American Institute of Mathematical Sciences

Advanced
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).  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

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.  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

[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

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences