3,402 research outputs found
Theory of weakly nonlinear self sustained detonations
We propose a theory of weakly nonlinear multi-dimensional self sustained
detonations based on asymptotic analysis of the reactive compressible
Navier-Stokes equations. We show that these equations can be reduced to a model
consisting of a forced, unsteady, small disturbance, transonic equation and a
rate equation for the heat release. In one spatial dimension, the model
simplifies to a forced Burgers equation. Through analysis, numerical
calculations and comparison with the reactive Euler equations, the model is
demonstrated to capture such essential dynamical characteristics of detonations
as the steady-state structure, the linear stability spectrum, the
period-doubling sequence of bifurcations and chaos in one-dimensional
detonations and cellular structures in multi- dimensional detonations
Oscillations of Hyperbolic Systems with Functional Arguments
Hyperbolic systems with functional arguments are studied, and sufficient conditions are obtained for every solution of boundary value problems to be weakly oscillatory (that is, at least one of its components is oscillatory) in a cylindrical domain. Robin-type boundary condition is considered. The approach used is to reduce the multi-dimensional oscillation problems to one-dimensional oscillation problems by using some integral means of solutions
Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation
The spectrum of the generator (Kolmogorov operator) of a diffusion process,
referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed
characterization of correlation functions and power spectra of stochastic
systems via decomposition formulas in terms of RP resonances. Stochastic
analysis techniques relying on the theory of Markov semigroups for the study of
the RP spectrum and a rigorous reduction method is presented in Part I. This
framework is here applied to study a stochastic Hopf bifurcation in view of
characterizing the statistical properties of nonlinear oscillators perturbed by
noise, depending on their stability. In light of the H\"ormander theorem, it is
first shown that the geometry of the unperturbed limit cycle, in particular its
isochrons, is essential to understand the effect of noise and the phenomenon of
phase diffusion. In addition, it is shown that the spectrum has a spectral gap,
even at the bifurcation point, and that correlations decay exponentially fast.
Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are
then obtained, away from the bifurcation point, based on the knowledge of the
linearized deterministic dynamics and the characteristics of the noise. These
formulas allow one to understand how the interaction of the noise with the
deterministic dynamics affect the decay of correlations. Numerical results
complement the study of the RP spectrum at the bifurcation, revealing useful
scaling laws. The analysis of the Markov semigroup for stochastic bifurcations
is thus promising in providing a complementary approach to the more geometric
random dynamical system approach. This approach is not limited to
low-dimensional systems and the reduction method presented in part I is applied
to a stochastic model relevant to climate dynamics in part III
Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum
We investigate the asymptotic properties of inertial modes confined in a
spherical shell when viscosity tends to zero. We first consider the mapping
made by the characteristics of the hyperbolic equation (Poincar\'e's equation)
satisfied by inviscid solutions. Characteristics are straight lines in a
meridional section of the shell, and the mapping shows that, generically, these
lines converge towards a periodic orbit which acts like an attractor.
We then examine the relation between this characteristic path and
eigensolutions of the inviscid problem and show that in a purely
two-dimensional problem, convergence towards an attractor means that the
associated velocity field is not square-integrable. We give arguments which
generalize this result to three dimensions. We then consider the viscous
problem and show how viscosity transforms singularities into internal shear
layers which in general betray an attractor expected at the eigenfrequency of
the mode. We find that there are nested layers, the thinnest and most internal
layer scaling with -scale, being the Ekman number. Using an
inertial wave packet traveling around an attractor, we give a lower bound on
the thickness of shear layers and show how eigenfrequencies can be computed in
principle. Finally, we show that as viscosity decreases, eigenfrequencies tend
towards a set of values which is not dense in , contrary to the
case of the full sphere ( is the angular velocity of the system).
Hence, our geometrical approach opens the possibility of describing the
eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers
(), which are out of reach numerically, and this for a wide
class of containers.Comment: 42 pages, 20 figures, abstract shortene
Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model
Pattern formation in biological, chemical and physical problems has received
considerable attention, with much attention paid to dissipative systems. For
example, the Ginzburg--Landau equation is a normal form that describes pattern
formation due to the appearance of a single mode of instability in a wide
variety of dissipative problems. In a similar vein, a certain "single-wave
model" arises in many physical contexts that share common pattern forming
behavior. These systems have Hamiltonian structure, and the single-wave model
is a kind of Hamiltonian mean-field theory describing the patterns that form in
phase space. The single-wave model was originally derived in the context of
nonlinear plasma theory, where it describes the behavior near threshold and
subsequent nonlinear evolution of unstable plasma waves. However, the
single-wave model also arises in fluid mechanics, specifically shear-flow and
vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter
physics, and other Hamiltonian theories characterized by mean field
interaction. We demonstrate, by a suitable asymptotic analysis, how the
single-wave model emerges from a large class of nonlinear advection-transport
theories. An essential ingredient for the reduction is that the Hamiltonian
system has a continuous spectrum in the linear stability problem, arising not
from an infinite spatial domain but from singular resonances along curves in
phase space whereat wavespeeds match material speeds (wave-particle resonances
in the plasma problem, or critical levels in fluid problems). The dynamics of
the continuous spectrum is manifest as the phenomenon of Landau damping when
the system is ... Such dynamical phenomena have been rediscovered in different
contexts, which is unsurprising in view of the normal-form character of the
single-wave model
Coupled Oscillators on a Circle
We consider a continuum of diffusively coupled oscillators on a circle. When each oscillator is of Lienard type, very little is known about the corresponding hyperbolic POE. When each oscillator is represented by a lossless transmission line, we obtain a partial neutral delay differential equation and give the beginnings of a qualitative theory for the dynamics. In particular, we discuss the properties of the solution map, the existence of the global attractor, behavior near an equilibrium point including the Hopf bifurcation, and some elementary properties near a periodic orbit
Gravitating lumps
Recent progress in the study of solitons and black holes in non-Abelian field
theories coupled to gravity is reviewed. New topics include gravitational
binding of monopoles, black holes with non-trivial topology, Lue-Weinberg
bifurcation, asymptotically AdS lumps, solutions to the Freedman-Schwarz model
with applications to holography, non-Abelian Born-Infeld solutionsComment: A written version of the talk given at the 16th International
Conference on General Relativity and Gravitation, held on July 15-21, 2001,
in Durban, South Africa. Latex error on the title page corrected. New
references adde
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