27 research outputs found
Hamilton-Poisson Realizations of the Integrable Deformations of the Rikitake System
Integrable deformations of an integrable case of the Rikitake system are constructed by modifying its constants of motions. Hamilton-Poisson realizations of these integrable deformations are given. Considering two concrete deformation functions, a Hamilton-Poisson approach of the obtained system is presented. More precisely, the stability of the equilibrium points and the existence of the periodic orbits are proved. Furthermore, the image of the energy-Casimir mapping is determined and its connections with the dynamical elements of the considered system are pointed out
Coupled spin models for magnetic variation of planets and stars
Geomagnetism is characterized by intermittent polarity reversals and rapid
fluctuations. We have recently proposed a coupled macro-spin model to describe
these dynamics based on the idea that the whole dynamo mechanism is described
by the coherent interactions of many small dynamo elements. In this paper, we
further develop this idea and construct a minimal model for magnetic
variations. This simple model naturally yields many of the observed features of
geomagnetism: its time evolution, the power spectrum, the frequency
distribution of stable polarity periods, etc. This model has coexistent two
phases; i.e. the cluster phase which determines the global dipole magnetic
moment and the expanded phase which gives random perpetual perturbations that
yield intermittent polarity flip of the dipole moment. This model can also
describe the synchronization of the spin oscillation. This corresponds to the
case of sun and the model well describes the quasi-regular cycles of the solar
magnetism. Furthermore, by analyzing the relevant terms of MHD equation based
on our model, we have obtained a scaling relation for the magnetism for
planets, satellites, sun, and stars. Comparing it with various observations, we
can estimate the scale of the macro-spins.Comment: 16 pages, 9 figure
On the generalized Hamiltonian structure of 3D dynamical systems
The Poisson structures for 3D systems possessing one constant of motion can
always be constructed from the solution of a linear PDE. When two constants of
the motion are available the problem reduces to a quadrature and the structure
functions include an arbitrary function of them
Attempts to relate the Navier-Stokes equations to turbulence
The present talk is designed as a survey, is slanted to my personal tastes, but I hope it is still representative. My intention is to keep the whole discussion pretty elementary by touching large numbers of topics and avoiding details as well as technical difficulties in any one of them. Subsequent talks will go deeper into some of the subjects we discuss today.
The main goal is to link up the statistics, entropy, correlation functions, etc., in the engineering side with a "nice" mathematical model of turbulence