49 research outputs found
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