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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 1, issue 1
Nonlin. Processes Geophys., 1, 31–40, 1994
https://doi.org/10.5194/npg-1-31-1994
© Author(s) 1994. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

Special issue: Including papers presented at the AGU Chapman Conference on...

Nonlin. Processes Geophys., 1, 31–40, 1994
https://doi.org/10.5194/npg-1-31-1994
© Author(s) 1994. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

  31 Mar 1994

31 Mar 1994

Periodic and homoclinic orbits in a toy climate model

M. Toner1 and A. D. Kirwan, Jr.2 M. Toner and A. D. Kirwan, Jr.
  • 1Dept. of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia, USA
  • 2Center for Coastal Physical Oceanography, Old Dominion University, Norfolk, Virginia, USA

Abstract. A two dimensional system of autonomous nonlinear ordinary differential equations models glacier growth and temperature changes on an idealized planet. We apply standard perturbative techniques from dynamical systems theory to study small amplitude periodic orbits about a constant equilibrium. The equations are put in cononical form and the local phase space topology is examined. Maximum and minimum periods of oscillation are obtained and related to the radius of the orbit. An adjacent equilibrium is shown to have saddle character and the inflowing and outflowing manifolds of this saddle are studied using numerical integration. The inflowing manifolds show the region of attraction for the periodic orbit. As the frequency gets small, the adjacent (saddle) equilibrium approaches the radius of the periodic orbit. The bifurcation of the periodic orbit to a stable homoclinic orbit is observed when an inflowing manifold and an outflowing manifold of the adjacent equilibrium cross.

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