2,341 research outputs found
From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data
One way to analyze complicated non-autonomous flows is through trying to
understand their transport behavior. In a quantitative, set-oriented approach
to transport and mixing, finite time coherent sets play an important role.
These are time-parametrized families of sets with unlikely transport to and
from their surroundings under small or vanishing random perturbations of the
dynamics. Here we propose, as a measure of transport and mixing for purely
advective (i.e., deterministic) flows, (semi)distances that arise under
vanishing perturbations in the sense of large deviations. Analogously, for
given finite Lagrangian trajectory data we derive a discrete-time and space
semidistance that comes from the "best" approximation of the randomly perturbed
process conditioned on this limited information of the deterministic flow. It
can be computed as shortest path in a graph with time-dependent weights.
Furthermore, we argue that coherent sets are regions of maximal farness in
terms of transport and mixing, hence they occur as extremal regions on a
spanning structure of the state space under this semidistance---in fact, under
any distance measure arising from the physical notion of transport. Based on
this notion we develop a tool to analyze the state space (or the finite
trajectory data at hand) and identify coherent regions. We validate our
approach on idealized prototypical examples and well-studied standard cases.Comment: J Nonlinear Sci, 201
Entropy in Dynamic Systems
In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed
Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System
Peer reviewedPublisher PD
Deterministic Dynamics and Chaos: Epistemology and Interdisciplinary Methodology
We analyze, from a theoretical viewpoint, the bidirectional interdisciplinary
relation between mathematics and psychology, focused on the mathematical theory
of deterministic dynamical systems, and in particular, on the theory of chaos.
On one hand, there is the direct classic relation: the application of
mathematics to psychology. On the other hand, we propose the converse relation
which consists in the formulation of new abstract mathematical problems
appearing from processes and structures under research of psychology. The
bidirectional multidisciplinary relation from-to pure mathematics, largely
holds with the "hard" sciences, typically physics and astronomy. But it is
rather new, from the social and human sciences, towards pure mathematics
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