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September  2008, 10(2&3, September): 495-509. doi: 10.3934/dcdsb.2008.10.495

On the stability of periodic orbits for differential systems in $\mathbb{R}^n$

Armengol Gasull 1, , Héctor Giacomini 2, and  Maite Grau 3,

1. 

Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain
2. 

Lab. de Mathématiques et Physique Théorique, CNRS UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200 Tours, France
3. 

Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain

Received  October 2006 Revised  April 2007 Published  June 2008

We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of $n-1$ codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.
Keywords: rigid body dynamics, Periodic orbit, characteristic multipliers, Steklov periodic orbit., invariant curve, Mathieu's equation.
Mathematics Subject Classification: Primary: 34D08, 37D05; Secondary: 70E5.
Citation: Armengol Gasull, Héctor Giacomini, Maite Grau. On the stability of periodic orbits for differential systems in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 495-509. doi: 10.3934/dcdsb.2008.10.495
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