201 research outputs found
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
On the dynamical origin of asymptotic t^2 dispersion of a nondiffusive tracer in incompressible laminar flows
Using an elementary application of Birkhoff's ergodic theorem, necessary and sufficient conditions are given for the existence of asymptotically t^2 dispersion of a distribution of nondiffusive passive tracer in a class of incompressible laminar flows. Nonergodicity is shown to be the dynamical mechanism giving rise to this behavior
A method for visualization of invariant sets of dynamical systems based on the ergodic partition
We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al
Invariant Sets in Quasiperiodically Forced Dynamical Systems
This paper addresses structures of state space in quasiperiodically forced
dynamical systems. We develop a theory of ergodic partition of state space in a
class of measure-preserving and dissipative flows, which is a natural extension
of the existing theory for measure-preserving maps. The ergodic partition
result is based on eigenspace at eigenvalue 0 of the associated Koopman
operator, which is realized via time-averages of observables, and provides a
constructive way to visualize a low-dimensional slice through a
high-dimensional invariant set. We apply the result to the systems with a
finite number of attractors and show that the time-average of a continuous
observable is well-defined and reveals the invariant sets, namely, a finite
number of basins of attraction. We provide a characterization of invariant sets
in the quasiperiodically forced systems. A theoretical result on uniform
boundedness of the invariant sets is presented. The series of theoretical
results enables numerical analysis of invariant sets in the quasiperiodically
forced systems based on the ergodic partition and time-averages. Using this, we
analyze a nonlinear model of complex power grids that represents the short-term
swing instability, named the coherent swing instability. We show that our
theoretical results can be used to understand stability regions in such complex
systems.Comment: 23 pages, 4 figure
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