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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
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.  Google Scholar

[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.  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), 1695-1721. 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), 495-509. doi: 10.3934/dcdsb.2008.10.495.  Google Scholar

[5]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[6]

J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12, American Mathematical Society, Providence, 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), London Mathematical Society Lecture Notes Series, 306, Cambridge University Press, Cambridge, 2005, 23-156. 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, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.  Google Scholar

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.  Google Scholar

[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.  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), 1695-1721. 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), 495-509. doi: 10.3934/dcdsb.2008.10.495.  Google Scholar

[5]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[6]

J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12, American Mathematical Society, Providence, 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), London Mathematical Society Lecture Notes Series, 306, Cambridge University Press, Cambridge, 2005, 23-156. 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, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.  Google Scholar

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