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

Daniela Cárcamo-Díaz; Jesús F. Palacián; Claudio Vidal; Patricia Yanguas

doi:10.3934/dcds.2021073

Article Contents

2021, Volume 41, Issue 11: 5183-5208. Doi: 10.3934/dcds.2021073

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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
^* Corresponding author: Jesús F. Palacián

Received: July 31, 2020

Revised: February 28, 2021

Early access: April 2021

Published: November 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37J25, 70H14; Secondary: 70K42.

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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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