American Institute of Mathematical Sciences

Advanced
December  2015, 8(6): 1165-1211. doi: 10.3934/dcdss.2015.8.1165

Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$

Fatima Ezzahra Lembarki 1, and  Jaume Llibre 1,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Received  July 2015 Revised  September 2015 Published  December 2015

A generalized Friedmann-Robertson-Walker Hamiltonian system is studied in dimension $6$. The averaging theory is the tool used to provide sufficient conditions on the six parameters of the system which guarantee the existence of continuous families of period orbits parameterized by the energy.
Keywords: family of periodic orbits, Friedmann-Robertson-Walker, periodic orbits parameterized by the energy., averaging theory, Periodic orbits.
Mathematics Subject Classification: Primary: 37G15, 37C80, 37C3.
Citation: Fatima Ezzahra Lembarki, Jaume Llibre. Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1165-1211. doi: 10.3934/dcdss.2015.8.1165
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics,, Benjamin, (1978).   Google Scholar

[2]

C. Belomte, D. Bocaaletti and G. Pucacco, On the orbit structure of the logarithm potential,, Astrophys. J., 669 (2007), 202.   Google Scholar

[3]

E. Calzeta and C. E. Hasi, Chaotic Friedmann-Robertson-Walker cosmology,, Class. Quantum Gravity, 10 (1993), 1825.  doi: 10.1088/0264-9381/10/9/022.  Google Scholar

[4]

S. W. Hawkings, Arrow of time in cosmology,, Phys. Rev. D, 32 (1985), 2489.  doi: 10.1103/PhysRevD.32.2489.  Google Scholar

[5]

F. Lembarki and J. Llibre, Periodic orbits for the generalized Yang-Mills Hamiltonian system in dimension $6$,, Nonlinear Dyn., 76 (2014), 1807.  doi: 10.1007/s11071-014-1249-9.  Google Scholar

[6]

J. Llibre and A. Makhlouf, Periodic orbits of the generalized Friedmann-Robertson-Walker Hamiltonian systems,, Astrophys, 344 (2013), 46.   Google Scholar

[7]

D. Merrit and M. Valluri, Chaos and mixing in triaxial stellar systems,, Astrophys, 471 (1996), 82.   Google Scholar

[8]

D. Page, Will entroy decrease if the universe recollapses?,, Phys. Rev. D, 32 (1991), 2496.  doi: 10.1103/PhysRevD.32.2496.  Google Scholar

[9]

Y. Papaphilippou and J. Laskar, Frequency map analysis and global dynamics in a galactic potential with two degrees of freedom,, Astron. Astrophys., 307 (1996), 427.   Google Scholar

[10]

Y. Papaphilippou and J. Laskar, Global dynamics of triaxial galactic models though frequency analysis,, Astron. Astrophys., 329 (1998), 451.   Google Scholar

[11]

G. Pucacco, D. Boccaletti and C. Belmonte, Quantitative predictions with detuned normal forms,, Celest. Mech. Dyn. Astron., 102 (2008), 163.  doi: 10.1007/s10569-008-9141-x.  Google Scholar

[12]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996).  doi: 10.1007/978-3-642-61453-8.  Google Scholar

[13]

H. S. Zhao, C. M. Carollo and T. De Zeeuw, Can galactic nuclei be non-axisymmetric? The parameter space of power-law discs,, Mon. Not. R. Astron. Soc., 304 (1999), 457.   Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics,, Benjamin, (1978).   Google Scholar

[2]

C. Belomte, D. Bocaaletti and G. Pucacco, On the orbit structure of the logarithm potential,, Astrophys. J., 669 (2007), 202.   Google Scholar

[3]

E. Calzeta and C. E. Hasi, Chaotic Friedmann-Robertson-Walker cosmology,, Class. Quantum Gravity, 10 (1993), 1825.  doi: 10.1088/0264-9381/10/9/022.  Google Scholar

[4]

S. W. Hawkings, Arrow of time in cosmology,, Phys. Rev. D, 32 (1985), 2489.  doi: 10.1103/PhysRevD.32.2489.  Google Scholar

[5]

F. Lembarki and J. Llibre, Periodic orbits for the generalized Yang-Mills Hamiltonian system in dimension $6$,, Nonlinear Dyn., 76 (2014), 1807.  doi: 10.1007/s11071-014-1249-9.  Google Scholar

[6]

J. Llibre and A. Makhlouf, Periodic orbits of the generalized Friedmann-Robertson-Walker Hamiltonian systems,, Astrophys, 344 (2013), 46.   Google Scholar

[7]

D. Merrit and M. Valluri, Chaos and mixing in triaxial stellar systems,, Astrophys, 471 (1996), 82.   Google Scholar

[8]

D. Page, Will entroy decrease if the universe recollapses?,, Phys. Rev. D, 32 (1991), 2496.  doi: 10.1103/PhysRevD.32.2496.  Google Scholar

[9]

Y. Papaphilippou and J. Laskar, Frequency map analysis and global dynamics in a galactic potential with two degrees of freedom,, Astron. Astrophys., 307 (1996), 427.   Google Scholar

[10]

Y. Papaphilippou and J. Laskar, Global dynamics of triaxial galactic models though frequency analysis,, Astron. Astrophys., 329 (1998), 451.   Google Scholar

[11]

G. Pucacco, D. Boccaletti and C. Belmonte, Quantitative predictions with detuned normal forms,, Celest. Mech. Dyn. Astron., 102 (2008), 163.  doi: 10.1007/s10569-008-9141-x.  Google Scholar

[12]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996).  doi: 10.1007/978-3-642-61453-8.  Google Scholar

[13]

H. S. Zhao, C. M. Carollo and T. De Zeeuw, Can galactic nuclei be non-axisymmetric? The parameter space of power-law discs,, Mon. Not. R. Astron. Soc., 304 (1999), 457.   Google Scholar

[1]

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
[2]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505
[3]

Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[4]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[5]

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
[6]

Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005
[7]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[8]

Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997
[9]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[10]

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
[11]

Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261
[12]

Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103
[13]

Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439
[14]

Piotr Oprocha, Xinxing Wu. On averaged tracing of periodic average pseudo orbits. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4943-4957. doi: 10.3934/dcds.2017212
[15]

Jeremias Epperlein, Vladimír Švígler. On arbitrarily long periodic orbits of evolutionary games on graphs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1895-1915. doi: 10.3934/dcdsb.2018187
[16]

Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595
[17]

Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399
[18]

Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
[19]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197
[20]

Manfred Denker, Samuel Senti, Xuan Zhang. Fluctuations of ergodic sums on periodic orbits under specification. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4665-4687. doi: 10.3934/dcds.2020197

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences