) Symmetry and perturbation theory II – SPT98. In: Proceedings of Roma workshop, Universitá La Sapienza, Dec 1998. World Scientific, SingaporeGoogle Scholar
  63. Deprit A (1969) Canonical transformation depending on a small parameter. Celest Mech 1:12–30ADSzbMATHMathSciNetGoogle Scholar
  64. Elphick C, Tirapegui E, Brachet ME, Coullet P, Iooss G (1987a) A simple global characterization for normal forms of singular vector fields. Physica D 29:95–127ADSzbMATHMathSciNetGoogle Scholar
  65. Elphick C, Tirapegui E, Brachet ME, Coullet P, Iooss G (1987b) Addendum. Physica D 32:488MathSciNetGoogle Scholar
  66. Fassò F (1990) Lie series method for vector fields and Hamiltonian perturbation theory. ZAMP 41:843–864ADSzbMATHGoogle Scholar
  67. Fassò F, Guzzo M, Benettin G (1998) Nekhoroshev stability of elliptic equilibria of Hamiltonian systems. Comm Math Phys 197:347–360ADSzbMATHMathSciNetGoogle Scholar
  68. Field MJ (1989) Equivariant bifurcation theory and symmetry breaking. J Dyn Dif Eqs 1:369–421zbMATHMathSciNetGoogle Scholar
  69. Field MJ (1996a) Lectures on bifurcations, dynamics and symmetry. Research notes in mathematics, vol 356. Pitman, BostonGoogle Scholar
  70. Field MJ (1996b) Symmetry breaking for compact Lie groups. Mem AMS 574:1–170Google Scholar
  71. Field MJ, Richardson RW (1989) Symmetry breaking and the maximal isotropy subgroup conjecture for reflection groups. Arch Rat Mech Anal 105:61–94zbMATHMathSciNetGoogle Scholar
  72. Field MJ, Richardson RW (1990) Symmetry breaking in equivariant bifurcation problems. Bull Am Math Soc 22:79–84zbMATHMathSciNetGoogle Scholar
  73. Field MJ, Richardson RW (1992a) Symmetry breaking and branching patterns in equivariant bifurcation theory I. Arch Rat Mech Anal 118:297–348zbMATHMathSciNetGoogle Scholar
  74. Field MJ, Richardson RW (1992b) Symmetry breaking and branching patterns in equivariant bifurcation theory II. Arch Rat Mech Anal 120:147–190zbMATHMathSciNetGoogle Scholar
  75. Fokas AS (1979a) Generalized symmetries and constants of motion of evolution equations. Lett Math Phys 3:467–473ADSzbMATHMathSciNetGoogle Scholar
  76. Fokas AS (1979b) Group theoretical aspects of constants of motion and separable solutions in classical mechanics. J Math Anal Appl 68:347–370zbMATHMathSciNetGoogle Scholar
  77. Fokas AS (1980) A symmetry approach to exactly solvable evolution equations. J Math Phys 21:1318–1326ADSzbMATHMathSciNetGoogle Scholar
  78. Fokas AS (1987) Symmetries and integrability. Stud Appl Math 77:253–299zbMATHMathSciNetGoogle Scholar
  79. Fokas AS, Gelfand IM (1996) Surfaces on Lie groups, Lie algebras, and their integrability. Comm Math Phys 177:203–220ADSzbMATHMathSciNetGoogle Scholar
  80. Fontich E, Gelfreich VG (1997) On analytical properties of normal forms. Nonlinearity 10:467–477ADSzbMATHMathSciNetGoogle Scholar
  81. Forest E, Murray D (1994) Freedom in minimal normal forms. Physica D 74:181–196ADSzbMATHMathSciNetGoogle Scholar
  82. Fushchich WI, Nikitin AG (1987) Symmetries of Maxwell equations. Reidel, DordrechtzbMATHGoogle Scholar
  83. Fushchich WI, Shtelen WM, Slavutsky SL (1989) Symmetry analysis and exact solutions of nonlinear equations of mathematical physics. Naukova Dumka, KievzbMATHGoogle Scholar
  84. Gaeta G (1990) Bifurcation and symmetry breaking. Phys Rep 189:1–87ADSzbMATHMathSciNetGoogle Scholar
  85. Gaeta G (1994) Nonlinear symmetries and nonlinear equations. Kluwer, DordrechtzbMATHGoogle Scholar
  86. Gaeta G (1997) Reduction of Poincaré normal forms. Lett Math Phys 42:103–114 & 235Google Scholar
  87. Gaeta G (1999a) An equivariant branching lemma for relative equilibria. Nuovo Cimento B 114:973–982ADSGoogle Scholar
  88. Gaeta G (1999b) Poincaré renormalized forms. Ann IHP Phys Theor 70:461–514ADSzbMATHMathSciNetGoogle Scholar
  89. Gaeta G (2001) Algorithmic reduction of Poincaré-Dulac normal forms and Lie algebraic structure. Lett Math Phys 57:41–60zbMATHMathSciNetGoogle Scholar
  90. Gaeta G (2002a) Poincaré normal and renormalized forms. Acta Appl Math 70:113–131zbMATHMathSciNetGoogle Scholar
  91. Gaeta G (2002b) Poincaré normal forms and simple compact Lie groups. Int J Mod Phys A 17:3571–3587ADSzbMATHMathSciNetGoogle Scholar
  92. Gaeta G (2002c) The Poincaré-Lyapunov-Nekhoroshev theorem. Ann Phys 297:157–173ADSzbMATHMathSciNetGoogle Scholar
  93. Gaeta G (2003) The Poincaré-Nekhoroshev map. J Nonlin Math Phys 10:51–64zbMATHMathSciNetGoogle Scholar
  94. Gaeta G (2006a) Finite group symmetry breaking. In: Francoise JP, Naber G, Tsou ST (eds) Encyclopedia of mathematical physics. Kluwer, DordrechtGoogle Scholar
  95. Gaeta G (2006b) Non-quadratic additional conserved quantities in Birkhoff normal forms. Cel Mech Dyn Astr 96:63–81ADSzbMATHMathSciNetGoogle Scholar
  96. Gaeta G (2006c) The Poincaré-Lyapunov-Nekhoroshev theorem for involutory systems of vector fields. Ann Phys NY 321:1277–1295ADSzbMATHMathSciNetGoogle Scholar
  97. Gaeta G, Marmo G (1996) Nonperturbative linearization of dynamical systems. J Phys A 29:5035–5048ADSzbMATHMathSciNetGoogle Scholar
  98. Gaeta G, Morando P (1997) Michel theory of symmetry breaking and gauge theories. Ann Phys NY 260:149–170ADSzbMATHMathSciNetGoogle Scholar
  99. Gaeta G, Walcher S (2005) Dimension increase and splitting for Poincaré-Dulac normal forms. J Nonlin Math Phys 12:S1327–S1342MathSciNetGoogle Scholar
  100. Gaeta G, Walcher S (2006) Embedding and splitting ordinary differential equations in normal form. J Diff Eqs 224:98–119ADSzbMATHMathSciNetGoogle Scholar
  101. Gaeta G, Prinari B, Rauch S, Terracini S (eds) (2005) Symmetry and perturbation theory – SPT2004. In: Proceedings of Cala Gonone workshop, 30 May–6 June 2004. World Scientific, SingaporeGoogle Scholar
  102. Gaeta G, Vitolo R, Walcher S (eds) (2007) Symmetry and perturbation theory – SPT2007. In: Proceedings of Otranto workshop, 2–9 June 2007. World Scientific, SingaporeGoogle Scholar
  103. Gaeta G, Grosshans FD, Scheurle J, Walcher S (2008) Reduction and reconstruction for symmetric ordinary differential equations. J Diff Eqs 244:1810–1839ADSzbMATHMathSciNetGoogle Scholar
  104. Gallavotti G (1983) The elements of mechanics. Springer, BerlinzbMATHGoogle Scholar
  105. Giorgilli A (1988) Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium point. Ann IHP Phys Theor 48:423–439zbMATHMathSciNetGoogle Scholar
  106. Giorgilli A, Locatelli U (1997) Kolmogorov theorem and classical perturbation theory. ZAMP 48:220–261ADSzbMATHMathSciNetGoogle Scholar
  107. Giorgilli A, Morbidelli A (1997) Invariant KAM tori and global stability for Hamiltonian systems. ZAMP 48:102–134ADSzbMATHMathSciNetGoogle Scholar
  108. Giorgilli A, Zehnder E (1992) Exponential stability for time dependent potentials. ZAMP 43:827–855ADSzbMATHMathSciNetGoogle Scholar
  109. Glendinning P (1994) Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
  110. Golubitsky M, Stewart I, Schaeffer D (1988) Singularity and groups in bifurcation theory – vol II. Springer, BerlinGoogle Scholar
  111. Gramchev T, Yoshino M (1999) Rapidly convergent iteration methods for simultaneous normal forms of commuting maps. Math Z 231:745–770zbMATHMathSciNetGoogle Scholar
  112. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer, BerlinGoogle Scholar
  113. Gustavson FG (1964) On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron J 71:670–686ADSGoogle Scholar
  114. Guzzo M, Fassò F, Benettin G (1998) On the stability of elliptic equilibria. Math Phys El J 4(1):16Google Scholar
  115. Hamermesh M (1962) Group theory. Addison-Wesley, Reading; reprinted by Dover, New York (1991)zbMATHGoogle Scholar
  116. Hanssmann H (2007) Local and semi-local bifurcations in Hamiltonian dynamical systems results and examples. Springer, BerlinzbMATHGoogle Scholar
  117. Hermann R (1968) The formal linearization of a semisimple Lie algebra of vector fields about a singular point. Trans AMS 130:105–109zbMATHMathSciNetGoogle Scholar
  118. Hoveijn I (1996) Versal deformations and normal forms for reversible and Hamiltonian linear systems. J Diff Eq 126:408–442ADSzbMATHMathSciNetGoogle Scholar
  119. Hoveijn I, Verhulst F (1990) Chaos in the 1:2:3 Hamiltonian normal form. Physica D 44:397–406ADSzbMATHMathSciNetGoogle Scholar
  120. Hydon PE (2000) Symmetry methods for differential equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
  121. Ibragimov N (1992) Group analysis of ordinary differential equations and the invariance principle in mathematical physics. Russ Math Surv 47(4):89–156Google Scholar
  122. Il’yashenko YS, Yakovenko SY (1991) Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Russ Math Surv 46(1):1–43zbMATHMathSciNetGoogle Scholar
  123. Iooss G, Adelmeyer M (1992) Topics in bifurcation theory and applications. World Scientific, SingaporezbMATHGoogle Scholar
  124. Isham CJ (1999) Modern differential geometry for physicists. World Scientific, SingaporezbMATHGoogle Scholar
  125. Kinyon M, Walcher S (1997) On ordinary differential equations admitting a finite linear group of symmetries. J Math Anal Appl 216:180–196zbMATHMathSciNetGoogle Scholar
  126. Kirillov AA (1976, 1984) Elements of the theory of representations. Springer, BerlinGoogle Scholar
  127. Kodama Y (1994) Normal forms, symmetry and infinite dimensional Lie algebra for systems of ODE’s. Phys Lett A 191:223–228ADSzbMATHMathSciNetGoogle Scholar
  128. Kokubu H, Oka H, Wang D (1996) Linear grading function and further reduction of normal forms. J Diff Eq 132:293–318ADSzbMATHMathSciNetGoogle Scholar
  129. Krasil’shchik IS, Vinogradov AM (1984) Nonlocal symmetries and the theory of coverings. Acta Appl Math 2:79–96zbMATHMathSciNetGoogle Scholar
  130. Krasil’shchik IS, Vinogradov AM (1999) Symmetries and conservation laws for differential equations of mathematical physics. AMS, ProvidencezbMATHGoogle Scholar
  131. Kummer M (1971) How to avoid secular terms in classical and quantum mechanics. Nuovo Cimento B 1:123–148MathSciNetGoogle Scholar
  132. Kummer M (1976) On resonant nonlinearly coupled oscillators with two equal frequencies. Comm Math Phys 48:53–79ADSzbMATHMathSciNetGoogle Scholar
  133. Lamb J (1996) Local bifurcations in k-symmetric dynamical systems. Nonlinearity 9:537–557ADSzbMATHMathSciNetGoogle Scholar
  134. Lamb J (1998) k-symmetry and return maps of spacetime symmetric flows. Nonlinearity 11:601–630ADSzbMATHMathSciNetGoogle Scholar
  135. Lamb J, Melbourne I (2007) Normal form theory for relative equilibria and relative periodic solutions. Trans AMS 359:4537–4556zbMATHMathSciNetGoogle Scholar
  136. Lamb J, Roberts J (1998) Time reversal symmetry in dynamical systems: a survey. Physica D 112:1–39ADSzbMATHMathSciNetGoogle Scholar
  137. Levi D, Winternitz P (1989) Non-classical symmetry reduction: example of the Boussinesq equation. J Phys A 22:2915–2924ADSzbMATHMathSciNetGoogle Scholar
  138. Lin CM, Vittal V, Kliemann W, Fouad AA (1996) Investigation of modal interaction and its effect on control performance in stressed power systems using normal forms of vector fields. IEEE Trans Power Syst 11:781–787Google Scholar
  139. Marsden JE (1992) Lectures on mechanics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  140. Marsden JE, Ratiu T (1994) Introduction to mechanics and symmetry. Springer, BerlinzbMATHGoogle Scholar
  141. Meyer KR, Hall GR (1992) Introduction to Hamiltonian dynamical systems and the N-body problem. Springer, New YorkzbMATHGoogle Scholar
  142. Michel L (1971a) Points critiques de fonctions invariantes sur une G-variété. Comptes Rendus Acad Sci Paris 272-A:433–436Google Scholar
  143. Michel L (1971b) Nonlinear group action Smooth action of compact Lie groups on manifolds. In: Sen RN, Weil C (eds) Statistical mechanics and field theory. Israel University Press, JerusalemGoogle Scholar
  144. Michel L (1975) Les brisure spontanées de symétrie en physique. J Phys Paris 36-C7:41–51Google Scholar
  145. Michel L (1980) Symmetry defects and broken symmetry configurations hidden symmetry. Rev Mod Phys 52:617–651ADSGoogle Scholar
  146. Michel L, Radicati L (1971) Properties of the breaking of hadronic internal symmetry. Ann Phys NY 66:758–783ADSMathSciNetGoogle Scholar
  147. Michel L, Radicati L (1973) The geometry of the octet. Ann IHP 18:185–214zbMATHMathSciNetGoogle Scholar
  148. Michel L, Zhilinskii BI (2001) Symmetry, invariants, topology basic tools. Phys Rep 341:11–84ADSzbMATHMathSciNetGoogle Scholar
  149. Mikhailov AV, Shabat AB, Yamilov RI (1987) The symmetry approach to the classification of non-linear equations complete list of integrable systems. Russ Math Surv 42(4):1–63MathSciNetGoogle Scholar
  150. Mitropolsky YA, Lopatin AK (1995) Nonlinear mechanics, groups and symmetry. Kluwer, DordrechtzbMATHGoogle Scholar
  151. Nakahara M (1990) Geometry, topology and physics. IOP, BristolzbMATHGoogle Scholar
  152. Nash C, Sen S (1983) Topology and geometry for physicists. Academic, LondonzbMATHGoogle Scholar
  153. Nekhoroshev NN (1994) The Poincaré-Lyapunov-Liouville-Arnol’d theorem. Funct Anal Appl 28:128–129MathSciNetGoogle Scholar
  154. Nekhoroshev NN (2002) Generalizations of Gordon theorem. Regul Chaotic Dyn 7:239–247ADSzbMATHMathSciNetGoogle Scholar
  155. Nekhoroshev NN (2005) Types of integrability on a submanifold and generalizations of Gordons theorem. Trans Moscow Math Soc 66:169–241MathSciNetGoogle Scholar
  156. Olver PJ (1986) Applications of Lie groups to differential equations. Springer, BerlinzbMATHGoogle Scholar
  157. Olver PJ (1995) Equivalence, invariants, and symmetry. Cambridge University Press, CambridgezbMATHGoogle Scholar
  158. Ovsjiannikov LV (1982) Group analysis of differential equations. Academic, LondonGoogle Scholar
  159. Palacián J, Yanguas P (2000) Reduction of polynomial Hamiltonians by the construction of formal integrals. Nonlinearity 13:1021–1054ADSzbMATHMathSciNetGoogle Scholar
  160. Palacián J, Yanguas P (2001) Generalized normal forms for polynomial vector fields. J Math Pures Appl 80:445–469zbMATHMathSciNetGoogle Scholar
  161. Palacián J, Yanguas P (2003) Equivariant N-DOF Hamiltonians via generalized normal forms. Comm Cont Math 5:449–480zbMATHGoogle Scholar
  162. Palacián J, Yanguas P (2005) A universal procedure for normalizing n-degree-of-freedom polynomial Hamiltonian systems. SIAM J Appl Math 65:1130–1152zbMATHMathSciNetGoogle Scholar
  163. Pucci E, Saccomandi G (1992) On the weak symmetry group of partial differential equations. J Math Anal Appl 163:588–598zbMATHMathSciNetGoogle Scholar
  164. Ruelle D (1973) Bifurcation in the presence of a symmetry group. Arch Rat Mech Anal 51:136–152zbMATHMathSciNetGoogle Scholar
  165. Ruelle D (1989) Elements of differentiable dynamics and bifurcation theory. Academic, LondonzbMATHGoogle Scholar
  166. Sadovskii DA, Delos JB (1996) Bifurcation of the periodic orbits of Hamiltonian systems – an analysis using normal form theory. Phys Rev A 54:2033–2070ADSGoogle Scholar
  167. Sanders JA (2003) Normal form theory and spectral sequences. J Diff Eqs 192:536–552ADSzbMATHGoogle Scholar
  168. Sanders JA (2005) Normal forms in filtered Lie algebra representations. Acta Appl Math 87:165–189zbMATHMathSciNetGoogle Scholar
  169. Sanders JA, Verhulst F (1985) Averaging methods in nonlinear dynamical systems. Springer, BerlinzbMATHGoogle Scholar
  170. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems. Springer, BerlinzbMATHGoogle Scholar
  171. Sartori G (1991) Geometric invariant theory a model-independent approach to spontaneous symmetry and/or supersymmetry breaking. Riv Nuovo Cimento 14–11:1–120MathSciNetGoogle Scholar
  172. Sartori G (2002) Geometric invariant theory in a model-independent analysis of spontaneous symmetry and supersymmetry breaking. Acta Appl Math 70:183–207zbMATHMathSciNetGoogle Scholar
  173. Sartori G, Valente G (2005) Constructive axiomatic approach to the determination of the orbit spaces of coregular compact linear groups. Acta Appl Math 87:191–228zbMATHMathSciNetGoogle Scholar
  174. Sattinger DH (1979) Group theoretic methods in bifurcation theory. Lecture notes in mathematics, vol 762. Springer, BerlinGoogle Scholar
  175. Sattinger DH (1983) Branching in the presence of symmetry. SIAM, PhiladelphiaGoogle Scholar
  176. Sattinger DH, Weaver O (1986) Lie groups and algebras. Springer, BerlinzbMATHGoogle Scholar
  177. Siegel K, Moser JK (1971) Lectures on celestial mechanics. Springer, Berlin; reprinted in Classics in mathematics. Springer, Berlin (1995)zbMATHGoogle Scholar
  178. Sokolov VV (1988) On the symmetries of evolutions equations. Russ Math Surv 43(5):165–204zbMATHGoogle Scholar
  179. Stephani H (1989) Differential equations their solution using symmetries. Cambridge University Press, CambridgezbMATHGoogle Scholar
  180. Stewart I (1988) Bifurcation with symmetry. In: Bedford T, Swift J (eds) New directions in dynamical systems. Cambridge University Press, CambridgeGoogle Scholar
  181. Tondl A, Ruijgrok T, Verhulst F, Nabergoj R (2000) Autoparametric resonance in mechanical systems. Cambridge University Press, CambridgezbMATHGoogle Scholar
  182. Ushiki S (1984) Normal forms for singularities of vector fields. Jpn J Appl Math 1:1–34zbMATHMathSciNetGoogle Scholar
  183. Vanderbauwhede A (1982) Local bifurcation and symmetry. Pitman, BostonzbMATHGoogle Scholar
  184. Verhulst F (1989) Nonlinear differential equations and dynamical systems. Springer, Berlin; (1996)Google Scholar
  185. Verhulst F (1998) Symmetry and integrability in Hamiltonian normal form. In: Bambusi D, Gaeta G (eds) Symmetry and perturbation theory. CNR, RomaGoogle Scholar
  186. Verhulst F (1999) On averaging methods for partial differential equations. In: Degasperis A, Gaeta G (eds) Symmetry and perturbation theory II. World Scientific, SingaporeGoogle Scholar
  187. Vinogradov AM (1984) Local symmetries and conservation laws. Acta Appl Math 2:21–78zbMATHMathSciNetGoogle Scholar
  188. Vittal V, Kliemann W, Ni YX, Chapman DG, Silk AD, Sobajic DJ (1998) Determination of generator groupings for an islanding scheme in the Manitoba hydro system using the method of normal forms. IEEE Trans Power Syst 13:1346–1351Google Scholar
  189. Vorob’ev EM (1986) Partial symmetries of systems of differential equations. Soviet Math Dokl 33:408–411zbMATHMathSciNetGoogle Scholar
  190. Vorob’ev EM (1991) Reduction and quotient equations for differential equations with symmetries. Acta Appl Math 23:1–24zbMATHMathSciNetGoogle Scholar
  191. Walcher S (1991) On differential equations in normal form. Math Ann 291:293–314zbMATHMathSciNetGoogle Scholar
  192. Walcher S (1993) On transformation into normal form. J Math Anal Appl 180:617–632zbMATHMathSciNetGoogle Scholar
  193. Walcher S (1999) Orbital symmetries of first order ODEs. In: Degasperis A, Gaeta G (eds) Symmetry and perturbation theory II. World Scientific, SingaporeGoogle Scholar
  194. Walcher S (2000) On convergent normal form transformations in the presence of symmetry. J Math Anal Appl 244:17–26zbMATHMathSciNetGoogle Scholar
  195. Wei J, Norman E (1963) Lie algebraic solution of linear differential equations. J Math Phys 4:575–581ADSzbMATHMathSciNetGoogle Scholar
  196. Winternitz P (1987) What is new in the study of differential equations by group theoretical methods? In: Gilmore R (ed) Group theoretical methods in physics proceedings of the XV ICGTMP. World Scientific, SingaporeGoogle Scholar
  197. Winternitz P (1993) Lie groups and solutions of nonlinear PDEs. In: Ibort LA, Rodriguez MA (eds) Integrable systems, quantum groups, and quantum field theory. NATO ASI 9009. Kluwer, DordrechtGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly