Existence of KAM tori for presymplectic vector fields
We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2020-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/126/abstr.html |
| _version_ | 1818870484547141632 |
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| author | Sean Bauer Nikola P. Petrov |
| author_facet | Sean Bauer Nikola P. Petrov |
| author_sort | Sean Bauer |
| collection | DOAJ |
| description | We prove the existence of a torus that is invariant with respect to the flow of a
vector field that preserves the presymplectic form in an exact presymplectic manifold.
The flow on this invariant torus is conjugate to a linear flow on a torus with a
Diophantine velocity vector.
The proof has an "a posteriori" format, the the invariant torus is constructed by
using a Newton method in a space of functions, starting from a torus that is approximately
invariant. The geometry of the problem plays a major role in the construction
by allowing us to construct a special adapted basis in which the equations that need to
be solved in each step of the iteration have a simple structure.
In contrast to the classical methods of proof, this method does not assume that
the system is close to integrable, and does not rely on using action-angle variables. |
| first_indexed | 2024-12-19T12:07:45Z |
| format | Article |
| id | doaj.art-9b988629095f465ba9a9b0b06a12aec2 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-19T12:07:45Z |
| publishDate | 2020-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-9b988629095f465ba9a9b0b06a12aec22022-12-21T20:22:18ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912020-12-012020126,126Existence of KAM tori for presymplectic vector fieldsSean Bauer0Nikola P. Petrov1 Univ. of Oklahoma, Norman, OK, USA Univ. of Oklahoma, Norman, OK, USA We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an "a posteriori" format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable, and does not rely on using action-angle variables.http://ejde.math.txstate.edu/Volumes/2020/126/abstr.htmlkam theoryinvariant toruspresymplectic manifoldstability |
| spellingShingle | Sean Bauer Nikola P. Petrov Existence of KAM tori for presymplectic vector fields Electronic Journal of Differential Equations kam theory invariant torus presymplectic manifold stability |
| title | Existence of KAM tori for presymplectic vector fields |
| title_full | Existence of KAM tori for presymplectic vector fields |
| title_fullStr | Existence of KAM tori for presymplectic vector fields |
| title_full_unstemmed | Existence of KAM tori for presymplectic vector fields |
| title_short | Existence of KAM tori for presymplectic vector fields |
| title_sort | existence of kam tori for presymplectic vector fields |
| topic | kam theory invariant torus presymplectic manifold stability |
| url | http://ejde.math.txstate.edu/Volumes/2020/126/abstr.html |
| work_keys_str_mv | AT seanbauer existenceofkamtoriforpresymplecticvectorfields AT nikolappetrov existenceofkamtoriforpresymplecticvectorfields |