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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
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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

G. Gaeta, Normal forms of reversible dynamical systems, International J. of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar

[6]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[7]

J. Hale, "Ordinary Differential Equations," Dover Publications, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

J. Murdock, J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Applied Mathematical Sciences, 59, Springer, New York, 2007.  Google Scholar

[15]

M. Sevryuk, Lower-dimensional tori in reversible systems, Chaos, 1 (1991), 160-167. doi: 10.1063/1.165858.  Google Scholar

[16]

M. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.  Google Scholar

[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.  Google Scholar

[18]

A. Vanderbauwhede, "Local Bifurcation and Symmetry," Res. Notes in Math., 75, Pitman (Advanced Publishing Program), Boston, MA, 1982.  Google Scholar

[19]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Universitext, Springer-Verlag, Berlin, 1990.  Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

G. Gaeta, Normal forms of reversible dynamical systems, International J. of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar

[6]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[7]

J. Hale, "Ordinary Differential Equations," Dover Publications, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

J. Murdock, J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Applied Mathematical Sciences, 59, Springer, New York, 2007.  Google Scholar

[15]

M. Sevryuk, Lower-dimensional tori in reversible systems, Chaos, 1 (1991), 160-167. doi: 10.1063/1.165858.  Google Scholar

[16]

M. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1998), 132-147. doi: 10.1016/S0167-2789(97)00207-8.  Google Scholar

[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.  Google Scholar

[18]

A. Vanderbauwhede, "Local Bifurcation and Symmetry," Res. Notes in Math., 75, Pitman (Advanced Publishing Program), Boston, MA, 1982.  Google Scholar

[19]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Universitext, Springer-Verlag, Berlin, 1990.  Google Scholar

[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. Google Scholar

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