333 research outputs found
Passive Scalars and Three-Dimensional Liouvillian Maps
Global aspects of the motion of passive scalars in time-dependent
incompressible fluid flows are well described by volume-preserving
(Liouvillian) three-dimensional maps. In this paper the possible invariant
structures in Liouvillian maps and the two most interesting nearly-integrable
cases are investigated. In addition, the fundamental role of invariant lines in
organizing the dynamics of this type of system is exposed. Bifurcations
involving the destruction of some invariant lines and tubes and the creation of
new ones are described in detail.Comment: 18 pages, plain TeX, appears in Physica D, 76, 22-33, 1994. (Lack of
figures in original submission corrected in this new upload.
A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems
In this paper we consider a representative a priori unstable Hamiltonian
system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism
for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc.
2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide
explicit, concrete and easily verifiable conditions for the existence of
diffusing orbits.
The simplification of the hypotheses allows us to perform explicitly the
computations along the proof, which contribute to present in an easily
understandable way the geometric mechanism of diffusion. In particular, we
fully describe the construction of the scattering map and the combination of
two types of dynamics on a normally hyperbolic invariant manifol
Resonances and Twist in Volume-Preserving Mappings
The phase space of an integrable, volume-preserving map with one action and
angles is foliated by a one-parameter family of -dimensional invariant
tori. Perturbations of such a system may lead to chaotic dynamics and
transport. We show that near a rank-one, resonant torus these mappings can be
reduced to volume-preserving "standard maps." These have twist only when the
image of the frequency map crosses the resonance curve transversely. We show
that these maps can be approximated---using averaging theory---by the usual
area-preserving twist or nontwist standard maps. The twist condition
appropriate for the volume-preserving setting is shown to be distinct from the
nondegeneracy condition used in (volume-preserving) KAM theory.Comment: Many typos fixed and notation simplified. New order
averaging theorem and volume-preserving variant. Numerical comparison with
averaging adde
Classical Mechanics
An overview of the foundations of Classical Mechanic
KdV equation under periodic boundary conditions and its perturbations
In this paper we discuss properties of the KdV equation under periodic
boundary conditions, especially those which are important to study
perturbations of the equation. Next we review what is known now about long-time
behaviour of solutions for perturbed KdV equations
Contributions of plasma physics to chaos and nonlinear dynamics
This topical review focusses on the contributions of plasma physics to chaos
and nonlinear dynamics bringing new methods which are or can be used in other
scientific domains. It starts with the development of the theory of Hamiltonian
chaos, and then deals with order or quasi order, for instance adiabatic and
soliton theories. It ends with a shorter account of dissipative and high
dimensional Hamiltonian dynamics, and of quantum chaos. Most of these
contributions are a spin-off of the research on thermonuclear fusion by
magnetic confinement, which started in the fifties. Their presentation is both
exhaustive and compact. [15 April 2016
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