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.

[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.

[3]

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

[4]

J. Hale, "Ordinary Differential Equations,", $1^{st}$ edition, (1969).

[5]

G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications,", Adv. Ser. Nonlinear Dynamics, 3 (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.

[7]

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

[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.

[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.

[10]

T. Wagenknecht, "An analytical Study of a Two Degrees of Freedom Hamiltonian System Associated the Reversible Hyperbolic Umbilic,", Ph. D thesis, (1999).

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. 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.

[3]

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

[4]

J. Hale, "Ordinary Differential Equations,", $1^{st}$ edition, (1969).

[5]

G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications,", Adv. Ser. Nonlinear Dynamics, 3 (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.

[7]

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

[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.

[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.

[10]

T. Wagenknecht, "An analytical Study of a Two Degrees of Freedom Hamiltonian System Associated the Reversible Hyperbolic Umbilic,", Ph. D thesis, (1999).

[1]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51

[2]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092

[3]

Haihua Liang, Yulin Zhao. Quadratic perturbations of a class of quadratic reversible systems with one center. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 325-335. doi: 10.3934/dcds.2010.27.325

[4]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703

[5]

A. Carati. Center manifold of unstable periodic orbits of helium atom: numerical evidence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 97-104. doi: 10.3934/dcdsb.2003.3.97

[6]

D. Ruiz, J. R. Ward. Some notes on periodic systems with linear part at resonance. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 337-350. doi: 10.3934/dcds.2004.11.337

[7]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623

[8]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331

[9]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361

[10]

Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251

[11]

Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835

[12]

Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2227-2272. doi: 10.3934/dcds.2015.35.2227

[13]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345

[14]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[15]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363

[16]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

[17]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

[18]

Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367

[19]

Paul H. Rabinowitz. On a class of reversible elliptic systems. Networks & Heterogeneous Media, 2012, 7 (4) : 927-939. doi: 10.3934/nhm.2012.7.927

[20]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]