782 research outputs found
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We are concerned with the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (known as stochastic heat equations with Markovian switching) in an infinite state space. The fixed point theory is utilized to discuss the existence, uniqueness, and pth moment exponential stability of the mild solution. Moreover, we also acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality. At last, two examples are provided to verify the effectiveness of our obtained results
Low-frequency variability and heat transport in a low-order nonlinear coupled ocean-atmosphere model
We formulate and study a low-order nonlinear coupled ocean-atmosphere model
with an emphasis on the impact of radiative and heat fluxes and of the
frictional coupling between the two components. This model version extends a
previous 24-variable version by adding a dynamical equation for the passive
advection of temperature in the ocean, together with an energy balance model.
The bifurcation analysis and the numerical integration of the model reveal
the presence of low-frequency variability (LFV) concentrated on and near a
long-periodic, attracting orbit. This orbit combines atmospheric and oceanic
modes, and it arises for large values of the meridional gradient of radiative
input and of frictional coupling. Chaotic behavior develops around this orbit
as it loses its stability; this behavior is still dominated by the LFV on
decadal and multi-decadal time scales that is typical of oceanic processes.
Atmospheric diagnostics also reveals the presence of predominant low- and
high-pressure zones, as well as of a subtropical jet; these features recall
realistic climatological properties of the oceanic atmosphere.
Finally, a predictability analysis is performed. Once the decadal-scale
periodic orbits develop, the coupled system's short-term instabilities --- as
measured by its Lyapunov exponents --- are drastically reduced, indicating the
ocean's stabilizing role on the atmospheric dynamics. On decadal time scales,
the recurrence of the solution in a certain region of the invariant subspace
associated with slow modes displays some extended predictability, as reflected
by the oscillatory behavior of the error for the atmospheric variables at long
lead times.Comment: v1: 41 pages, 17 figures; v2-: 42 pages, 15 figure
Hamiltonian dynamics and geometry of phase transitions in classical XY models
The Hamiltonian dynamics associated to classical, planar, Heisenberg XY
models is investigated for two- and three-dimensional lattices. Besides the
conventional signatures of phase transitions, here obtained through time
averages of thermodynamical observables in place of ensemble averages,
qualitatively new information is derived from the temperature dependence of
Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests
to consider other observables of geometric meaning tightly related with the
largest Lyapunov exponent. The numerical computation of these observables -
unusual in the study of phase transitions - sheds a new light on the
microscopic dynamical counterpart of thermodynamics also pointing to the
existence of some major change in the geometry of the mechanical manifolds at
the thermodynamical transition. Through the microcanonical definition of the
entropy, a relationship between thermodynamics and the extrinsic geometry of
the constant energy surfaces of phase space can be naturally
established. In this framework, an approximate formula is worked out,
determining a highly non-trivial relationship between temperature and topology
of the . Whence it can be understood that the appearance of a phase
transition must be tightly related to a suitable major topology change of the
. This contributes to the understanding of the origin of phase
transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22
PostScript figure
Hamiltonian Dynamics of Yang-Mills Fields on a Lattice
We review recent results from studies of the dynamics of classical Yang-Mills
fields on a lattice. We discuss the numerical techniques employed in solving
the classical lattice Yang-Mills equations in real time, and present results
exhibiting the universal chaotic behavior of nonabelian gauge theories. The
complete spectrum of Lyapunov exponents is determined for the gauge group
SU(2). We survey results obtained for the SU(3) gauge theory and other
nonlinear field theories. We also discuss the relevance of these results to the
problem of thermalization in gauge theories.Comment: REVTeX, 51 pages, 20 figure
Completely Mixing Quantum Open Systems and Quantum Fractals
Departing from classical concepts of ergodic theory, formulated in terms of
probability densities, measures describing the chaotic behavior and the loss of
information in quantum open systems are proposed. As application we discuss the
chaotic outcomes of continuous measurement processes in the EEQT framework.
Simultaneous measurement of four noncommuting spin components is shown to lead
to a chaotic jump on quantum spin sphere and to generate specific fractal
images - nonlinear ifs (iterated function system). The model is purely
theoretical at this stage, and experimental confirmation of the chaotic
behavior of measuring instruments during simultaneous continuous measurement of
several noncommuting quantum observables would constitute a quantitative
verification of Event Enhanced Quantum Theory.Comment: Latex format, 20 pages, 6 figures in jpg format. New replacement has
two more references (including one to a paper by G. Casati et al on quantum
fractal eigenstates), adds example and comments concerning mixing properties
of of a two-level atom driven by a laser field, and also adds a number of
other remarks which should make it easier to follow mathematical argument
Amplitude Death: The emergence of stationarity in coupled nonlinear systems
When nonlinear dynamical systems are coupled, depending on the intrinsic
dynamics and the manner in which the coupling is organized, a host of novel
phenomena can arise. In this context, an important emergent phenomenon is the
complete suppression of oscillations, formally termed amplitude death (AD).
Oscillations of the entire system cease as a consequence of the interaction,
leading to stationary behavior. The fixed points that the coupling stabilizes
can be the otherwise unstable fixed points of the uncoupled system or can
correspond to novel stationary points. Such behaviour is of relevance in areas
ranging from laser physics to the dynamics of biological systems. In this
review we discuss the characteristics of the different coupling strategies and
scenarios that lead to AD in a variety of different situations, and draw
attention to several open issues and challenging problems for further study.Comment: Physics Reports (2012
Global Demons in Field Theory : Critical Slowing Down in the Xy Model
We investigate the use of global demons, a `canonical dynamics', as an
approach to simulating lattice regularized field theories. This
deterministically chaotic dynamics is non-local and non-Hamiltonian, and
preserves the canonical measure rather than . We apply this
inexact dynamics to the 2D XY model, comparing to various implementations of
hybrid Monte Carlo, focusing on critical exponents and critical slowing down.
In addition, we discuss a scheme for making energy non-conserving dynamical
algorithms exact without the use of a Metropolis hit.Comment: 23 pages text plus 12 figures [Submitted to Nuc. Phys. B, 7/92
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