November  2021, 41(11): 5183-5208. doi: 10.3934/dcds.2021073

Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates

1. 

Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, VIII Región, Chile

2. 

Departamento de Estadística, Informática y Matemáticas, Institute for Advanced Materials and Mathematics (INAMAT2), Universidad Pública de Navarra, 31006 Pamplona, Spain

3. 

Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, VIII Región, Chile

* Corresponding author: Jesús F. Palacián

Received  August 2020 Revised  March 2021 Published  November 2021 Early access  April 2021

Fund Project: The authors are partially supported by Projects MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain and MTM 2017-88137-C2-1-P of the Ministry of Science, Innovation and Universities of Spain. D. C.-D. acknowledges support from CONICYT PhD/2016-21161143. C. Vidal is partially supported by Fondecyt, grant 1180288

In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with $ n $ degrees of freedom we provide a criterion to obtain a type of formal stability, called Lie stability. Our result generalises previous approaches, as exponential stability in the sense of Nekhoroshev (excepting a few situations) and other classical results on formal stability of equilibria. In case of Lie stable systems we bound the solutions near the equilibrium over exponentially long times. Some examples are provided to illustrate our main contributions.

Citation: Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073
References:
[1]

G. BenettinF. Fassò and M. Guzzo, Nekhoroshev stability of $L_4$ and $L_5$ in the spatial restricted three body problem, Regul. Chaotic Dyn., 3 (1998), 56-72.  doi: 10.1070/rd1998v003n03ABEH000080.  Google Scholar

[2]

A. Bounemoura, B. Fayad and L. Niederman, Super-exponential stability for generic real-analytic elliptic equilibrium points, Adv. Math. 366 (2020), 107088, 30 pp. doi: 10.1016/j.aim.2020.107088.  Google Scholar

[3]

A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330.   Google Scholar

[4]

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.  Google Scholar

[5]

D. Cárcamo-Díaz, Stability and Estimates near Elliptic Equilibrium Points in Hamiltonian Systems and Applications, Ph.D thesis, Universidad del Bío-Bío in Concepción, Chile, 2019. Google Scholar

[6]

D. Cárcamo-Díaz and C. Vidal, Instability of equilibrium solutions of Hamiltonian systems with $n$-degrees of freedom under the existence of a single resonance and an invariant ray, J. Differential Equations, 265 (2018), 6295-6315.  doi: 10.1016/j.jde.2018.07.022.  Google Scholar

[7]

D. Cárcamo-Díaz and C. Vidal, Instability of equilibrium solutions of Hamiltonian systems with $n$-degrees of freedom under the existence of multiple resonances and an application to the spatial satellite problem, J. Dynam. Differential Equations, 31 (2019), 853-882.  doi: 10.1007/s10884-018-9679-6.  Google Scholar

[8]

D. Cárcamo-DíazJ. F. PalaciánC. Vidal and P. Yanguas, On the nonlinear stability of the triangular points in the circular spatial restricted three-body problem, Regul. Chaotic Dyn., 25 (2020), 131-148.  doi: 10.1134/S156035472002001X.  Google Scholar

[9]

D. Cárcamo-Díaz, J. F. Palacián, C. Vidal and P. Yanguas, Nonlinear stability in the spatial attitude motion of a satellite in a circular orbit, preprint, 2021. Google Scholar

[10]

P. ChartierA. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅲ: Error bounds, Found. Comput. Math., 15 (2015), 591-612.  doi: 10.1007/s10208-013-9175-7.  Google Scholar

[11]

H. S. DumasK. R. MeyerJ. F. Palacián and P. Yanguas, Asymptotic stability estimates near an equilibrium point, J. Differential Equations, 263 (2017), 1125-1139.  doi: 10.1016/j.jde.2017.03.011.  Google Scholar

[12]

F. FassòM. Guzzo and G. Benettin, Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, Commun. Math. Phys., 197 (1998), 347-360.  doi: 10.1007/s002200050454.  Google Scholar

[13]

B. Fayad, Lyapunov unstable elliptic equilibria, preprint, hal-02414963. Google Scholar

[14]

A. Giorgilli, Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium point, Ann. Inst. H. Poincaré Phys. Théor., 48 (1988), 423-439.   Google Scholar

[15]

J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 17 (1964), 509-526.  doi: 10.1002/cpa.3160170408.  Google Scholar

[16]

M. GuzzoL. Chierchia and G. Benettin, The steep Nekhoroshev's theorem, Commun. Math. Phys., 342 (2016), 569-601.  doi: 10.1007/s00220-015-2555-x.  Google Scholar

[17]

M. GuzzoE. Lega and C. Froeschlé, Diffusion and stability in perturbed non-convex integrable systems, Nonlinearity, 19 (2006), 1049-1067.  doi: 10.1088/0951-7715/19/5/003.  Google Scholar

[18]

L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances, Prikl. Mat. Mekh. 35 1971,423–431. doi: 10.1016/0021-8928(71)90006-2.  Google Scholar

[19]

L. G. Khazin, Interaction of third-order resonances in problems of the stability of Hamiltonian systems, Prikl. Mat. Mekh., 48 (1984), 496-498.  doi: 10.1016/0021-8928(84)90146-1.  Google Scholar

[20]

A. L. Kunitsyn and A. A. Perezhogin, The stability of neutral systems in the case of a multiple fourth-order resonance, Prikl. Mat. Mekh., 49 (1985), 57-63.  doi: 10.1016/0021-8928(85)90127-3.  Google Scholar

[21]

P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Math. Surveys, 47 (1992), 57-133.  doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar

[22]

A. P. Markeev, On the stability of the triangular libration points in the circular bounded three body problem, Prikl. Mat. Mekh., 33 (1969), 105-110.  doi: 10.1016/0021-8928(69)90117-8.  Google Scholar

[23]

A. P. Markeev, Libration Points in Celestial Mechanics and Space Dynamics, Moscow, Nauka (in Russian), 1978. Google Scholar

[24]

K. R. Meyer and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Applied Mathematical Sciences, 90, Springer, New York, 2017. doi: 10.1007/978-3-319-53691-0.  Google Scholar

[25]

K. R. MeyerJ. F. Palacián and P. Yanguas, Normally stable Hamiltonian systems, Discrete Contin. Dynam. Syst., 33 (2013), 1201-1214.  doi: 10.3934/dcds.2013.33.1201.  Google Scholar

[26]

J. Moser, Stabilitätsverhalten kanonischer Differentialgleichungssysteme, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, 6 (1955), 87-120.   Google Scholar

[27]

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.  Google Scholar

[28]

A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., 48 (1984), 197-204.  doi: 10.1016/0021-8928(84)90078-9.  Google Scholar

[29]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. I, Usp. Mat. Nauk., 32 (1977), 5-66.   Google Scholar

[30]

L. Niederman, Nonlinear stability around an elliptic equilibrium point in a Hamiltonian system, Nonlinearity, 11 (1998), 1465-1479.  doi: 10.1088/0951-7715/11/6/002.  Google Scholar

[31]

L. Niederman, Prevalence of exponential stability among nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, 27 (2007), 905-928.  doi: 10.1017/S0143385706000927.  Google Scholar

[32]

J. Pöschel, On Nekhoroshev's estimate at an elliptic equilibrium, Int. Math. Res. Not., 1999 (1999), 203-215.  doi: 10.1155/S1073792899000100.  Google Scholar

[33]

F. dos SantosJ. 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.  Google Scholar

[34]

F. dos Santos and C. Vidal, Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems in the case of multiple resonances, J. Differential Equations, 258 (2015), 3880-3901.  doi: 10.1016/j.jde.2015.01.044.  Google Scholar

[35]

F. dos Santos and C. Vidal, Stability of equilibrium solutions of Hamiltonian systems with $n$-degrees of freedom and single resonance in the critical case, J. Differential Equations, 264 (2018), 5152-5179.  doi: 10.1016/j.jde.2017.12.033.  Google Scholar

[36]

G. Schirinzi and M. Guzzo, On the formulation of new explicit conditions for steepness from a former result of N. N. Nekhoroshev, J. Math. Phys., 54 (2013), 072702, 23 pp. doi: 10.1063/1.4813059.  Google Scholar

[37]

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.  Google Scholar

[38]

V. E. Zhavnerchik, On the stability of autonomous systems in the presence of several resonances, Prikl. Mat. Mekh., 43 (1979), 229-234.   Google Scholar

show all references

References:
[1]

G. BenettinF. Fassò and M. Guzzo, Nekhoroshev stability of $L_4$ and $L_5$ in the spatial restricted three body problem, Regul. Chaotic Dyn., 3 (1998), 56-72.  doi: 10.1070/rd1998v003n03ABEH000080.  Google Scholar

[2]

A. Bounemoura, B. Fayad and L. Niederman, Super-exponential stability for generic real-analytic elliptic equilibrium points, Adv. Math. 366 (2020), 107088, 30 pp. doi: 10.1016/j.aim.2020.107088.  Google Scholar

[3]

A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330.   Google Scholar

[4]

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.  Google Scholar

[5]

D. Cárcamo-Díaz, Stability and Estimates near Elliptic Equilibrium Points in Hamiltonian Systems and Applications, Ph.D thesis, Universidad del Bío-Bío in Concepción, Chile, 2019. Google Scholar

[6]

D. Cárcamo-Díaz and C. Vidal, Instability of equilibrium solutions of Hamiltonian systems with $n$-degrees of freedom under the existence of a single resonance and an invariant ray, J. Differential Equations, 265 (2018), 6295-6315.  doi: 10.1016/j.jde.2018.07.022.  Google Scholar

[7]

D. Cárcamo-Díaz and C. Vidal, Instability of equilibrium solutions of Hamiltonian systems with $n$-degrees of freedom under the existence of multiple resonances and an application to the spatial satellite problem, J. Dynam. Differential Equations, 31 (2019), 853-882.  doi: 10.1007/s10884-018-9679-6.  Google Scholar

[8]

D. Cárcamo-DíazJ. F. PalaciánC. Vidal and P. Yanguas, On the nonlinear stability of the triangular points in the circular spatial restricted three-body problem, Regul. Chaotic Dyn., 25 (2020), 131-148.  doi: 10.1134/S156035472002001X.  Google Scholar

[9]

D. Cárcamo-Díaz, J. F. Palacián, C. Vidal and P. Yanguas, Nonlinear stability in the spatial attitude motion of a satellite in a circular orbit, preprint, 2021. Google Scholar

[10]

P. ChartierA. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅲ: Error bounds, Found. Comput. Math., 15 (2015), 591-612.  doi: 10.1007/s10208-013-9175-7.  Google Scholar

[11]

H. S. DumasK. R. MeyerJ. F. Palacián and P. Yanguas, Asymptotic stability estimates near an equilibrium point, J. Differential Equations, 263 (2017), 1125-1139.  doi: 10.1016/j.jde.2017.03.011.  Google Scholar

[12]

F. FassòM. Guzzo and G. Benettin, Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, Commun. Math. Phys., 197 (1998), 347-360.  doi: 10.1007/s002200050454.  Google Scholar

[13]

B. Fayad, Lyapunov unstable elliptic equilibria, preprint, hal-02414963. Google Scholar

[14]

A. Giorgilli, Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium point, Ann. Inst. H. Poincaré Phys. Théor., 48 (1988), 423-439.   Google Scholar

[15]

J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 17 (1964), 509-526.  doi: 10.1002/cpa.3160170408.  Google Scholar

[16]

M. GuzzoL. Chierchia and G. Benettin, The steep Nekhoroshev's theorem, Commun. Math. Phys., 342 (2016), 569-601.  doi: 10.1007/s00220-015-2555-x.  Google Scholar

[17]

M. GuzzoE. Lega and C. Froeschlé, Diffusion and stability in perturbed non-convex integrable systems, Nonlinearity, 19 (2006), 1049-1067.  doi: 10.1088/0951-7715/19/5/003.  Google Scholar

[18]

L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances, Prikl. Mat. Mekh. 35 1971,423–431. doi: 10.1016/0021-8928(71)90006-2.  Google Scholar

[19]

L. G. Khazin, Interaction of third-order resonances in problems of the stability of Hamiltonian systems, Prikl. Mat. Mekh., 48 (1984), 496-498.  doi: 10.1016/0021-8928(84)90146-1.  Google Scholar

[20]

A. L. Kunitsyn and A. A. Perezhogin, The stability of neutral systems in the case of a multiple fourth-order resonance, Prikl. Mat. Mekh., 49 (1985), 57-63.  doi: 10.1016/0021-8928(85)90127-3.  Google Scholar

[21]

P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Math. Surveys, 47 (1992), 57-133.  doi: 10.1070/RM1992v047n06ABEH000965.  Google Scholar

[22]

A. P. Markeev, On the stability of the triangular libration points in the circular bounded three body problem, Prikl. Mat. Mekh., 33 (1969), 105-110.  doi: 10.1016/0021-8928(69)90117-8.  Google Scholar

[23]

A. P. Markeev, Libration Points in Celestial Mechanics and Space Dynamics, Moscow, Nauka (in Russian), 1978. Google Scholar

[24]

K. R. Meyer and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Applied Mathematical Sciences, 90, Springer, New York, 2017. doi: 10.1007/978-3-319-53691-0.  Google Scholar

[25]

K. R. MeyerJ. F. Palacián and P. Yanguas, Normally stable Hamiltonian systems, Discrete Contin. Dynam. Syst., 33 (2013), 1201-1214.  doi: 10.3934/dcds.2013.33.1201.  Google Scholar

[26]

J. Moser, Stabilitätsverhalten kanonischer Differentialgleichungssysteme, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, 6 (1955), 87-120.   Google Scholar

[27]

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.  Google Scholar

[28]

A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., 48 (1984), 197-204.  doi: 10.1016/0021-8928(84)90078-9.  Google Scholar

[29]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. I, Usp. Mat. Nauk., 32 (1977), 5-66.   Google Scholar

[30]

L. Niederman, Nonlinear stability around an elliptic equilibrium point in a Hamiltonian system, Nonlinearity, 11 (1998), 1465-1479.  doi: 10.1088/0951-7715/11/6/002.  Google Scholar

[31]

L. Niederman, Prevalence of exponential stability among nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, 27 (2007), 905-928.  doi: 10.1017/S0143385706000927.  Google Scholar

[32]

J. Pöschel, On Nekhoroshev's estimate at an elliptic equilibrium, Int. Math. Res. Not., 1999 (1999), 203-215.  doi: 10.1155/S1073792899000100.  Google Scholar

[33]

F. dos SantosJ. 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.  Google Scholar

[34]

F. dos Santos and C. Vidal, Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems in the case of multiple resonances, J. Differential Equations, 258 (2015), 3880-3901.  doi: 10.1016/j.jde.2015.01.044.  Google Scholar

[35]

F. dos Santos and C. Vidal, Stability of equilibrium solutions of Hamiltonian systems with $n$-degrees of freedom and single resonance in the critical case, J. Differential Equations, 264 (2018), 5152-5179.  doi: 10.1016/j.jde.2017.12.033.  Google Scholar

[36]

G. Schirinzi and M. Guzzo, On the formulation of new explicit conditions for steepness from a former result of N. N. Nekhoroshev, J. Math. Phys., 54 (2013), 072702, 23 pp. doi: 10.1063/1.4813059.  Google Scholar

[37]

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.  Google Scholar

[38]

V. E. Zhavnerchik, On the stability of autonomous systems in the presence of several resonances, Prikl. Mat. Mekh., 43 (1979), 229-234.   Google Scholar

Figure 1.  On the left we plot the curves $ I_2 = 3 I_3 $ (blue) and $ 40 I_3^3 = I_2^3/2 $ (orange) showing that $ {\mathcal H}_6 $ changes sign in $ S $, hence Lie stability cannot be accomplished. On the right we consider $ {\mathcal H}_6 = 4 I_3^3 - I_2^3/3 $ and plot the curves $ I_2 = 3 I_3 $ (blue) and $ 4 I_3^3 = I_2^3/3 $ (orange) showing that the origin of $ \mathbb{R}^{6} $ is Lie stable for the Hamiltonian $ H_2 +{\mathcal H}_6 + \cdots $
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