Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit
The purpose of thiswork is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant periodic orbit, as a function of the distance to the orbit. We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian s...
| Authors: | ; ; |
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| Format: | Article |
| Publication date: | 2003 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/912 |
| Online Access: | https://hdl.handle.net/2117/912 |
| Access Level: | Open access |
| Keyword: | Differential equations Bifurcation theory Hamiltonian systems Quantitative estimates periodic orbit Equacions diferencials ordinàries Bifurcació, Teoria de la Hamilton, Sistemes de Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory |
| _version_ | 1878732175271002112 |
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| author | Ollé Torner, Mercè|||0000-0002-8050-9055 |
| author2 | Pacha Andújar, Juan Ramón|||0000-0003-4599-3141 Villanueva Castelltort, Jordi|||0000-0001-8725-2785 |
| author2_role | author author |
| author_browse | Ollé Torner, Mercè|||0000-0002-8050-9055 Pacha Andújar, Juan Ramón|||0000-0003-4599-3141 Villanueva Castelltort, Jordi|||0000-0001-8725-2785 |
| author_facet | Ollé Torner, Mercè|||0000-0002-8050-9055 Pacha Andújar, Juan Ramón|||0000-0003-4599-3141 Villanueva Castelltort, Jordi|||0000-0001-8725-2785 |
| author_role | author |
| collection | UPCommons. Portal del coneixement obert de la UPC |
| description | The purpose of thiswork is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant periodic orbit, as a function of the distance to the orbit. We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian system having a pairwise collision of its non-trivial characteristic multipliers on the unit circle. Under generic hypotheses of non-resonance and non-degeneracy of the collision, we present a constructive methodology to reduce the Hamiltonian around the orbit to its (integrable) normal form, up to any given order. This constructive process allows to obtain quantitative estimates for the size of the remainder of the normal form, as a function of the normalizing order. By selecting appropriately this order in terms of the distance R to the resonant orbit (measured using suitable coordinates), r = ropt(R) := 2 + ?exp(W(log(1/R1/(τ+1+ε))))?, we have proved that the size of the remainder can be bounded (for small R) by Rropt(R)/2. Here, W(·) stands for Lambert’s W function and verifies that W(z) exp(W(z)) = z, τ ? 1 is the exponent of the required Diophantine condition and ε > 0 is any small quantity. The reasons leading to this bound instead of classical exponentially small estimates are also discussed. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_3dfd8f68d73d751f5ed2cd7a40d95705 |
| institution | Universitat Politècnica de Catalunya (UPC) |
| instname_str | Universitat Politècnica de Catalunya (UPC) |
| language | eng |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:upcommons.upc.edu:2117/912 |
| publishDate | 2003 |
| publishDateSort | 2003 |
| reponame_str | UPCommons. Portal del coneixement obert de la UPC |
| spelling | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit Ollé Torner, Mercè|||0000-0002-8050-9055 Pacha Andújar, Juan Ramón|||0000-0003-4599-3141 Villanueva Castelltort, Jordi|||0000-0001-8725-2785 Differential equations Bifurcation theory Hamiltonian systems Quantitative estimates periodic orbit Equacions diferencials ordinàries Bifurcació, Teoria de la Hamilton, Sistemes de Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory The purpose of thiswork is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant periodic orbit, as a function of the distance to the orbit. We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian system having a pairwise collision of its non-trivial characteristic multipliers on the unit circle. Under generic hypotheses of non-resonance and non-degeneracy of the collision, we present a constructive methodology to reduce the Hamiltonian around the orbit to its (integrable) normal form, up to any given order. This constructive process allows to obtain quantitative estimates for the size of the remainder of the normal form, as a function of the normalizing order. By selecting appropriately this order in terms of the distance R to the resonant orbit (measured using suitable coordinates), r = ropt(R) := 2 + ?exp(W(log(1/R1/(τ+1+ε))))?, we have proved that the size of the remainder can be bounded (for small R) by Rropt(R)/2. Here, W(·) stands for Lambert’s W function and verifies that W(z) exp(W(z)) = z, τ ? 1 is the exponent of the required Diophantine condition and ε > 0 is any small quantity. The reasons leading to this bound instead of classical exponentially small estimates are also discussed. https://hdl.handle.net/2117/912 |
| spellingShingle | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit Ollé Torner, Mercè|||0000-0002-8050-9055 Differential equations Bifurcation theory Hamiltonian systems Quantitative estimates periodic orbit Equacions diferencials ordinàries Bifurcació, Teoria de la Hamilton, Sistemes de Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory |
| title | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit |
| title_full | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit |
| title_fullStr | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit |
| title_full_unstemmed | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit |
| title_short | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit |
| title_sort | Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit |
| topic | Differential equations Bifurcation theory Hamiltonian systems Quantitative estimates periodic orbit Equacions diferencials ordinàries Bifurcació, Teoria de la Hamilton, Sistemes de Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory |
| topic_facet | Differential equations Bifurcation theory Hamiltonian systems Quantitative estimates periodic orbit Equacions diferencials ordinàries Bifurcació, Teoria de la Hamilton, Sistemes de Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory |
| url | https://hdl.handle.net/2117/912 |