Tailoring Phase Space : A Way to Control Hamiltonian Transport

We present a method to control transport in Hamiltonian systems. We provide an algorithm - based on a perturbation of the original Hamiltonian localized in phase space - to design small control terms that are able to create isolated barriers of transport without modifying other parts of phase space. We apply this method of localized control to a forced pendulum model and to a system describing the motion of charged particles in a model of turbulent electric field

Local control of Hamiltonian chaos

We review a method of control for Hamiltonian systems which is able to create smooth invariant tori. This method of control is based on an apt modification of the perturbation which is small and localized in phase space

Perturbation Theory and Control in Classical or Quantum Mechanics by an Inversion Formula

We consider a perturbation of an ``integrable'' Hamiltonian and give an expression for the canonical or unitary transformation which ``simplifies'' this perturbed system. The problem is to invert a functional defined on the Lie- algebra of observables. We give a bound for the perturbation in order to solve this inversion. And apply this result to a particular case of the control theory, as a first example, and to the ``quantum adiabatic transformation'', as another example.Comment: Version 8.0. 26 pages, Latex2e, final version published in J. Phys.

Controlling chaotic transport in a Hamiltonian model of interest to magnetized plasmas

We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a model that reproduces turbulent ExB drift and show numerically that the control is able to drastically reduce chaotic transport

Lifting of the Vlasov-Maxwell Bracket by Lie-transform Method

The Vlasov-Maxwell equations possess a Hamiltonian structure expressed in terms of a Hamiltonian functional and a functional bracket. In the present paper, the transformation ("lift") of the Vlasov-Maxwell bracket induced by the dynamical reduction of single-particle dynamics is investigated when the reduction is carried out by Lie-transform perturbation methods. The ultimate goal of this work is to derive explicit Hamiltonian formulations for the guiding-center and gyrokinetic Vlasov-Maxwell equations that have important applications in our understanding of turbulent magnetized plasmas. Here, it is shown that the general form of the reduced Vlasov-Maxwell equations possesses a Hamiltonian structure defined in terms of a reduced Hamiltonian functional and a reduced bracket that automatically satisfies the standard bracket properties.Comment: 39 page

Controlling chaos in area-preserving maps

We describe a method of control of chaos that occurs in area-preserving maps. This method is based on small modifications of the original map by addition of a small control term. We apply this control technique to the standard map and to the tokamap

Weakly regular Floquet Hamiltonians with pure point spectrum

We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on the parameter omega. We assume that the spectrum of H is discrete, {h_m (m = 1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m - h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show that in that case there exist a suitable norm to measure the regularity of V, denoted epsilon, and positive constants, epsilon_* & delta_*, such that: if epsilon |Omega_0| - delta_* epsilon and the Floquet Hamiltonian has a pure point spectrum for all omega in Omega_infinity.Comment: 35 pages, Latex with AmsAr