January  2016, 21(1): 1-11. doi: 10.3934/dcdsb.2016.21.1

Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems

1. 

Departamento de Matemática, Faculdade de Ciências, UNESP, Av. Eng. Luiz Edmundo Carrijo Coube 14-01, CEP 17033-360, Bauru, SP, Brazil

2. 

Departamento de Matemática, IMECC-UNICAMP, R. Sérgio Buarque de Holanda, 651, CEP 13083-970, Campinas, SP, Brazil

3. 

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

4. 

Universidade Federal de Goiás, IME, CEP 74001-970, Caixa Postal 131, Goiânia, Goiás, Brazil

Received  April 2015 Revised  May 2015 Published  November 2015

Using the averaging theory we study the periodic solutions and their linear stability of the $3$--dimensional chaotic quadratic polynomial differential systems without equilibria studied in [3]. All these differential systems depend only on one--parameter.
Citation: Tiago de Carvalho, Rodrigo Donizete Euzébio, Jaume Llibre, Durval José Tonon. Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 1-11. doi: 10.3934/dcdsb.2016.21.1
References:
[1]

R. D. Euzébio and J. Llibre, Periodic solutions of El Niño model through the Vallis differential system,, DCDS-A, 34 (2014), 3455.  doi: 10.3934/dcds.2014.34.3455.  Google Scholar

[2]

S. Jafari and J. C. Sprott, Simple chaotic flows with a line equilibrium,, Chaos Solit. Fract., 57 (2013), 79.  doi: 10.1016/j.chaos.2013.08.018.  Google Scholar

[3]

S. Jafari, J. C. Sprott and S. M. R. H. Golpayegani, Elementary quadratic chaotic flows with no equilibria,, Physics Letters A, 377 (2013), 699.  doi: 10.1016/j.physleta.2013.01.009.  Google Scholar

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F. Legoll, M. Luskin and R. Moeckel, Non-ergodicity of Nosé-Hoover dynamics,, Nonlinearity, 22 (2009), 1673.  doi: 10.1088/0951-7715/22/7/011.  Google Scholar

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A. Mahdi and C. Valls, Integrability of the Nosé-Hoover equation,, J. Geo. Phy., 61 (2011), 1348.  doi: 10.1016/j.geomphys.2011.02.018.  Google Scholar

[6]

V. T. Pham, C. Volos, S. Jafari, Z. Wei and X. Wang, Constructing a novel no-equilibrium chaotic system,, Int. J. Bifuc. Chaos, 24 (2014).  doi: 10.1142/S0218127414500734.  Google Scholar

[7]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems,, Second edition, 59 (2007).   Google Scholar

[8]

J. C. Sprott, Some simple chaotic flows,, Phys. Rev. A, 50 (1994).  doi: 10.1103/PhysRevE.50.R647.  Google Scholar

[9]

P. Swinnerton-Dyer and T. Wagenknecht, Some third-order ordinary differential equations,, London Mathematical Society, 40 (2008), 725.  doi: 10.1112/blms/bdn046.  Google Scholar

[10]

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

show all references

References:
[1]

R. D. Euzébio and J. Llibre, Periodic solutions of El Niño model through the Vallis differential system,, DCDS-A, 34 (2014), 3455.  doi: 10.3934/dcds.2014.34.3455.  Google Scholar

[2]

S. Jafari and J. C. Sprott, Simple chaotic flows with a line equilibrium,, Chaos Solit. Fract., 57 (2013), 79.  doi: 10.1016/j.chaos.2013.08.018.  Google Scholar

[3]

S. Jafari, J. C. Sprott and S. M. R. H. Golpayegani, Elementary quadratic chaotic flows with no equilibria,, Physics Letters A, 377 (2013), 699.  doi: 10.1016/j.physleta.2013.01.009.  Google Scholar

[4]

F. Legoll, M. Luskin and R. Moeckel, Non-ergodicity of Nosé-Hoover dynamics,, Nonlinearity, 22 (2009), 1673.  doi: 10.1088/0951-7715/22/7/011.  Google Scholar

[5]

A. Mahdi and C. Valls, Integrability of the Nosé-Hoover equation,, J. Geo. Phy., 61 (2011), 1348.  doi: 10.1016/j.geomphys.2011.02.018.  Google Scholar

[6]

V. T. Pham, C. Volos, S. Jafari, Z. Wei and X. Wang, Constructing a novel no-equilibrium chaotic system,, Int. J. Bifuc. Chaos, 24 (2014).  doi: 10.1142/S0218127414500734.  Google Scholar

[7]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems,, Second edition, 59 (2007).   Google Scholar

[8]

J. C. Sprott, Some simple chaotic flows,, Phys. Rev. A, 50 (1994).  doi: 10.1103/PhysRevE.50.R647.  Google Scholar

[9]

P. Swinnerton-Dyer and T. Wagenknecht, Some third-order ordinary differential equations,, London Mathematical Society, 40 (2008), 725.  doi: 10.1112/blms/bdn046.  Google Scholar

[10]

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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