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

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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

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

Abstract
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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

References:

[1]	P. Birtea, M. Boleanţu, M. Puta and R. M. Tudoran, Asymptotic stability for a class of metriplectic systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2771420. Google Scholar
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[7]	T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, II: A crash course in geometric mechanics,, in Geometric Mechanics and Symmetry* (eds. J. Montaldi and T. S. Ratiu)*, (2005), 23. doi: 10.1017/CBO9780511526367.003. Google Scholar
[8]	R. M. Tudoran, Affine Distributions on Riemannian Manifolds with Applications to Dissipative Dynamics,, J. Geom. Phys., (2015). doi: 10.1016/j.geomphys.2015.01.017. Google Scholar
[9]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, $2^{nd}$ edition, (1996). doi: 10.1007/978-3-642-61453-8. Google Scholar

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References:

[1]	P. Birtea, M. Boleanţu, M. Puta and R. M. Tudoran, Asymptotic stability for a class of metriplectic systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2771420. Google Scholar
[2]	P. Birtea and D. Comănescu, Asymptotic stability of dissipated Hamilton-Poisson systems,, SIAM J. Appl. Dyn. Syst., 8 (2009), 967. doi: 10.1137/080735217. Google Scholar
[3]	C. Dăniasă, A. Gîrban and R. M. Tudoran, New aspects on the geometry and dynamics of quadratic Hamiltonian systems on $(\mathfrak{so}(3))^{}$,, Int. J. Geom. Methods Mod. Phys.*, 8 (2011), 1695. doi: 10.1142/S0219887811005889. Google Scholar
[4]	A. Gasull, H. Giacomini and M. Grau, On the stability of periodic orbits for differential systems on $\mathbbR^n$,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 495. doi: 10.3934/dcdsb.2008.10.495. Google Scholar
[5]	P. Hartman, Ordinary Differential Equations,, Classics in Applied Mathematics, (2002). doi: 10.1137/1.9780898719222. Google Scholar
[6]	J. Moser and E. Zehnder, Notes on Dynamical Systems,, Courant Lecture Notes in Mathematics, (2005). Google Scholar
[7]	T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, II: A crash course in geometric mechanics,, in Geometric Mechanics and Symmetry* (eds. J. Montaldi and T. S. Ratiu)*, (2005), 23. doi: 10.1017/CBO9780511526367.003. Google Scholar
[8]	R. M. Tudoran, Affine Distributions on Riemannian Manifolds with Applications to Dissipative Dynamics,, J. Geom. Phys., (2015). doi: 10.1016/j.geomphys.2015.01.017. Google Scholar
[9]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, $2^{nd}$ edition, (1996). doi: 10.1007/978-3-642-61453-8. Google Scholar

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