Scalar autonomous second order ordinary differential equations

• Autores: Clodoaldo Grotta Ragazzo
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 11, Nº 2, 2012, págs. 277-415
• Idioma: inglés
• Enlaces
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• Resumen

  • This paper is about equations of the form u˙=v,v˙=F(u,v)where(u,v)∈R2 and F is an infinitely differentiable function. Its main theorem states that if F(u, −v) = F(u, v) then, under some additional conditions, there exists an infinitely differentiable change of variables (u, v) → (x, y) onto R2 such that in the new variables the equation becomes x˙=y,y˙=g(x) .

• Referencias bibliográficas
  • 1. Belitskii, G.R., Kopanskii, A.Ya.: Sternberg theorem for equivariant Hamiltonian vector fields. Nonlinear Anal. 47, 4491–4499 (2001)
  • 2. Eliasson, L.H.: Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case. Comment. Math. Helv. 65, 4–35 (1990)
  • 3. Fiedler, B., Rocha, C., Wolfrum, M.: Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle. J. Differ....
  • 4. Gorenflo, R., Vessella, S.: Abel Integral Equations. Lect. Notes Math., vol. 1461. Springer-Verlag, Berlin (1991)
  • 5. Hale, J. K.: Ordinary Differential Equations, 2nd edn. Robert E. Krieger Publishing Company, Malabar (1980)
  • 6. Markus, L.: Global integrals of f Zx + gZh = h. Acad. Royale de Belgique Bull. de la Classe de Sciences 38, 311–332 (1952)
  • 7. Markus, L.: Global structure of ordinary differential equations in the plane. Trans. A. M. S. 76, 127– 148 (1954)
  • 8. Rocha, C.: Realization of period maps of planar Hamiltonian systems. J. Dyn. Differ. Equ. 19, 571– 591 (2007)
  • 9. Sabatini, M.: On the period function of x + f (x)x2 +g(x) = 0,. J. Differ. Equ. 196, 151–168 (2004)
  • 10. Whitney, H.: Differentiable even functions. Duke Math. J. 10, 159–160 (1943)