3,211 research outputs found
Geometric singular perturbation theory for stochastic differential equations
We consider slow-fast systems of differential equations, in which both the
slow and fast variables are perturbed by noise. When the deterministic system
admits a uniformly asymptotically stable slow manifold, we show that the sample
paths of the stochastic system are concentrated in a neighbourhood of the slow
manifold, which we construct explicitly. Depending on the dynamics of the
reduced system, the results cover time spans which can be exponentially long in
the noise intensity squared (that is, up to Kramers' time). We obtain
exponentially small upper and lower bounds on the probability of exceptional
paths. If the slow manifold contains bifurcation points, we show similar
concentration properties for the fast variables corresponding to
non-bifurcating modes. We also give conditions under which the system can be
approximated by a lower-dimensional one, in which the fast variables contain
only bifurcating modes.Comment: 43 pages. Published version. Remarks added, minor correction
Discrete Geometric Singular Perturbation Theory
We propose a mathematical formalism for discrete multi-scale dynamical
systems induced by maps which parallels the established geometric singular
perturbation theory for continuous-time fast-slow systems. We identify limiting
maps corresponding to both 'fast' and 'slow' iteration under the map. A notion
of normal hyperbolicity is defined by a spectral gap requirement for the
multipliers of the fast limiting map along a critical fixed-point manifold .
We provide a set of Fenichel-like perturbation theorems by reformulating
pre-existing results so that they apply near compact, normally hyperbolic
submanifolds of . The persistence of the critical manifold , local
stable/unstable manifolds and foliations of
by stable/unstable fibers is described in detail. The
practical utility of the resulting discrete geometric singular perturbation
theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify
singular geometry corresponding to excitability, relaxation, chaotic and
non-chaotic bursting in a map-based neural model. Second, we derive results
which relate the geometry and dynamics of fast-slow ODEs with non-trivial
time-scale separation and their Euler-discretized counterpart. Finally, we show
that fast-slow ODE systems with fast rotation give rise to fast-slow Poincar\'e
maps, the geometry and dynamics of which can be described in detail using
DGSPT.Comment: Updated to include minor corrections made during the review process
(no major changes
Singular perturbation theory for interacting fermions in two dimensions
We consider a system of interacting fermions in two dimensions beyond the
second-order perturbation theory in the interaction. It is shown that the
mass-shell singularities in the self-energy, arising already at the second
order of the perturbation theory, manifest a non-perturbative effect: an
interaction with the zero-sound mode. Resumming the perturbation theory for a
weak, short-range interaction and accounting for a finite curvature of the
fermion spectrum, we eliminate the singularities and obtain the results for the
quasi-particle self-energy and the spectral function to all orders in the
interaction with the zero-sound mode. A threshold for emission of zero-sound
waves leads a non-monotonic variation of the self-energy with energy (or
momentum) near the mass shell. Consequently, the spectral function has a
kink-like feature. We also study in detail a non-analytic temperature
dependence of the specific heat, . It turns out that although
the interaction with the collective mode results in an enhancement of the
fermion self-energy, this interaction does not affect the non-analytic term in
due to a subtle cancellation between the contributions from the real and
imaginary parts of the self-energy. For a short-range and weak interaction,
this implies that the second-order perturbation theory suffices to determine
the non-analytic part of . We also obtain a general form of the
non-analytic term in , valid for the case of a generic Fermi liquid,
\emph{i.e.}, beyond the perturbation theory.Comment: 53 pages, 10 figure
A study of the application of singular perturbation theory
A hierarchical real time algorithm for optimal three dimensional control of aircraft is described. Systematic methods are developed for real time computation of nonlinear feedback controls by means of singular perturbation theory. The results are applied to a six state, three control variable, point mass model of an F-4 aircraft. Nonlinear feedback laws are presented for computing the optimal control of throttle, bank angle, and angle of attack. Real Time capability is assessed on a TI 9900 microcomputer. The breakdown of the singular perturbation approximation near the terminal point is examined Continuation methods are examined to obtain exact optimal trajectories starting from the singular perturbation solutions
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