Geometric shadowing in slow–fast Hamiltonian systems

Abstract

Abstract. We study a class of slow-fast Hamiltonian systems with any finite number of degrees of freedom, but with at least one slow one and two fast ones. At ε = 0 the slow dynamics is frozen. We assume that the frozen system (i.e. the unperturbed fast dynamics) has families of hyperbolic periodic orbits with transversal heteroclinics.For each periodic orbit we define an action J. This action may be viewed as an action Hamiltonian (in the slow variables). It has been shown in [4] that there are orbits of the full dynamics which shadow any finite combination of forward orbits of J for a time t = O(ε −1 ).We introduce an assumption on the mutual relationship between the actions J. This assumption enables us to shadow any continuous curve (of arbitrary length) in the slow phase space for any time. The slow dynamics shadows the curve as a purely geometrical object, thus the time on the slow dynamics has to be reparameterised. ‡ The research of NB was funded by the Academy of Finland and the EU research training network CODY. § Funded by the Academy of Finland.

Fermi acceleration in non-autonomous billiards

“…a / b varies with time), that the boundary motion is slow and that the phase space of the frozen billiard retains the Smale horseshoe for all times. Then, both periodic and non-periodic cases can be treated by the proposed method, see the corresponding theory in [13,17]. We note that the trajectories constructed above have the largest acceleration.…”

Локальные Адиабатические Инварианты В Окрестности Гомоклинического Множества Быстро-Медленной Гамильтоновой Системы

“…Результаты настоящей работы тесно связаны с работой В. Гельфрейха и Д. Тураева [13] (см. также [10]), где рассматривается случай, когда замороженная система имеет гиперболические хаотические инвариантные множества на регулярных уровнях энергии Σ w,E . Точнее, предположим, что для всех w ∈ D замороженная система имеет несколько гиперболических периодических траекторий γ k w,E на уровне энергии Σ w,E , соединенных трансверсальными гетероклиническими траекториями.…”