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March  2015, 7(1): 109-124. doi: 10.3934/jgm.2015.7.109

On the control of stability of periodic orbits of completely integrable systems

Răzvan M. Tudoran 1,

1. 

The West University of Timişoara, Faculty of Mathematics and C.S., Department of Mathematics, B-dul. Vasile Pârvan, No. 4, 300223 - Timişoara, Romania

Received  February 2014 Revised  January 2015 Published  March 2015

We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.
Keywords: characteristic multipliers, stability, periodic orbits, Dissipative dynamics, control..
Mathematics Subject Classification: Primary: 37C27, 37C75; Secondary: 34C2.
Citation: Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109
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show all references

References:
[1]

J. Math. Phys., 48 (2007), 082703, 7pp. doi: 10.1063/1.2771420.  Google Scholar

[2]

SIAM J. Appl. Dyn. Syst., 8 (2009), 967-976. doi: 10.1137/080735217.  Google Scholar

[3]

Int. J. Geom. Methods Mod. Phys., 8 (2011), 1695-1721. doi: 10.1142/S0219887811005889.  Google Scholar

[4]

Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 495-509. doi: 10.3934/dcdsb.2008.10.495.  Google Scholar

[5]

Classics in Applied Mathematics, 38, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[6]

Courant Lecture Notes in Mathematics, 12, American Mathematical Society, Providence, 2005.  Google Scholar

[7]

in Geometric Mechanics and Symmetry (eds. J. Montaldi and T. S. Ratiu), London Mathematical Society Lecture Notes Series, 306, Cambridge University Press, Cambridge, 2005, 23-156. doi: 10.1017/CBO9780511526367.003.  Google Scholar

[8]

J. Geom. Phys., (2015). doi: 10.1016/j.geomphys.2015.01.017.  Google Scholar

[9]

$2^{nd}$ edition, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.  Google Scholar

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