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March  2013, 33(3): 1177-1199. doi: 10.3934/dcds.2013.33.1177

Reversibility and branching of periodic orbits

Ana Cristina Mereu 1, and  Marco Antonio Teixeira 2,

1. 

Departamento de Física, Química e Matemática, Universidade Federal de São Carlos, 18052-780, S.P., Brazil
2. 

Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P., Brazil

Received  April 2011 Revised  April 2012 Published  October 2012

We study the dynamics near an equilibrium point of a $2$-parameter family of a reversible system in $\mathbb{R}^6$. In particular, we exhibit conditions for the existence of periodic orbits near the equilibrium of systems having the form $x^{(vi)}+ \lambda_1 x^{(iv)} + \lambda_2 x'' +x = f(x,x',x'',x''',x^{(iv)},x^{(v)})$. The techniques used are Belitskii normal form combined with Lyapunov-Schmidt reduction.
Keywords: normal form, Lyapunov center theorem., resonance, reversible systems, Periodic orbits.
Mathematics Subject Classification: Primary: 34C29, 34C25; Secondary: 47H1.
Citation: Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177
References:
[1]

A. R. Champneys, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Physica D, 112 (1998), 158-186. doi: 10.1016/S0167-2789(97)00209-1.  Google Scholar

[2]

J. V. Chaparova, L. A. Peletier and S. A. Tersian, Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations, Advances in Differential Equations, 8 (2003), 1237-1258.  Google Scholar

[3]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

[4]

J. Hale, "Ordinary Differential Equations," $1^{st}$ edition, New York, Wiley-Interscience, 1969.  Google Scholar

[5]

G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," Adv. Ser. Nonlinear Dynamics, 3, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.  Google Scholar

[6]

A. Jacquemard, M. F. S. Lima and M. Teixeira, Degenerate resonances and branching of periodic orbits, Annali di Matematica ed Applicata, 187 (1992), 105-117.  Google Scholar

[7]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D, 112 (1998), 1-39.  Google Scholar

[8]

M. F. S. Lima and M. Teixeira, Families of periodic orbits in resonant reversible systems, Bull. Braz. Math. Soc., 40 (2009), 521-547. doi: 10.1007/s00574-009-0025-9.  Google Scholar

[9]

C. W. Shih, Bifurcations of Symmetric Periodic Orbits near Equilibrium in Reversible Systems, Int. J. Bifurcation and Chaos, 7 (1997), 569-584. doi: 10.1142/S0218127497000406.  Google Scholar

[10]

T. Wagenknecht, "An analytical Study of a Two Degrees of Freedom Hamiltonian System Associated the Reversible Hyperbolic Umbilic," Ph. D thesis, University Ilmenau, Germany, 1999. Google Scholar

show all references

References:
[1]

A. R. Champneys, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Physica D, 112 (1998), 158-186. doi: 10.1016/S0167-2789(97)00209-1.  Google Scholar

[2]

J. V. Chaparova, L. A. Peletier and S. A. Tersian, Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations, Advances in Differential Equations, 8 (2003), 1237-1258.  Google Scholar

[3]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

[4]

J. Hale, "Ordinary Differential Equations," $1^{st}$ edition, New York, Wiley-Interscience, 1969.  Google Scholar

[5]

G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," Adv. Ser. Nonlinear Dynamics, 3, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.  Google Scholar

[6]

A. Jacquemard, M. F. S. Lima and M. Teixeira, Degenerate resonances and branching of periodic orbits, Annali di Matematica ed Applicata, 187 (1992), 105-117.  Google Scholar

[7]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D, 112 (1998), 1-39.  Google Scholar

[8]

M. F. S. Lima and M. Teixeira, Families of periodic orbits in resonant reversible systems, Bull. Braz. Math. Soc., 40 (2009), 521-547. doi: 10.1007/s00574-009-0025-9.  Google Scholar

[9]

C. W. Shih, Bifurcations of Symmetric Periodic Orbits near Equilibrium in Reversible Systems, Int. J. Bifurcation and Chaos, 7 (1997), 569-584. doi: 10.1142/S0218127497000406.  Google Scholar

[10]

T. Wagenknecht, "An analytical Study of a Two Degrees of Freedom Hamiltonian System Associated the Reversible Hyperbolic Umbilic," Ph. D thesis, University Ilmenau, Germany, 1999. Google Scholar

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