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

Previous Article
Higher-order variational calculus on Lie algebroids
JGM Home
This Issue
Next Article

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
Related Papers

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.
[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.
[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.
[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.
[5]	P. Hartman, Ordinary Differential Equations,, Classics in Applied Mathematics, (2002). doi: 10.1137/1.9780898719222.
[6]	J. Moser and E. Zehnder, Notes on Dynamical Systems,, Courant Lecture Notes in Mathematics, (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)*, (2005), 23. 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, (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). 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. 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. doi: 10.1142/S0219887811005889.
[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.
[5]	P. Hartman, Ordinary Differential Equations,, Classics in Applied Mathematics, (2002). doi: 10.1137/1.9780898719222.
[6]	J. Moser and E. Zehnder, Notes on Dynamical Systems,, Courant Lecture Notes in Mathematics, (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)*, (2005), 23. 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, (1996). doi: 10.1007/978-3-642-61453-8.

[1]	Alessandra Celletti, Sara Di Ruzza. Periodic and quasi--periodic orbits of the dissipative standard map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151
[2]	Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595
[3]	Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[4]	Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264
[5]	Armengol Gasull, Héctor Giacomini, Maite Grau. On the stability of periodic orbits for differential systems in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 495-509. doi: 10.3934/dcdsb.2008.10.495
[6]	V. Afraimovich, T.R. Young. Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 691-704. doi: 10.3934/dcds.2000.6.691
[7]	Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015
[8]	Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177
[9]	Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505
[10]	Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[11]	Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[12]	Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621
[13]	Ciprian D. Coman. Dissipative effects in piecewise linear dynamics. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 163-177. doi: 10.3934/dcdsb.2003.3.163
[14]	Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331
[15]	Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005
[16]	Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889
[17]	Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301
[18]	Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997
[19]	Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[20]	Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261

2017 Impact Factor: 0.561

American Institute of Mathematical Sciences