NOMINAL CROSS RECURRENCE AS A GENERALIZED LAG SEQUENTIAL ANALYSIS FOR BEHAVIORAL STREAMS

We briefly present lag sequential analysis for behavioral streams, a commonly used method in psychology for quantifying the relationships between two nominal time series. Cross recurrence quantification analysis (CRQA) is shown as an extension of this technique, and we exemplify this nominal application of CRQA to eye-movement data in human interaction. In addition, we demonstrate nominal CRQA in a simple coupled logistic map simulation used in previous communication research, permitting the investigation of properties of nonlinear systems such as bifurcation and onset to chaos, even in the streams obtained by coarse-graining a coupled nonlinear model. We end with a summary of the importance of CRQA for exploring the relationship between two behavioral streams, and review a recent theoretical trend in the cognitive sciences that would be usefully informed by this and similar nonlinear methods. We hope this work will encourage scientists interested in general properties of complex, nonlinear dynamical systems to apply emerging methods to coarse-grained, nominal units of measure, as there is an immediate need for their application in the psychological domain.

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Linear and Nonlinear Dynamic Behaviour

10.1093/oso/9780198782933.003.0002 ◽

2018 ◽

Author(s):

Ray Huffaker ◽

Marco Bittelli ◽

Rodolfo Rosa

Keyword(s):

Dynamic Behavior ◽

Dynamic Systems ◽

Dynamic Behaviour ◽

Nonlinear Dynamical Systems ◽

Logistic Map ◽

Nonlinear Dynamical ◽

Deterministic Dynamic ◽

Random Behavior ◽

Single Solution ◽

Equivalent Systems

In this chapter, we describe how highly erratic dynamic behavior can arise from a nonlinear logistic map, and how this apparently random behavior is governed by a surprising order. With this lesson in mind, we should not be overly surprised that highly erratic and random appearing observed data might also be generated by parsimonious deterministic dynamic systems. At a minimum, we contend that researchers should apply NLTS to test for this possibility. We also introduced tools to analyze dynamic behavior that form the foundation for NLTS. In particular, we have stressed the quite unexpected capability to achieve some form of predictability even with only one trajectory at hand. In subsequent chapters, we treat known nonlinear dynamical systems as unknown, and investigate how NLTS methods rely on a single solution (or multiple solutions) generated by them to reconstruct equivalent systems. This is a conventional approach in the literature for seeing how NLTS methods work since we know what needs to be reconstructed.

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Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0754 ◽

2014 ◽

Vol 470 (2166) ◽

pp. 20130754 ◽

Cited By ~ 26

Author(s):

D. Venturi ◽

G. E. Karniadakis

Keyword(s):

Dynamical Systems ◽

Stochastic Analysis ◽

Evolution Equations ◽

Nonlinear Dynamical Systems ◽

Coarse Graining ◽

Perturbation Series ◽

Stochastic Nonlinear Systems ◽

Nonlinear Dynamical ◽

Major Interest ◽

Low Dimensional

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima–Zwanzig–Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection–reaction problems.

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Analysis of the equilibrium positions of nonlinear dynamical systems in the presence of coarse-graining disturbance in space

Journal of Applied Mathematics and Computing ◽

10.1007/s12190-009-0254-5 ◽

2009 ◽

Vol 32 (2) ◽

pp. 329-351 ◽

Cited By ~ 3

Author(s):

Guy Jumarie

Keyword(s):

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Coarse Graining ◽

Nonlinear Dynamical

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Superconvergence of topological entropy in the symbolic dynamics of substitution sequences

SciPost Physics ◽

10.21468/scipostphys.7.2.018 ◽

2019 ◽

Vol 7 (2) ◽

Cited By ~ 2

Author(s):

Leon Zaporski ◽

Felix Flicker

Keyword(s):

Topological Entropy ◽

Symbolic Dynamics ◽

Convergence Rates ◽

Nonlinear Dynamical Systems ◽

Logistic Map ◽

Universality Class ◽

Accumulation Point ◽

Nonlinear Dynamical ◽

Double Exponential ◽

Symbolic Sequences

We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.

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RECURRENCE PLOT AND RECURRENCE QUANTIFICATION ANALYSIS TECHNIQUES FOR DETECTING A CRITICAL REGIME.

International Journal of Modern Physics C ◽

10.1142/s0129183105007492 ◽

2005 ◽

Vol 16 (05) ◽

pp. 671-706 ◽

Cited By ~ 51

Author(s):

A. FABRETTI ◽

M. AUSLOOS

Keyword(s):

Nonlinear Dynamical Systems ◽

Recurrence Plot ◽

Recurrence Quantification Analysis ◽

Critical Regime ◽

Initial Time ◽

Nonlinear Dynamical ◽

Recurrence Quantification ◽

Quantification Analysis ◽

Set Up ◽

Financial Indices

Recurrence Plot (RP) and Recurrence Quantification Analysis (RQA) are signal numerical analysis methodologies able to work with nonlinear dynamical systems and nonstationarity. Moreover, they well evidence changes in the states of a dynamical system. We recall their features and give practical recipes. It is shown that RP and RQA detect the critical regime in financial indices (in analogy with phase transition) before a bubble bursts, whence allowing to estimate the bubble initial time. The analysis is made on DAX and NASDAQ daily closing price between January 1998 and November 2003. DAX is studied in order to set-up overall considerations, and as a support for deducing technical rules. The NASDAQ bubble initial time has been estimated to be on 19 October 1999.

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Ljapunov Exponents, Hyperchaos and Hurst Exponent

Zeitschrift für Naturforschung A ◽

10.1515/zna-2005-0406 ◽

2005 ◽

Vol 60 (4) ◽

pp. 252-254 ◽

Cited By ~ 1

Author(s):

Willi-Hans Steeb ◽

Eugenio Cosme Andrieu

Keyword(s):

Dynamical Systems ◽

Hurst Exponent ◽

Nonlinear Dynamical Systems ◽

Logistic Map ◽

Two Dimensional ◽

Nonlinear Dynamical

Abstract We consider nonlinear dynamical systems with chaotic and hyperchaotic behaviour.We investigate the behaviour of the Hurst exponent at the transition from chaos to hyperchaos. A two-dimensional coupled logistic map is studied.

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Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

10.31235/osf.io/g8nsf ◽

2018 ◽

Author(s):

Geoff Boeing

Keyword(s):

Dynamical Systems ◽

Phase Diagrams ◽

Nonlinear Dynamical Systems ◽

Nonlinear Models ◽

Logistic Map ◽

Three Dimensional ◽

Basins Of Attraction ◽

Nonlinear Dynamical ◽

One Dimensional ◽

One Dimensional Maps

Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

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