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

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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

Abstract
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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).
[2]	C. Belomte, D. Bocaaletti and G. Pucacco, On the orbit structure of the logarithm potential,, Astrophys. J., 669 (2007), 202.
[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.
[4]	S. W. Hawkings, Arrow of time in cosmology,, Phys. Rev. D, 32 (1985), 2489. doi: 10.1103/PhysRevD.32.2489.
[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.
[6]	J. Llibre and A. Makhlouf, Periodic orbits of the generalized Friedmann-Robertson-Walker Hamiltonian systems,, Astrophys, 344* (2013), 46.*
[7]	D. Merrit and M. Valluri, Chaos and mixing in triaxial stellar systems,, Astrophys, 471* (1996), 82.*
[8]	D. Page, Will entroy decrease if the universe recollapses?,, Phys. Rev. D, 32 (1991), 2496. doi: 10.1103/PhysRevD.32.2496.
[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.*
[10]	Y. Papaphilippou and J. Laskar, Global dynamics of triaxial galactic models though frequency analysis,, Astron. Astrophys., 329* (1998), 451.*
[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.
[12]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996). doi: 10.1007/978-3-642-61453-8.
[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.

show all references

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, Benjamin, (1978).
[2]	C. Belomte, D. Bocaaletti and G. Pucacco, On the orbit structure of the logarithm potential,, Astrophys. J., 669 (2007), 202.
[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.
[4]	S. W. Hawkings, Arrow of time in cosmology,, Phys. Rev. D, 32 (1985), 2489. doi: 10.1103/PhysRevD.32.2489.
[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.
[6]	J. Llibre and A. Makhlouf, Periodic orbits of the generalized Friedmann-Robertson-Walker Hamiltonian systems,, Astrophys, 344* (2013), 46.*
[7]	D. Merrit and M. Valluri, Chaos and mixing in triaxial stellar systems,, Astrophys, 471* (1996), 82.*
[8]	D. Page, Will entroy decrease if the universe recollapses?,, Phys. Rev. D, 32 (1991), 2496. doi: 10.1103/PhysRevD.32.2496.
[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.*
[10]	Y. Papaphilippou and J. Laskar, Global dynamics of triaxial galactic models though frequency analysis,, Astron. Astrophys., 329* (1998), 451.*
[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.
[12]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996). doi: 10.1007/978-3-642-61453-8.
[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.

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