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September  2014, 34(9): 3455-3469. doi: 10.3934/dcds.2014.34.3455

Periodic solutions of El Niño model through the Vallis differential system

1. 

Department of Mathematics, IBILCE, UNESP - Univ Estadual Paulista, Rua Cristovão Colombo, 2265, Jardim Nazareth, CEP 15.054-000, Sao José de Rio Preto, SP, Brazil

2. 

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

Received  April 2013 Revised  December 2013 Published  March 2014

By rescaling the variables, the parameters and the periodic function of the Vallis differential system we provide sufficient conditions for the existence of periodic solutions and we also characterize their kind of stability. The results are obtained using averaging theory.
Citation: Rodrigo Donizete Euzébio, Jaume Llibre. Periodic solutions of El Niño model through the Vallis differential system. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3455-3469. doi: 10.3934/dcds.2014.34.3455
References:
[1]

A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Comm. on Pure and Appl. Anal., 6 (2007), 103-111. doi: 10.3934/cpaa.2007.6.103.

[2]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Revised and Corrected Reprint of the 1983 Original, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990.

[3]

A. Kanatnikov and A. Krishchenko, Localization of invariant compact sets of nonautonomous systems, Differ. Equ., 45 (2009), 46-52. doi: 10.1134/S0012266109010054.

[4]

A. P. Krishchenko and K. E. Starkov, Localization of compact invariant sets of nonlinear time-varying systems, Intern. Journal of Bifurcation and Chaos, 18 (2008), 1599-1604. doi: 10.1142/S021812740802121X.

[5]

J. Llibre and C. Vidal, Periodic solutions of a periodic FitzHugh-Nagumo differential system,, to appear., (). 

[6]

I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.

[7]

M. Roseau, Vibrations Non Linéaires et Théorie de la Stabilité, (French) Springer Tracts in Natural Philosophy, Vol. 8 Springer-Verlag, Berlin-New York, 1966.

[8]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Method in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59. Springer, New York, 2007. xxii+431 pp.

[9]

D. Strozzi, On the Origin of Interannual and Irregular Behaviour in the El Niño Properties, Report of Department of Physics, Princeton University, available at the WEB, 1999.

[10]

G. K. Vallis, Conceptual models of El Niño and the southern oscillation, Geophys. Res., 93 (1988), 13979-13991. doi: 10.1029/JC093iC11p13979.

show all references

References:
[1]

A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Comm. on Pure and Appl. Anal., 6 (2007), 103-111. doi: 10.3934/cpaa.2007.6.103.

[2]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Revised and Corrected Reprint of the 1983 Original, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990.

[3]

A. Kanatnikov and A. Krishchenko, Localization of invariant compact sets of nonautonomous systems, Differ. Equ., 45 (2009), 46-52. doi: 10.1134/S0012266109010054.

[4]

A. P. Krishchenko and K. E. Starkov, Localization of compact invariant sets of nonlinear time-varying systems, Intern. Journal of Bifurcation and Chaos, 18 (2008), 1599-1604. doi: 10.1142/S021812740802121X.

[5]

J. Llibre and C. Vidal, Periodic solutions of a periodic FitzHugh-Nagumo differential system,, to appear., (). 

[6]

I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.

[7]

M. Roseau, Vibrations Non Linéaires et Théorie de la Stabilité, (French) Springer Tracts in Natural Philosophy, Vol. 8 Springer-Verlag, Berlin-New York, 1966.

[8]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Method in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59. Springer, New York, 2007. xxii+431 pp.

[9]

D. Strozzi, On the Origin of Interannual and Irregular Behaviour in the El Niño Properties, Report of Department of Physics, Princeton University, available at the WEB, 1999.

[10]

G. K. Vallis, Conceptual models of El Niño and the southern oscillation, Geophys. Res., 93 (1988), 13979-13991. doi: 10.1029/JC093iC11p13979.

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