8,592 research outputs found
Anomalous Thermostat and Intraband Discrete Breathers
We investigate the dynamics of a macroscopic system which consists of an
anharmonic subsystem embedded in an arbitrary harmonic lattice, including
quenched disorder. Elimination of the harmonic degrees of freedom leads to a
nonlinear Langevin equation for the anharmonic coordinates. For zero
temperature, we prove that the support of the Fourier transform of the memory
kernel and of the time averaged velocity-velocity correlations functions of the
anharmonic system can not overlap. As a consequence, the asymptotic solutions
can be constant, periodic,quasiperiodic or almost periodic, and possibly weakly
chaotic. For a sinusoidal trajectory with frequency we find that the
energy transferred to the harmonic system up to time is proportional
to . If equals one of the phonon frequencies ,
it is . We prove that there is a full measure set such that for
in this set it is , i.e. there is no energy dissipation.
Under certain conditions there exists a zero measure set such that for (0 \leq
\alpha < 1)(1 <\alpha \leq 2)\Omega\Omega\mathcal{C}(k)\Omega\in\mathcal{C}(k)t$.Comment: Physica D in prin
Least Squares Shadowing method for sensitivity analysis of differential equations
For a parameterized hyperbolic system the derivative
of the ergodic average to the parameter can be computed via
the Least Squares Shadowing algorithm (LSS). We assume that the sytem is
ergodic which means that depends only on (not on the
initial condition of the hyperbolic system). After discretizing this continuous
system using a fixed timestep, the algorithm solves a constrained least squares
problem and, from the solution to this problem, computes the desired derivative
. The purpose of this paper is to prove that the
value given by the LSS algorithm approaches the exact derivative when the
discretization timestep goes to and the timespan used to formulate the
least squares problem grows to infinity.Comment: 21 pages, this article complements arXiv:1304.3635 and analyzes LSS
for the case of continuous hyperbolic system
Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations
We prove that the attractor of the 1D quintic complex Ginzburg-Landau
equation with a broken phase symmetry has strictly positive space-time entropy
for an open set of parameter values. The result is obtained by studying chaotic
oscillations in grids of weakly interacting solitons in a class of
Ginzburg-Landau type equations. We provide an analytic proof for the existence
of two-soliton configurations with chaotic temporal behavior, and construct
solutions which are closed to a grid of such chaotic soliton pairs, with every
pair in the grid well spatially separated from the neighboring ones for all
time. The temporal evolution of the well-separated multi-soliton structures is
described by a weakly coupled lattice dynamical system (LDS) for the
coordinates and phases of the solitons. We develop a version of normal
hyperbolicity theory for the weakly coupled LDSs with continuous time and
establish for them the existence of space-time chaotic patterns similar to the
Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory
may be of independent interest, the main difficulty addressed in the paper
concerns with lifting the space-time chaotic solutions of the LDS back to the
initial PDE. The equations we consider here are space-time autonomous, i.e. we
impose no spatial or temporal modulation which could prevent the individual
solitons in the grid from drifting towards each other and destroying the
well-separated grid structure in a finite time. We however manage to show that
the set of space-time chaotic solutions for which the random soliton drift is
arrested is large enough, so the corresponding space-time entropy is strictly
positive
Statistical properties of Lorenz like flows, recent developments and perspectives
We comment on mathematical results about the statistical behavior of Lorenz
equations an its attractor, and more generally to the class of singular
hyperbolic systems. The mathematical theory of such kind of systems turned out
to be surprisingly difficult. It is remarkable that a rigorous proof of the
existence of the Lorenz attractor was presented only around the year 2000 with
a computer assisted proof together with an extension of the hyperbolic theory
developed to encompass attractors robustly containing equilibria. We present
some of the main results on the statisitcal behavior of such systems. We show
that for attractors of three-dimensional flows, robust chaotic behavior is
equivalent to the existence of certain hyperbolic structures, known as
singular-hyperbolicity. These structures, in turn, are associated to the
existence of physical measures: \emph{in low dimensions, robust chaotic
behavior for flows ensures the existence of a physical measure}. We then give
more details on recent results on the dynamics of singular-hyperbolic
(Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial
conditions, physical measure, singular-hyperbolicity, expansiveness, robust
attractor, robust chaotic flow, positive Lyapunov exponent, large deviations,
hitting and recurrence times. Minor typos corrected and precise
acknowledgments of financial support added. To appear in Int J of Bif and
Chaos in App Sciences and Engineerin
Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations
This work continues the analysis of complex dynamics in a class of
bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal
voltage dynamics with adaptation and spike emission. We show that these models
can generically display a form of mixed-mode oscillations (MMOs), which are
trajectories featuring an alternation of small oscillations with spikes or
bursts (multiple consecutive spikes). The mechanism by which these are
generated relies fundamentally on the hybrid structure of the flow: invariant
manifolds of the continuous dynamics govern small oscillations, while discrete
resets govern the emission of spikes or bursts, contrasting with classical MMO
mechanisms in ordinary differential equations involving more than three
dimensions and generally relying on a timescale separation. The decomposition
of mechanisms reveals the geometrical origin of MMOs, allowing a relatively
simple classification of points on the reset manifold associated to specific
numbers of small oscillations. We show that the MMO pattern can be described
through the study of orbits of a discrete adaptation map, which is singular as
it features discrete discontinuities with unbounded left- and
right-derivatives. We study orbits of the map via rotation theory for
discontinuous circle maps and elucidate in detail complex behaviors arising in
the case where MMOs display at most one small oscillation between each
consecutive pair of spikes
On the policy function in continuos time economic models
In this paper, I consider a general class of continuous-time economic models with unbounded horizon. I study the sets of conditions under which the policy function is continuous, Lipschitz continuous, and Cl differentiable. 1 also single out certain postulates which may prevent higher-order differentiability. The analysis provides, therefore, a fmn foundation to the use of dynamic programming methods in continuous time models with unbounded horizo
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