March  2013, 33(3): 1201-1214. doi: 10.3934/dcds.2013.33.1201

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

Received  May 2011 Revised  November 2011 Published  October 2012

We study the stability of an equilibrium point of a Hamiltonian system with $n$ degrees of freedom. A new concept of stability called normal stability is given which applies to a system in normal form and relies on the existence of a formal integral whose quadratic part is positive definite. We give a necessary and sufficient condition for normal stability. This condition depends only on the quadratic terms of the Hamiltonian. We relate normal stability with formal stability and Liapunov stability. An application to the stability of the $L_4$ and $L_5$ equilibrium points of the spatial circular restricted three body problem is given.
Citation: Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201
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.   Google Scholar

[2]

A. D. Bryuno, Formal stability of Hamiltonian systems,, Mat. Zametki, 1 (1967), 325.   Google Scholar

[3]

H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems,, Nonlinearity, 12 (1999), 1351.  doi: 10.1088/0951-7715/12/5/309.  Google Scholar

[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.  doi: 10.1098/rsta.1928.0005.  Google Scholar

[5]

N. G. Chetaev, Un théorème sur l'instabilité,, Dokl. Akad. Nauk SSSR, 1 (1934), 529.   Google Scholar

[6]

G. L. Dirichlet, Über die stabilität des Gleichgewichts,, J. Reine Angew. Math., 32 (1846), 85.  doi: 10.1515/crll.1846.32.85.  Google Scholar

[7]

F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids,, Arch. Ration. Mech. Anal., 158 (2001), 259.  doi: 10.1007/PL00004245.  Google Scholar

[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.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[9]

J. Glimm, Formal stability of Hamiltonian systems,, Comm. Pure Appl. Math., 16 (1963), 509.   Google Scholar

[10]

L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances,, Prikl. Mat. Mekh., 35 (1971), 423.   Google Scholar

[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.   Google Scholar

[12]

A. Liapounoff, Problème général de la stabilité du mouvement,, Ann. Fac. Sci. Toulouse, 9 (1907), 203.   Google Scholar

[13]

K. R. Meyer, Normal forms for Hamiltonian systems,, Celestial Mech., 9 (1974), 517.   Google Scholar

[14]

K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem,", $2^{nd}$ edition, (2009).   Google Scholar

[15]

J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87.   Google Scholar

[16]

J. Moser, New aspects in the theory of stability of Hamiltonian systems,, Comm. Pure Appl. Math., 11 (1958), 81.  doi: 10.1002/cpa.3160110105.  Google Scholar

[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.   Google Scholar

[18]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein,, Comm. Pure Appl. Math., 29 (1976), 727.  doi: 10.1002/cpa.3160290613.  Google Scholar

[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.  doi: 10.1007/s10884-010-9176-z.  Google Scholar

[20]

C. L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Differential gleichungen in der Nähe einer Gleichgewichtslösung,, Math. Ann., 128 (1954), 144.  doi: 10.1007/BF01360131.  Google Scholar

[21]

V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies,", Academic Press, (1967).   Google Scholar

[22]

C. Vidal, Stability of equilibrium positions of Hamiltonian systems,, Qual. Theory Dyn. Syst., 7 (2008), 253.   Google Scholar

[23]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems,, Invent. Math., 20 (1973), 47.  doi: 10.1007/BF01405263.  Google Scholar

[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.  doi: 10.1002/cpa.3160300207.  Google Scholar

[25]

A. Weinstein, Bifurcations and Hamilton's principle,, Math. Z., 159 (1978), 235.  doi: 10.1007/BF01214573.  Google Scholar

[26]

V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients,", Vol. 1 and 2, (1975).   Google Scholar

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.   Google Scholar

[2]

A. D. Bryuno, Formal stability of Hamiltonian systems,, Mat. Zametki, 1 (1967), 325.   Google Scholar

[3]

H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems,, Nonlinearity, 12 (1999), 1351.  doi: 10.1088/0951-7715/12/5/309.  Google Scholar

[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.  doi: 10.1098/rsta.1928.0005.  Google Scholar

[5]

N. G. Chetaev, Un théorème sur l'instabilité,, Dokl. Akad. Nauk SSSR, 1 (1934), 529.   Google Scholar

[6]

G. L. Dirichlet, Über die stabilität des Gleichgewichts,, J. Reine Angew. Math., 32 (1846), 85.  doi: 10.1515/crll.1846.32.85.  Google Scholar

[7]

F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids,, Arch. Ration. Mech. Anal., 158 (2001), 259.  doi: 10.1007/PL00004245.  Google Scholar

[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.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[9]

J. Glimm, Formal stability of Hamiltonian systems,, Comm. Pure Appl. Math., 16 (1963), 509.   Google Scholar

[10]

L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances,, Prikl. Mat. Mekh., 35 (1971), 423.   Google Scholar

[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.   Google Scholar

[12]

A. Liapounoff, Problème général de la stabilité du mouvement,, Ann. Fac. Sci. Toulouse, 9 (1907), 203.   Google Scholar

[13]

K. R. Meyer, Normal forms for Hamiltonian systems,, Celestial Mech., 9 (1974), 517.   Google Scholar

[14]

K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem,", $2^{nd}$ edition, (2009).   Google Scholar

[15]

J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87.   Google Scholar

[16]

J. Moser, New aspects in the theory of stability of Hamiltonian systems,, Comm. Pure Appl. Math., 11 (1958), 81.  doi: 10.1002/cpa.3160110105.  Google Scholar

[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.   Google Scholar

[18]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein,, Comm. Pure Appl. Math., 29 (1976), 727.  doi: 10.1002/cpa.3160290613.  Google Scholar

[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.  doi: 10.1007/s10884-010-9176-z.  Google Scholar

[20]

C. L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Differential gleichungen in der Nähe einer Gleichgewichtslösung,, Math. Ann., 128 (1954), 144.  doi: 10.1007/BF01360131.  Google Scholar

[21]

V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies,", Academic Press, (1967).   Google Scholar

[22]

C. Vidal, Stability of equilibrium positions of Hamiltonian systems,, Qual. Theory Dyn. Syst., 7 (2008), 253.   Google Scholar

[23]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems,, Invent. Math., 20 (1973), 47.  doi: 10.1007/BF01405263.  Google Scholar

[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.  doi: 10.1002/cpa.3160300207.  Google Scholar

[25]

A. Weinstein, Bifurcations and Hamilton's principle,, Math. Z., 159 (1978), 235.  doi: 10.1007/BF01214573.  Google Scholar

[26]

V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients,", Vol. 1 and 2, (1975).   Google Scholar

[1]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[2]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[3]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[4]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[5]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[6]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079

[7]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[8]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[9]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[10]

Akio Matsumot, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021069

[11]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068

[12]

Yongjian Liu, Qiujian Huang, Zhouchao Wei. Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3357-3380. doi: 10.3934/dcdsb.2020235

[13]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

[14]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

[15]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[16]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[17]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[18]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[19]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[20]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (3)

[Back to Top]