1,549 research outputs found
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
Lagrangian coherent structures in n-dimensional systems
Numerical simulations and experimental observations reveal that unsteady fluid systems can be divided into regions of qualitatively different dynamics. The key to understanding transport and stirring is to identify the dynamic boundaries between these almost-invariant regions. Recently, ridges in finite-time Lyapunov exponent fields have been used to define such hyperbolic, almost material, Lagrangian coherent structures in two-dimensional systems. The objective of this paper is to develop and apply a similar theory in higher dimensional spaces. While the separatrix nature of these structures is their most important property, a necessary condition is their almost material nature. This property is addressed in this paper. These results are applied to a model of Rayleigh-Bénard convection based on a three-dimensional extension of the model of Solomon and Gollub
Time-Dependent 2-D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures
This paper presents an approach to a time-dependent variant of the concept of
vector field topology for 2-D vector fields. Vector field topology is defined
for steady vector fields and aims at discriminating the domain of a vector
field into regions of qualitatively different behaviour. The presented approach
represents a generalization for saddle-type critical points and their
separatrices to unsteady vector fields based on generalized streak lines, with
the classical vector field topology as its special case for steady vector
fields. The concept is closely related to that of Lagrangian coherent
structures obtained as ridges in the finite-time Lyapunov exponent field. The
proposed approach is evaluated on both 2-D time-dependent synthetic and vector
fields from computational fluid dynamics
Do Finite-Size Lyapunov Exponents Detect Coherent Structures?
Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as
indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous
mathematical link between the FSLE and LCSs, however, has been missing. Here we
prove that an FSLE ridge satisfying certain conditions does signal a nearby
ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn
indicates a hyperbolic LCS under further conditions. Other FSLE ridges
violating our conditions, however, are seen to be false positives for LCSs. We
also find further limitations of the FSLE in Lagrangian coherence detection,
including ill-posedness, artificial jump-discontinuities, and sensitivity with
respect to the computational time step.Comment: 22 pages, 7 figures, v3: corrects the z-axis labels of Fig. 2 (left)
that appears in the version published in Chao
Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents
We consider issues associated with the Lagrangian characterisation of flow
structures arising in aperiodically time-dependent vector fields that are only
known on a finite time interval. A major motivation for the consideration of
this problem arises from the desire to study transport and mixing problems in
geophysical flows where the flow is obtained from a numerical solution, on a
finite space-time grid, of an appropriate partial differential equation model
for the velocity field. Of particular interest is the characterisation,
location, and evolution of "transport barriers" in the flow, i.e. material
curves and surfaces. We argue that a general theory of Lagrangian transport has
to account for the effects of transient flow phenomena which are not captured
by the infinite-time notions of hyperbolicity even for flows defined for all
time. Notions of finite-time hyperbolic trajectories, their finite time stable
and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields
and associated Lagrangian coherent structures have been the main tools for
characterizing transport barriers in the time-aperiodic situation. In this
paper we consider a variety of examples, some with explicit solutions, that
illustrate, in a concrete manner, the issues and phenomena that arise in the
setting of finite-time dynamical systems. Of particular significance for
geophysical applications is the notion of "flow transition" which occurs when
finite-time hyperbolicity is lost, or gained. The phenomena discovered and
analysed in our examples point the way to a variety of directions for rigorous
mathematical research in this rapidly developing, and important, new area of
dynamical systems theory
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