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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 & 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.  Google Scholar [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.  Google Scholar [3] A. Kanatnikov and A. Krishchenko, Localization of invariant compact sets of nonautonomous systems, Differ. Equ., 45 (2009), 46-52. doi: 10.1134/S0012266109010054.  Google Scholar [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.  Google Scholar [5] J. Llibre and C. Vidal, Periodic solutions of a periodic FitzHugh-Nagumo differential system,, to appear., ().   Google Scholar [6] I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [10] G. K. Vallis, Conceptual models of El Niño and the southern oscillation, Geophys. Res., 93 (1988), 13979-13991. doi: 10.1029/JC093iC11p13979.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] A. Kanatnikov and A. Krishchenko, Localization of invariant compact sets of nonautonomous systems, Differ. Equ., 45 (2009), 46-52. doi: 10.1134/S0012266109010054.  Google Scholar [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.  Google Scholar [5] J. Llibre and C. Vidal, Periodic solutions of a periodic FitzHugh-Nagumo differential system,, to appear., ().   Google Scholar [6] I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [10] G. K. Vallis, Conceptual models of El Niño and the southern oscillation, Geophys. Res., 93 (1988), 13979-13991. doi: 10.1029/JC093iC11p13979.  Google Scholar
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