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

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

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), 082703, 7pp. doi: 10.1063/1.2771420.
[2]	P. Birtea and D. Comănescu, Asymptotic stability of dissipated Hamilton-Poisson systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 967-976. doi: 10.1137/080735217.
[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), 1695-1721. doi: 10.1142/S0219887811005889.
[4]	A. Gasull, H. Giacomini and M. Grau, On the stability of periodic orbits for differential systems on $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 495-509. doi: 10.3934/dcdsb.2008.10.495.
[5]	P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222.
[6]	J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12, American Mathematical Society, Providence, 2005.
[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), London Mathematical Society Lecture Notes Series, 306, Cambridge University Press, Cambridge, 2005, 23-156. doi: 10.1017/CBO9780511526367.003.
[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.
[9]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, $2^{nd}$ edition, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.

show all references

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), 082703, 7pp. doi: 10.1063/1.2771420.
[2]	P. Birtea and D. Comănescu, Asymptotic stability of dissipated Hamilton-Poisson systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 967-976. doi: 10.1137/080735217.
[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), 1695-1721. doi: 10.1142/S0219887811005889.
[4]	A. Gasull, H. Giacomini and M. Grau, On the stability of periodic orbits for differential systems on $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 495-509. doi: 10.3934/dcdsb.2008.10.495.
[5]	P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222.
[6]	J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12, American Mathematical Society, Providence, 2005.
[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), London Mathematical Society Lecture Notes Series, 306, Cambridge University Press, Cambridge, 2005, 23-156. doi: 10.1017/CBO9780511526367.003.
[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.
[9]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, $2^{nd}$ edition, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.

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