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.

[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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505

[17]

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

[18]

Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439

[19]

Piotr Oprocha, Xinxing Wu. On averaged tracing of periodic average pseudo orbits. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4943-4957. doi: 10.3934/dcds.2017212

[20]

Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399

2016 Impact Factor: 0.857

Metrics

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

Other articles
by authors

[Back to Top]