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

Tiago de  Carvalho; Rodrigo Donizete  Euzébio; Jaume  Llibre; Durval José  Tonon

doi:10.3934/dcdsb.2016.21.1

Article Contents

2016, Volume 21, Issue 1: 1-11. Doi: 10.3934/dcdsb.2016.21.1

This issue Previous Article Next Article A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat

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
2.
Departamento de Matemática, IMECC-UNICAMP, R. Sérgio Buarque de Holanda, 651, CEP 13083-970, Campinas, SP
3.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
4.
Universidade Federal de Goiás, IME, CEP 74001-970, Caixa Postal 131, Goiânia, Goiás

Received: April 2015

Revised: May 2015

Published online: November 2015

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Periodic solutions,
averaging theory,
quadratic polynomial differential system,
chaotic systems,
limit cycles.

Mathematics Subject Classification: Primary: 34C05, 37C27.

Citation:

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References

[1]	R. D. Euzébio and J. Llibre, Periodic solutions of El Niño model through the Vallis differential system, DCDS-A, 34 (2014), 3455-3469.doi: 10.3934/dcds.2014.34.3455.
[2]	S. Jafari and J. C. Sprott, Simple chaotic flows with a line equilibrium, Chaos Solit. Fract., 57 (2013), 79-84.doi: 10.1016/j.chaos.2013.08.018.
[3]	S. Jafari, J. C. Sprott and S. M. R. H. Golpayegani, Elementary quadratic chaotic flows with no equilibria, Physics Letters A, 377 (2013), 699-702.doi: 10.1016/j.physleta.2013.01.009.
[4]	F. Legoll, M. Luskin and R. Moeckel, Non-ergodicity of Nosé-Hoover dynamics, Nonlinearity, 22 (2009), 1673-1694.doi: 10.1088/0951-7715/22/7/011.
[5]	A. Mahdi and C. Valls, Integrability of the Nosé-Hoover equation, J. Geo. Phy., 61 (2011), 1348-1352.doi: 10.1016/j.geomphys.2011.02.018.
[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), 1450073, 6pp.doi: 10.1142/S0218127414500734.
[7]	J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Second edition, Applied Mathematical Sciences 59, Springer, New York, 2007.
[8]	J. C. Sprott, Some simple chaotic flows, Phys. Rev. A, 50 (1994), R647-R650.doi: 10.1103/PhysRevE.50.R647.
[9]	P. Swinnerton-Dyer and T. Wagenknecht, Some third-order ordinary differential equations, London Mathematical Society, 40 (2008), 725-748.doi: 10.1112/blms/bdn046.
[10]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext Springer Verlag, 1996.doi: 10.1007/978-3-642-61453-8.