283 research outputs found
Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems
We consider a class of Hamiltonians with three degrees of freedom that can be
mapped into quasi-periodically driven pendulums. The purpose of this paper is
to determine the threshold of the break-up of invariant tori with a specific
frequency vector. We apply two techniques: the frequency map analysis and
renormalization-group methods. The renormalization transformation acting on a
Hamiltonian is a canonical change of coordinates which is a combination of a
partial elimination of the irrelevant modes of the Hamiltonian and a rescaling
of phase space around the considered torus. We give numerical evidence that the
critical coupling at which the renormalization transformation starts to diverge
is the same as the value given by the frequency map analysis for the break-up
of invariant tori. Furthermore, we obtain by these methods numerical values of
the threshold of the break-up of the last invariant torus.Comment: 18 pages, 4 figure
Hamiltonian formulation of the modified Hasegawa Mima equation
We derive the Hamiltonian structure of the modified Hasegawa-Mima equation
from the ion fluid equations applying Dirac's theory of constraints. We discuss
the Casimirs obtained from the corresponding Poisson structure
Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments
We consider Hamiltonian closures of the Vlasov equation using the phase-space
moments of the distribution function. We provide some conditions on the
closures imposed by the Jacobi identity. We completely solve some families of
examples. As a result, we show that imposing that the resulting reduced system
preserves the Hamiltonian character of the parent model shapes its phase space
by creating a set of Casimir invariants as a direct consequence of the Jacobi
identity
Stochastic ionization through noble tori: Renormalization results
We find that chaos in the stochastic ionization problem develops through the
break-up of a sequence of noble tori. In addition to being very accurate, our
method of choice, the renormalization map, is ideally suited for analyzing
properties at criticality. Our computations of chaos thresholds agree closely
with the widely used empirical Chirikov criterion
Hamiltonian closures for fluid models with four moments by dimensional analysis
Fluid reductions of the Vlasov-Amp{\`e}re equations that preserve the
Hamiltonian structure of the parent kinetic model are investigated. Hamiltonian
closures using the first four moments of the Vlasov distribution are obtained,
and all closures provided by a dimensional analysis procedure for satisfying
the Jacobi identity are identified. Two Hamiltonian models emerge, for which
the explicit closures are given, along with their Poisson brackets and Casimir
invariants
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
Derivation of reduced two-dimensional fluid models via Dirac's theory of constrained Hamiltonian systems
We present a Hamiltonian derivation of a class of reduced plasma
two-dimensional fluid models, an example being the Charney-Hasegawa-Mima
equation. These models are obtained from the same parent Hamiltonian model,
which consists of the ion momentum equation coupled to the continuity equation,
by imposing dynamical constraints. It is shown that the Poisson bracket
associated with these reduced models is the Dirac bracket obtained from the
Poisson bracket of the parent model
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