-
Previous Article
Computing collinear 4-Body Problem central configurations with given masses
- DCDS Home
- This Issue
-
Next Article
Reversibility and branching of periodic orbits
Normally stable hamiltonian systems
1. | Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, United States |
2. | Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain, Spain |
References:
[1] |
G. Benettin, F. Fassò and M. Guzzo, Nekhoroshev-stability of $L_4$ and $L_5$ in the spatial restricted three-body problem, Regul. Chaotic Dyn., 3 (1998), 56-71. |
[2] |
A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330; English translation, Math. Notes, 1 (1967), 216-219. |
[3] |
H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems, Nonlinearity, 12 (1999), 1351-1362.
doi: 10.1088/0951-7715/12/5/309. |
[4] |
T. M. Cherry, On periodic solutions of Hamiltonian systems of differential equations, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 227 (1928), 137-221.
doi: 10.1098/rsta.1928.0005. |
[5] |
N. G. Chetaev, Un théorème sur l'instabilité, Dokl. Akad. Nauk SSSR, 1 (1934), 529-531. |
[6] |
G. L. Dirichlet, Über die stabilität des Gleichgewichts, J. Reine Angew. Math., 32 (1846), 85-88.
doi: 10.1515/crll.1846.32.85. |
[7] |
F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Arch. Ration. Mech. Anal., 158 (2001), 259-292.
doi: 10.1007/PL00004245. |
[8] |
A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, J. Differential Equations, 77 (1989), 167-198.
doi: 10.1016/0022-0396(89)90161-7. |
[9] |
J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 16 (1963), 509-526. |
[10] |
L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances, Prikl. Mat. Mekh., 35 (1971), 423-431; English translation, J. Appl. Math. Mech., 35 (1971), 384-391. |
[11] |
A. L. Kunitsyn and A. A. Tuyakbayev, The stability of Hamiltonian systems in the case of a multiple fourth-order resonance, Prikl. Mat. Mekh., 56 (1992), 672-675; English translation, J. Appl. Math. Mech., 56 (1992), 572-576. |
[12] |
A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203-474; French translation of the 1892 Russian article. |
[13] |
K. R. Meyer, Normal forms for Hamiltonian systems, Celestial Mech., 9 (1974), 517-522. |
[14] |
K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," $2^{nd}$ edition, Springer-Verlag, New York, 2009. |
[15] |
J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87-120. |
[16] |
J. Moser, New aspects in the theory of stability of Hamiltonian systems, Comm. Pure Appl. Math., 11 (1958), 81-114.
doi: 10.1002/cpa.3160110105. |
[17] |
J. Moser, On the elimination of the irrationality condition and Birkhoff's concept of complete stability, Bol. Soc. Mat. Mexicana (2), 5 (1960), 167-175. |
[18] |
J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747; addendum in: Comm. Pure Appl. Math., 31 (1978), 529-530.
doi: 10.1002/cpa.3160290613. |
[19] |
F. dos Santos, J. E. Mansilla and C. Vidal, Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems with $n$-degrees of freedom in the case of single resonance, J. Dynam. Differential Equations, 22 (2010), 805-821.
doi: 10.1007/s10884-010-9176-z. |
[20] |
C. L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Differential gleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., 128 (1954), 144-170.
doi: 10.1007/BF01360131. |
[21] |
V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, New York, 1967. |
[22] |
C. Vidal, Stability of equilibrium positions of Hamiltonian systems, Qual. Theory Dyn. Syst., 7 (2008), 253-294. |
[23] |
A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[24] |
A. Weinstein, Symplectic $V$-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math., 30 (1977), 265-271.
doi: 10.1002/cpa.3160300207. |
[25] |
A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), 235-248.
doi: 10.1007/BF01214573. |
[26] |
V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," Vol. 1 and 2, John Wiley & Sons, New York, 1975. |
show all references
References:
[1] |
G. Benettin, F. Fassò and M. Guzzo, Nekhoroshev-stability of $L_4$ and $L_5$ in the spatial restricted three-body problem, Regul. Chaotic Dyn., 3 (1998), 56-71. |
[2] |
A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330; English translation, Math. Notes, 1 (1967), 216-219. |
[3] |
H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems, Nonlinearity, 12 (1999), 1351-1362.
doi: 10.1088/0951-7715/12/5/309. |
[4] |
T. M. Cherry, On periodic solutions of Hamiltonian systems of differential equations, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 227 (1928), 137-221.
doi: 10.1098/rsta.1928.0005. |
[5] |
N. G. Chetaev, Un théorème sur l'instabilité, Dokl. Akad. Nauk SSSR, 1 (1934), 529-531. |
[6] |
G. L. Dirichlet, Über die stabilität des Gleichgewichts, J. Reine Angew. Math., 32 (1846), 85-88.
doi: 10.1515/crll.1846.32.85. |
[7] |
F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Arch. Ration. Mech. Anal., 158 (2001), 259-292.
doi: 10.1007/PL00004245. |
[8] |
A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, J. Differential Equations, 77 (1989), 167-198.
doi: 10.1016/0022-0396(89)90161-7. |
[9] |
J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 16 (1963), 509-526. |
[10] |
L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances, Prikl. Mat. Mekh., 35 (1971), 423-431; English translation, J. Appl. Math. Mech., 35 (1971), 384-391. |
[11] |
A. L. Kunitsyn and A. A. Tuyakbayev, The stability of Hamiltonian systems in the case of a multiple fourth-order resonance, Prikl. Mat. Mekh., 56 (1992), 672-675; English translation, J. Appl. Math. Mech., 56 (1992), 572-576. |
[12] |
A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203-474; French translation of the 1892 Russian article. |
[13] |
K. R. Meyer, Normal forms for Hamiltonian systems, Celestial Mech., 9 (1974), 517-522. |
[14] |
K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," $2^{nd}$ edition, Springer-Verlag, New York, 2009. |
[15] |
J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87-120. |
[16] |
J. Moser, New aspects in the theory of stability of Hamiltonian systems, Comm. Pure Appl. Math., 11 (1958), 81-114.
doi: 10.1002/cpa.3160110105. |
[17] |
J. Moser, On the elimination of the irrationality condition and Birkhoff's concept of complete stability, Bol. Soc. Mat. Mexicana (2), 5 (1960), 167-175. |
[18] |
J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747; addendum in: Comm. Pure Appl. Math., 31 (1978), 529-530.
doi: 10.1002/cpa.3160290613. |
[19] |
F. dos Santos, J. E. Mansilla and C. Vidal, Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems with $n$-degrees of freedom in the case of single resonance, J. Dynam. Differential Equations, 22 (2010), 805-821.
doi: 10.1007/s10884-010-9176-z. |
[20] |
C. L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Differential gleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., 128 (1954), 144-170.
doi: 10.1007/BF01360131. |
[21] |
V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, New York, 1967. |
[22] |
C. Vidal, Stability of equilibrium positions of Hamiltonian systems, Qual. Theory Dyn. Syst., 7 (2008), 253-294. |
[23] |
A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[24] |
A. Weinstein, Symplectic $V$-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math., 30 (1977), 265-271.
doi: 10.1002/cpa.3160300207. |
[25] |
A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), 235-248.
doi: 10.1007/BF01214573. |
[26] |
V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," Vol. 1 and 2, John Wiley & Sons, New York, 1975. |
[1] |
Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 |
[2] |
Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612 |
[3] |
Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317 |
[4] |
Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529 |
[5] |
Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure and Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703 |
[6] |
Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations and Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207 |
[7] |
Mark A. Pinsky, Alexandr A. Zevin. Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 243-250. doi: 10.3934/dcds.2005.12.243 |
[8] |
Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734 |
[9] |
Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67 |
[10] |
Luca Biasco, Luigi Chierchia. On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 233-262. doi: 10.3934/dcds.2003.9.233 |
[11] |
Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073 |
[12] |
Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 279-314. doi: 10.3934/dcdsb.2007.8.279 |
[13] |
Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357 |
[14] |
Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661 |
[15] |
Nguyen Hai Son. Solution stability to parametric distributed optimal control problems with finite unilateral constraints. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021047 |
[16] |
Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1 |
[17] |
Ugo Boscain, Grégoire Charlot, Mario Sigalotti. Stability of planar nonlinear switched systems. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 415-432. doi: 10.3934/dcds.2006.15.415 |
[18] |
Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations and Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493 |
[19] |
J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625 |
[20] |
D. J. W. Simpson. On the stability of boundary equilibria in Filippov systems. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3093-3111. doi: 10.3934/cpaa.2021097 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]