9,952 research outputs found
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
Response Characterization for Auditing Cell Dynamics in Long Short-term Memory Networks
In this paper, we introduce a novel method to interpret recurrent neural
networks (RNNs), particularly long short-term memory networks (LSTMs) at the
cellular level. We propose a systematic pipeline for interpreting individual
hidden state dynamics within the network using response characterization
methods. The ranked contribution of individual cells to the network's output is
computed by analyzing a set of interpretable metrics of their decoupled step
and sinusoidal responses. As a result, our method is able to uniquely identify
neurons with insightful dynamics, quantify relationships between dynamical
properties and test accuracy through ablation analysis, and interpret the
impact of network capacity on a network's dynamical distribution. Finally, we
demonstrate generalizability and scalability of our method by evaluating a
series of different benchmark sequential datasets
Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: pro or contra?
The phenomenon of a topological monodromy in integrable Hamiltonian and
nonholonomic systems is discussed. An efficient method for computing and
visualizing the monodromy is developed. The comparative analysis of the
topological monodromy is given for the rolling ellipsoid of revolution problem
in two cases, namely, on a smooth and on a rough plane. The first of these
systems is Hamiltonian, the second is nonholonomic. We show that, from the
viewpoint of monodromy, there is no difference between the two systems, and
thus disprove the conjecture by Cushman and Duistermaat stating that the
topological monodromy gives a topological obstruction for Hamiltonization of
the rolling ellipsoid of revolution on a rough plane.Comment: 31 pages, 11 figure
Visualizing the geometry of state space in plane Couette flow
Motivated by recent experimental and numerical studies of coherent structures
in wall-bounded shear flows, we initiate a systematic exploration of the
hierarchy of unstable invariant solutions of the Navier-Stokes equations. We
construct a dynamical, 10^5-dimensional state-space representation of plane
Couette flow at Re = 400 in a small, periodic cell and offer a new method of
visualizing invariant manifolds embedded in such high dimensions. We compute a
new equilibrium solution of plane Couette flow and the leading eigenvalues and
eigenfunctions of known equilibria at this Reynolds number and cell size. What
emerges from global continuations of their unstable manifolds is a surprisingly
elegant dynamical-systems visualization of moderate-Reynolds turbulence. The
invariant manifolds tessellate the region of state space explored by
transiently turbulent dynamics with a rigid web of continuous and discrete
symmetry-induced heteroclinic connections.Comment: 32 pages, 13 figures submitted to Journal of Fluid Mechanic
Visualizing the logistic map with a microcontroller
The logistic map is one of the simplest nonlinear dynamical systems that
clearly exhibit the route to chaos. In this paper, we explored the evolution of
the logistic map using an open-source microcontroller connected to an array of
light emitting diodes (LEDs). We divided the one-dimensional interval
into ten equal parts, and associated and LED to each segment. Every time an
iteration took place a corresponding LED turned on indicating the value
returned by the logistic map. By changing some initial conditions of the
system, we observed the transition from order to chaos exhibited by the map.Comment: LaTeX, 6 pages, 3 figures, 1 listin
Geometrical Models of the Phase Space Structures Governing Reaction Dynamics
Hamiltonian dynamical systems possessing equilibria of stability type display \emph{reaction-type
dynamics} for energies close to the energy of such equilibria; entrance and
exit from certain regions of the phase space is only possible via narrow
\emph{bottlenecks} created by the influence of the equilibrium points. In this
paper we provide a thorough pedagogical description of the phase space
structures that are responsible for controlling transport in these problems. Of
central importance is the existence of a \emph{Normally Hyperbolic Invariant
Manifold (NHIM)}, whose \emph{stable and unstable manifolds} have sufficient
dimensionality to act as separatrices, partitioning energy surfaces into
regions of qualitatively distinct behavior. This NHIM forms the natural
(dynamical) equator of a (spherical) \emph{dividing surface} which locally
divides an energy surface into two components (`reactants' and `products'), one
on either side of the bottleneck. This dividing surface has all the desired
properties sought for in \emph{transition state theory} where reaction rates
are computed from the flux through a dividing surface. In fact, the dividing
surface that we construct is crossed exactly once by reactive trajectories, and
not crossed by nonreactive trajectories, and related to these properties,
minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space
structures contained in it for 2-degree-of-freedom (DoF) systems in the
threedimensional space , and two schematic models which capture many of
the essential features of the dynamics for -DoF systems. In addition, we
elucidate the structure of the NHIM.Comment: 44 pages, 38 figures, PDFLaTe
A Hopf variables view on the libration points dynamics
The dynamics about the libration points of the Hill problem is investigated
analytically. In particular, the use of Lissajous variables and perturbation
theory allows to reduce the problem to a one degree of freedom Hamiltonian
depending on two physical parameters. The invariant manifolds structure of the
Hill problem is then disclosed, yet accurate computations are limited to energy
values close to that of the libration points
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