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Scalar Autonomous Second Order Ordinary Differential Equations

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Abstract

This paper is about equations of the form \({\dot u=v,\dot v = F(u, v) \, \, {\rm where} \, \, (u,v) \in \mathbb {R}^{2}}\) 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 \({\mathbb {R}^{2}}\) such that in the new variables the equation becomes \({\dot x=y, \dot y =g(x)}\).

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References

  1. Belitskii G.R., Kopanskii A.Ya.: Sternberg theorem for equivariant Hamiltonian vector fields. Nonlinear Anal. 47, 4491–4499 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Eliasson L.H.: Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case. Comment. Math. Helv. 65, 4–35 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fiedler B., Rocha C., Wolfrum M.: Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle. J. Differ. Equ. 201, 99–138 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  6. Markus L.: Global integrals of fZ x  + gZ h  = h. Acad. Royale de Belgique Bull. de la Classe de Sciences 38, 311–332 (1952)

    MathSciNet  MATH  Google Scholar 

  7. Markus L.: Global structure of ordinary differential equations in the plane. Trans. A. M. S. 76, 127–148 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rocha C.: Realization of period maps of planar Hamiltonian systems. J. Dyn. Differ. Equ. 19, 571–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sabatini M.: On the period function of x′′ + f(x)x ′ 2 + g(x) = 0,. J. Differ. Equ. 196, 151–168 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Whitney H.: Differentiable even functions. Duke Math. J. 10, 159–160 (1943)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática e Estatí stica, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, 05508-090, Brazil

    C. Ragazzo

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  1. C. Ragazzo
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Correspondence to C. Ragazzo.

Additional information

Partially supported by CNPq (Brazil) Grant 305089/2009-9.

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Ragazzo, C. Scalar Autonomous Second Order Ordinary Differential Equations. Qual. Theory Dyn. Syst. 11, 277–415 (2012). https://doi.org/10.1007/s12346-011-0063-8

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  • DOI: https://doi.org/10.1007/s12346-011-0063-8

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