Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System
- Авторлар: Bolotin S.1,2
-
Мекемелер:
- Steklov Mathematical Institute
- University of Wisconsin-Madison
- Шығарылым: Том 24, № 6 (2019)
- Беттер: 682-703
- Бөлім: Article
- URL: https://journals.rcsi.science/1560-3547/article/view/219410
- DOI: https://doi.org/10.1134/S1560354719060078
- ID: 219410
Дәйексөз келтіру
Аннотация
We consider a Hamiltonian system depending on a parameter which slowly changes with rate ε ≪ 1. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order ε. We prove a partial analog of Neishtadt’s result for a system with n degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order ε at time intervals of order ∣ln ε∣, so the energy may grow with rate ε/∣ln ε∣. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order ε.
Авторлар туралы
Sergey Bolotin
Steklov Mathematical Institute; University of Wisconsin-Madison
Хат алмасуға жауапты Автор.
Email: bolotin@mi.ras.ru
Ресей, ul. Gubkina 8, Moscow, 119991; 480 Lincoln Dr., Madison, WI, 53706-1325