Periodic solutions of El Niño model through the Vallis differential system. (English) Zbl 1302.86010
Summary: 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.
MSC:
| 86A10 | Meteorology and atmospheric physics |
| 37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
References:
| [1] | A. Buică, Periodic solutions of nonlinear periodic differential systems with a small parameter,, Comm. on Pure and Appl. Anal., 6, 103 (2007) · Zbl 1139.34036 · doi:10.3934/cpaa.2007.6.103 |
| [2] | J. Guckenheimer, <em>Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Revised and Corrected Reprint of the 1983 Original</em>,, Applied Mathematical Sciences (1990) |
| [3] | A. Kanatnikov, Localization of invariant compact sets of nonautonomous systems,, Differ. Equ., 45, 46 (2009) · Zbl 1183.34061 · doi:10.1134/S0012266109010054 |
| [4] | A. P. Krishchenko, Localization of compact invariant sets of nonlinear time-varying systems,, Intern. Journal of Bifurcation and Chaos, 18, 1599 (2008) · Zbl 1147.34326 · doi:10.1142/S021812740802121X |
| [5] | J. Llibre, Periodic solutions of a periodic FitzHugh-Nagumo differential system,, to appear. · Zbl 1330.34070 |
| [6] | I. G. Malkin, <em>Some Problems of the Theory of Nonlinear Oscillations</em>,, (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit. (1956) · Zbl 0070.08703 |
| [7] | M. Roseau, <em>Vibrations Non Linéaires et Théorie de la Stabilité</em>,, (French) Springer Tracts in Natural Philosophy (1966) · Zbl 0135.30603 |
| [8] | J. A. Sanders, <em>Averaging Method in Nonlinear Dynamical Systems</em>,, Applied Mathematical Sciences (2007) · Zbl 1128.34001 |
| [9] | D. Strozzi, <em>On the Origin of Interannual and Irregular Behaviour in the El Niño Properties</em>,, Report of Department of Physics (1999) |
| [10] | G. K. Vallis, Conceptual models of El Niño and the southern oscillation,, Geophys. Res., 93, 13979 (1988) · doi:10.1029/JC093iC11p13979 |
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