On the birth of minimal sets for perturbed reversible vector fields

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September 2011, 31(3): 763-777. doi: 10.3934/dcds.2011.31.763

On the birth of minimal sets for perturbed reversible vector fields

Jaume Llibre ^1, , Ricardo Miranda Martins ^2, and Marco Antonio Teixeira ^2,

1.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.	Instituto de Matemática, Estatística e Computação Cientíﬁca, Universidade Estadual de Campinas, 13083–859 Campinas, SP, Brazil, Brazil

Received June 2010 Revised May 2011 Published August 2011

Abstract
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The results in this paper fit into a program to study the existence of periodic orbits, invariant cylinders and tori filled with periodic orbits in perturbed reversible systems. Here we focus on bifurcations of one-parameter families of periodic orbits for reversible vector fields in $\mathbb{R}^4$. The main used tools are normal forms theory, Lyapunov-Schmidt method and averaging theory.

Keywords: Invariant torus, Limit cycle, Averaging method, Isochronous center, Reversible system., Periodic orbit.

Mathematics Subject Classification: Primary: 34C29, 34C25; Secondary: 47H1.

Citation: Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763

References:

[1]	G. Belitskii, $C^\infty$-normal forms of local vector fields. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 23-41. doi: 10.1023/A:1013909812387.
[2]	H. Broer, G. Huitema and M. Sevryuk, "Quasi-Periodic Motions in Families of Dynamical Systems. Order Amidst Chaos," Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996.
[3]	C. A. Buzzi, L. A. Roberto and M. A. Teixeira, Branching of periodic orbits in reversible Hamiltonian systems, in "Real and Complex Singularities" (eds. M. Manoel, M. C. Romero Fuster and C. T. C. Wall), Cambridge University Press, (2010), 380, 46-70.
[4]	R. L. Devaney, Reversible diffeomorphisms and flows, Transactions of the American Mathematical Society, 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.
[5]	G. Gaeta, Normal forms of reversible dynamical systems, International J. of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.
[6]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.
[7]	J. Hale, "Ordinary Differential Equations," Dover Publications, 2009.
[8]	A. Jacquemard, M. Firmino Silva Lima and M. A. Teixeira, Degenerate resonances and branching of periodic orbits, Annali di Matematica Pura ed Applicata, 187 (2008), 105-117. doi: 10.1007/s10231-006-0036-8.
[9]	J. Knobloch and A. Vanderbauwhede, A general reduction method for periodic solutions in conservative and reversible systems, J. Dynam. Differential Equations, 8 (1996), 71-102.
[10]	M. F. S. Lima and M. A. Teixeira, Families of periodic orbits in resonant reversible systems, Bull. Braz. Math. Soc. (N.S.), 40 (2009), 511-537.
[11]	J. Llibre, A. C. O. Mereu and M. A. Teixeira, Invariant tori filled with periodic orbits for $4$-dimensional $C^2$ differential systems in presence of resonance, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 20 (2010), 3341-3344. doi: 10.1142/S0218127410027738.
[12]	R. M. Martins and M. A. Teixeira, On the Similarity of Hamiltonian and Reversible Vector Fields in 4D, Communications on Pure and Applied Analysis, 10 (2011), 1257-1266.
[13]	M. Matveyev, Structure of the sets of invariant tori and problems of stability in reversible systems, in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995) (ed. C. Simó), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, (1999), 489-493.
[14]	J. Murdock, J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," 2^nd edition, Applied Mathematical Sciences, 59, Springer, New York, 2007.
[15]	M. Sevryuk, Lower-dimensional tori in reversible systems, Chaos, 1 (1991), 160-167. doi: 10.1063/1.165858.
[16]	M. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.
[17]	C.-W. Shih, Bifurcations of symmetric periodic orbits near equilibrium in reversible systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 7 (1997), 569-584. doi: 10.1142/S0218127497000406.
[18]	A. Vanderbauwhede, "Local Bifurcation and Symmetry," Res. Notes in Math., 75, Pitman (Advanced Publishing Program), Boston, MA, 1982.
[19]	F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Universitext, Springer-Verlag, Berlin, 1990.
[20]	T. Wagenknecht, "An Analytical Study of a Two Degrees of Freedom Hamiltonian System Associated to the Reversible Hyperbolic Umbilic," Ph.D. thesis, University Ilmenau, 1999.

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References:

[1]	G. Belitskii, $C^\infty$-normal forms of local vector fields. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 23-41. doi: 10.1023/A:1013909812387.
[2]	H. Broer, G. Huitema and M. Sevryuk, "Quasi-Periodic Motions in Families of Dynamical Systems. Order Amidst Chaos," Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996.
[3]	C. A. Buzzi, L. A. Roberto and M. A. Teixeira, Branching of periodic orbits in reversible Hamiltonian systems, in "Real and Complex Singularities" (eds. M. Manoel, M. C. Romero Fuster and C. T. C. Wall), Cambridge University Press, (2010), 380, 46-70.
[4]	R. L. Devaney, Reversible diffeomorphisms and flows, Transactions of the American Mathematical Society, 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.
[5]	G. Gaeta, Normal forms of reversible dynamical systems, International J. of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.
[6]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.
[7]	J. Hale, "Ordinary Differential Equations," Dover Publications, 2009.
[8]	A. Jacquemard, M. Firmino Silva Lima and M. A. Teixeira, Degenerate resonances and branching of periodic orbits, Annali di Matematica Pura ed Applicata, 187 (2008), 105-117. doi: 10.1007/s10231-006-0036-8.
[9]	J. Knobloch and A. Vanderbauwhede, A general reduction method for periodic solutions in conservative and reversible systems, J. Dynam. Differential Equations, 8 (1996), 71-102.
[10]	M. F. S. Lima and M. A. Teixeira, Families of periodic orbits in resonant reversible systems, Bull. Braz. Math. Soc. (N.S.), 40 (2009), 511-537.
[11]	J. Llibre, A. C. O. Mereu and M. A. Teixeira, Invariant tori filled with periodic orbits for $4$-dimensional $C^2$ differential systems in presence of resonance, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 20 (2010), 3341-3344. doi: 10.1142/S0218127410027738.
[12]	R. M. Martins and M. A. Teixeira, On the Similarity of Hamiltonian and Reversible Vector Fields in 4D, Communications on Pure and Applied Analysis, 10 (2011), 1257-1266.
[13]	M. Matveyev, Structure of the sets of invariant tori and problems of stability in reversible systems, in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995) (ed. C. Simó), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, (1999), 489-493.
[14]	J. Murdock, J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," 2^nd edition, Applied Mathematical Sciences, 59, Springer, New York, 2007.
[15]	M. Sevryuk, Lower-dimensional tori in reversible systems, Chaos, 1 (1991), 160-167. doi: 10.1063/1.165858.
[16]	M. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.
[17]	C.-W. Shih, Bifurcations of symmetric periodic orbits near equilibrium in reversible systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 7 (1997), 569-584. doi: 10.1142/S0218127497000406.
[18]	A. Vanderbauwhede, "Local Bifurcation and Symmetry," Res. Notes in Math., 75, Pitman (Advanced Publishing Program), Boston, MA, 1982.
[19]	F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Universitext, Springer-Verlag, Berlin, 1990.
[20]	T. Wagenknecht, "An Analytical Study of a Two Degrees of Freedom Hamiltonian System Associated to the Reversible Hyperbolic Umbilic," Ph.D. thesis, University Ilmenau, 1999.

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