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

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.
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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