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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

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 & Continuous Dynamical Systems, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763
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show all references

References:
[1]

Acta Appl. Math., 70 (2002), 23-41. doi: 10.1023/A:1013909812387.  Google Scholar

[2]

Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996.  Google Scholar

[3]

in "Real and Complex Singularities" (eds. M. Manoel, M. C. Romero Fuster and C. T. C. Wall), Cambridge University Press, (2010), 380, 46-70. Google Scholar

[4]

Transactions of the American Mathematical Society, 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

[5]

International J. of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar

[6]

Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[7]

Dover Publications, 2009. Google Scholar

[8]

Annali di Matematica Pura ed Applicata, 187 (2008), 105-117. doi: 10.1007/s10231-006-0036-8.  Google Scholar

[9]

J. Dynam. Differential Equations, 8 (1996), 71-102.  Google Scholar

[10]

Bull. Braz. Math. Soc. (N.S.), 40 (2009), 511-537.  Google Scholar

[11]

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 20 (2010), 3341-3344. doi: 10.1142/S0218127410027738.  Google Scholar

[12]

Communications on Pure and Applied Analysis, 10 (2011), 1257-1266. Google Scholar

[13]

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.  Google Scholar

[14]

2nd edition, Applied Mathematical Sciences, 59, Springer, New York, 2007.  Google Scholar

[15]

Chaos, 1 (1991), 160-167. doi: 10.1063/1.165858.  Google Scholar

[16]

Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.  Google Scholar

[17]

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 7 (1997), 569-584. doi: 10.1142/S0218127497000406.  Google Scholar

[18]

Res. Notes in Math., 75, Pitman (Advanced Publishing Program), Boston, MA, 1982.  Google Scholar

[19]

Universitext, Springer-Verlag, Berlin, 1990.  Google Scholar

[20]

Ph.D. thesis, University Ilmenau, 1999. Google Scholar

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