# American Institute of Mathematical Sciences

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
##### 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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##### 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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