Numerical and experimental investigation of the effect of filtering on chaotic symbolic dynamics

Motivated by the practical consideration of the measurement of chaotic signals in experiments or the transmission of these signals through a physical medium, we investigate the effect of filtering on chaotic symbolic dynamics. We focus on the linear, time-invariant filters that are used frequently in many applications, and on the two quantities characterizing chaotic symbolic dynamics: topological entropy and bit-error rate. Theoretical consideration suggests that the topological entropy is invariant under filtering. Since computation of this entropy requires that the generating partition for defining the symbolic dynamics be known, in practical situations the computed entropy may change as a filtering parameter is changed. We find, through numerical computations and experiments with a chaotic electronic circuit, that with reasonable care the computed or measured entropy values can be preserved for a wide range of the filtering parameter.     

Decline of minorities in stubborn societies

Opinion compromise models can give insight into how groups of individuals may either come to form consensus or clusters of opinion groups, corresponding to parties. We consider models where randomly selected individuals interact pairwise. If the opinions of the interacting agents are not within a certain confidence threshold, the agents retain their own point of view. Otherwise, they constructively dialogue and smooth their opinions. Persuasible agents are inclined to compromise with interacting individuals. Stubborn individuals slightly modify their opinion during the interaction. Collective states for persuasible societies include extremist minorities, which instead decline in stubborn societies. We derive a mean field approximation for the compromise model in stubborn populations. Bifurcation and clustering analysis of this model compares favorably with Monte Carlo analysis found in the literature.     

Deconstructing spatiotemporal chaos using local symbolic dynamics

We find that the global symbolic dynamics of a diffusively coupled map lattice is well approximated by a very small local model for weak to moderate coupling strengths. A local symbolic model is a truncation of the full symbolic model to one that considers only a single element and a few neighbors. Using interval analysis, we give rigorous results for a range of coupling strengths and different local model widths. Examples are presented of extracting a local symbolic model from data and of controlling spatiotemporal chaos.     

Dimensional implications of dynamical data on manifolds to empirical KL analysis

We explore the approximation of attracting manifolds of complex systems using dimension reducing methods. Complex systems having high-dimensional dynamics typically are initially analyzed by exploring techniques to reduce the dimension. Linear techniques, such as Galerkin projection methods, and nonlinear techniques, such as center manifold reduction are just some of the examples used to approximate the manifolds on which the attractors lie. In general, if the manifold is not highly curved, then both linear and nonlinear methods approximate the surface well. However, if the manifold curvature changes significantly with respect to parametric variations, then linear techniques may fail to give an accurate model of the manifold. This may not be a surprise in itself, but it is a fact so often overlooked or misunderstood when utilizing the popular KL method, that we offer this explicit study of the effects and consequences. Here we show that certain dimensions defined by linear methods are highly sensitive when modeled in situations where the attracting manifolds have large parametric curvature. Specifically, we show how manifold curvature mediates the dimension when using a linear basis set as a model. We punctuate our results with the definition of what we call, a “curvature induced parameter,” $d_{CI}$. Both finite- and infinite-dimensional models are used to illustrate the theory.     

Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion

We introduce variants of the variational image denoising method proposed by Blomgren et al. (In: Numerical Analysis 1999 (Dundee), pp. 43–67. Chapman & Hall, Boca Raton, FL, 2000), which interpolates between total-variation denoising and isotropic diffusion denoising. We study how parameter choices affect results and allow tuning between TV denoising and isotropic diffusion for respecting texture on one spatial scale while denoising features assumed to be noise on finer spatial scales. Furthermore, we prove existence and (where appropriate) uniqueness of minimizers. We consider both L ² and L ¹ data fidelity terms.     

Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series

A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for determining the symbols. We address to what extent the symbolic dynamics so obtained can faithfully represent the phase-space dynamics. Our principal result is that such a practice can lead to a severe misrepresentation of the dynamical system. The measured topological entropy is a Devil's staircase-like, but surprisingly nonmonotone, function of a parameter characterizing the amount of misplacement of the partition.     

The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: vector fields and transport barriers from movies

In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences.     

A concept of homeomorphic defect for defining mostly conjugate dynamical systems

A centerpiece of dynamical systems is comparison by an equivalence relationship called topological conjugacy. We present details of how a method to produce conjugacy functions based on a functional fixed point iteration scheme can be generalized to compare dynamical systems that are not conjugate. When applied to nonconjugate dynamical systems, we show that the fixed-point iteration scheme still has a limit point, which is a function we now call a "commuter"-a nonhomeomorphic change of coordinates translating between dissimilar systems. This translation is natural to the concepts of dynamical systems in that it matches the systems within the language of their orbit structures, meaning that orbits must be matched to orbits by some commuter function. We introduce methods to compare nonequivalent systems by quantifying how much the commuter function fails to be a homeomorphism, an approach that gives more respect to the dynamics than the traditional comparisons based on normed linear spaces, such as L(2). Our discussion addresses a fundamental issue-how does one make principled statements of the degree to which a "toy model" might be representative of a more complicated system?