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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, Reading, Masachusets, 1978.  Google Scholar

[2]

C. Belomte, D. Bocaaletti and G. Pucacco, On the orbit structure of the logarithm potential, Astrophys. J., 669 (2007), 202-217. Google Scholar

[3]

E. Calzeta and C. E. Hasi, Chaotic Friedmann-Robertson-Walker cosmology, Class. Quantum Gravity, 10 (1993), 1825-1841. doi: 10.1088/0264-9381/10/9/022.  Google Scholar

[4]

S. W. Hawkings, Arrow of time in cosmology, Phys. Rev. D, 32 (1985), 2489-2495. 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), 1807-1819. 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), 46-50. Google Scholar

[7]

D. Merrit and M. Valluri, Chaos and mixing in triaxial stellar systems, Astrophys, 471 (1996), 82-105. Google Scholar

[8]

D. Page, Will entroy decrease if the universe recollapses?, Phys. Rev. D, 32 (1991), 2496-2499. 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), 427-449. Google Scholar

[10]

Y. Papaphilippou and J. Laskar, Global dynamics of triaxial galactic models though frequency analysis, Astron. Astrophys., 329 (1998), 451-481. Google Scholar

[11]

G. Pucacco, D. Boccaletti and C. Belmonte, Quantitative predictions with detuned normal forms, Celest. Mech. Dyn. Astron., 102 (2008), 163-176. doi: 10.1007/s10569-008-9141-x.  Google Scholar

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

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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), 457-464. Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin, Reading, Masachusets, 1978.  Google Scholar

[2]

C. Belomte, D. Bocaaletti and G. Pucacco, On the orbit structure of the logarithm potential, Astrophys. J., 669 (2007), 202-217. Google Scholar

[3]

E. Calzeta and C. E. Hasi, Chaotic Friedmann-Robertson-Walker cosmology, Class. Quantum Gravity, 10 (1993), 1825-1841. doi: 10.1088/0264-9381/10/9/022.  Google Scholar

[4]

S. W. Hawkings, Arrow of time in cosmology, Phys. Rev. D, 32 (1985), 2489-2495. 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), 1807-1819. 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), 46-50. Google Scholar

[7]

D. Merrit and M. Valluri, Chaos and mixing in triaxial stellar systems, Astrophys, 471 (1996), 82-105. Google Scholar

[8]

D. Page, Will entroy decrease if the universe recollapses?, Phys. Rev. D, 32 (1991), 2496-2499. 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), 427-449. Google Scholar

[10]

Y. Papaphilippou and J. Laskar, Global dynamics of triaxial galactic models though frequency analysis, Astron. Astrophys., 329 (1998), 451-481. Google Scholar

[11]

G. Pucacco, D. Boccaletti and C. Belmonte, Quantitative predictions with detuned normal forms, Celest. Mech. Dyn. Astron., 102 (2008), 163-176. doi: 10.1007/s10569-008-9141-x.  Google Scholar

[12]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer-Verlag, 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), 457-464. Google Scholar

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