220
Views
9
CrossRef citations to date
0
Altmetric
Original Articles

Dimension Increase and Splitting for Poincaré-Dulac Normal Forms

&
Pages 327-342 | Published online: 21 Jan 2013

References

  • Arnold V.I. Geometrical methods in the theory of ordinary differential equations, Springer (Berlin) 1983, 2nd ed. 1989
  • Benoit E. Dynamic bifurcations (Lect. Notes Math. 1493), Springer, Berlin 1991
  • Calogero F. “Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials”, J. Math. Phys. 12 (1971), 419–436; “Exactly solvable one-dimensional many-body problems”, Lett. N. Cim. 13 (1975), 411–416
  • Cariñena J.F. Grabowski J. Marmo G. “Lie-Scheffers systems: a geometric approach”, Bibliopolis, Napoli 2000; J.F. Cariêna and A. Ramos, “A new geometric approach to Lie systems and physical applications”, Acta Appl. Math. 70 (2002), 43–69
  • Cicogna G. Gaeta G. Symmetry and perturbation theory in nonlinear dynamics, (Lecture Notes in Physics, vol. M57); Springer (Berlin) 1999
  • Dulac H. “Solution d’un systéme d’équations differéntiélles dans le voisinage des valeurs singuliérs”, Bull. Soc. Math. France 40 (1912), 324–383
  • Elphick C. et al. “A simple global characterization for normal forms of singular vector fields”, Physica D 29 (1987), 95–127; Addendum, Physica D 32 (1988), 488
  • Gaeta G. “A splitting lemma for equivariant dynamics”, Lett. Math. Phys. 33 (1995), 313–320
  • Gaeta G. “Algorithmic reduction of Poincaré-Dulac normal forms and Lie algebraic structure”, Lett. Math. Phys. 57 (2001), 41–60
  • Gaeta G. “Resonant Poincaré-Dulac normal forms as constrained linear systems”, Mod. Phys. Lett. A 17 (2002), 583–597
  • Gaeta G. Walcher S. “Linear Lie algebras with finite dimensional centralizer”, J. Math. Anal. Appl. 269 (2002), 578–587
  • Gaeta G. Walcher S. “Embedding and splitting ordinary differential equations in normal form”, preprint 2004
  • Kazhdan D. Kostant B. Sternberg S. “Hamiltonian group actions and dynamical systems of Calogero type” Comm. Pure Appl. Math. 31 (1978), 481–508
  • Lax P.D. “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Appl. Math. 21 (1968), 467–490
  • Marle C.M. “Symplectic manifolds, dynamical groups and Hamiltonian mechanics”, in Differential geometry and relativity, M. Cohen and M. Flato eds., Reidel, Boston 1976
  • Marmo G. Saletan E.J. Simoni A. Vitale B. Dynamical systems. A differential geometric approach to symmetry and reduction, Wiley, Chichester 1985
  • Neishtadt A.I. “On calculation of stability loss delay time for dynamical bifurcations”, in: XIth International Congress of Mathematical Physics, D. Iagolnitzer ed., International Press, Cambridge (MA-USA) 1995
  • Shnider S. Winternitz P. “Classification of systems of nonlinear ordinary differential equations with superposition principles”, J. Math. Phys. 25 (1984), 3155–3165
  • Thieme H. “Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations”, J. Math. Biol. 30 (1992), 755–763
  • Walcher S. “On differential equations in normal form”, Math. Ann. 291 (1991), 293–314
  • Wei J. Norman E. “Lie algebraic solution of linear differential equations”, J. Math. Phys. 4 (1963), 575–581; “On global representations of the solutions of linear differential equations as a product of exponentials”, Proc. A.M.S. 15 (1964), 327–334
  • Yanguas P. “Lowering the dimension of polynomial vector fields in R2 and R3”, Chaos 11 (2001), 306–318; J. Palacian, “Invariant manifolds of an autonomous ordinary differential equation from its generalized normal forms”, Chaos 13 (2003), 1188–1204

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.