Reversibility and branching of periodic orbits

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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

Abstract
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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 and 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.
[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.
[3]	R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.
[4]	J. Hale, "Ordinary Differential Equations," $1^{st}$ edition, New York, Wiley-Interscience, 1969.
[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.
[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.
[7]	J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D, 112 (1998), 1-39.
[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.
[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.
[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.

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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.
[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.
[3]	R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.
[4]	J. Hale, "Ordinary Differential Equations," $1^{st}$ edition, New York, Wiley-Interscience, 1969.
[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.
[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.
[7]	J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D, 112 (1998), 1-39.
[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.
[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.
[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.

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