# American Institute of Mathematical Sciences

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

## Reversibility and branching of periodic orbits

 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.
Citation: Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems - A, 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.  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.   Google Scholar [3] R. L. Devaney, Reversible diffeomorphisms and flows,, Trans. Am. Math. Soc., 218 (1976), 89.  doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar [4] J. Hale, "Ordinary Differential Equations,", $1^{st}$ edition, (1969).   Google Scholar [5] G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications,", Adv. Ser. Nonlinear Dynamics, 3 (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.   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.   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.  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.  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, (1999).   Google Scholar

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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.  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.   Google Scholar [3] R. L. Devaney, Reversible diffeomorphisms and flows,, Trans. Am. Math. Soc., 218 (1976), 89.  doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar [4] J. Hale, "Ordinary Differential Equations,", $1^{st}$ edition, (1969).   Google Scholar [5] G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications,", Adv. Ser. Nonlinear Dynamics, 3 (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.   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.   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.  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.  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, (1999).   Google Scholar
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