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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 - A, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763
References:
[1]

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C. A. Buzzi, L. A. Roberto and M. A. Teixeira, Branching of periodic orbits in reversible Hamiltonian systems,, in, 380 (2010), 46.   Google Scholar

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R. L. Devaney, Reversible diffeomorphisms and flows,, Transactions of the American Mathematical Society, 218 (1976), 89.  doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

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G. Gaeta, Normal forms of reversible dynamical systems,, International J. of Theoretical Physics, 33 (1994), 1917.  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 (1983).   Google Scholar

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

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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.  doi: 10.1007/s10231-006-0036-8.  Google Scholar

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J. Knobloch and A. Vanderbauwhede, A general reduction method for periodic solutions in conservative and reversible systems,, J. Dynam. Differential Equations, 8 (1996), 71.   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.   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.  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.   Google Scholar

[13]

M. Matveyev, Structure of the sets of invariant tori and problems of stability in reversible systems,, in, 533 (1999), 489.   Google Scholar

[14]

J. Murdock, J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems,", 2nd edition, 59 (2007).   Google Scholar

[15]

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

[16]

M. Sevryuk, The finite-dimensional reversible KAM theory,, Phys. D, 112 (1998), 132.  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.  doi: 10.1142/S0218127497000406.  Google Scholar

[18]

A. Vanderbauwhede, "Local Bifurcation and Symmetry,", Res. Notes in Math., 75 (1982).   Google Scholar

[19]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Universitext, (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, (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.  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 (1645).   Google Scholar

[3]

C. A. Buzzi, L. A. Roberto and M. A. Teixeira, Branching of periodic orbits in reversible Hamiltonian systems,, in, 380 (2010), 46.   Google Scholar

[4]

R. L. Devaney, Reversible diffeomorphisms and flows,, Transactions of the American Mathematical Society, 218 (1976), 89.  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.  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 (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.  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.   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.   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.  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.   Google Scholar

[13]

M. Matveyev, Structure of the sets of invariant tori and problems of stability in reversible systems,, in, 533 (1999), 489.   Google Scholar

[14]

J. Murdock, J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems,", 2nd edition, 59 (2007).   Google Scholar

[15]

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

[16]

M. Sevryuk, The finite-dimensional reversible KAM theory,, Phys. D, 112 (1998), 132.  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.  doi: 10.1142/S0218127497000406.  Google Scholar

[18]

A. Vanderbauwhede, "Local Bifurcation and Symmetry,", Res. Notes in Math., 75 (1982).   Google Scholar

[19]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Universitext, (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, (1999).   Google Scholar

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