The role of chaotic dynamics and fractal geometry in exploration

1991 ◽  
Vol 10 (1) ◽  
pp. 25-27 ◽  
Author(s):  
Pierre L. Goupillaud
Keyword(s):  
Fractal Geometry ◽  
Chaotic Dynamics

scholarly journals The Invariant Set Postulate: a new geometric framework for the foundations of quantum theory and the role played by gravity

2009 ◽  
Vol 465 (2110) ◽  
pp. 3165-3185 ◽  
Author(s):  
T. N. Palmer
Keyword(s):  
Quantum Theory ◽  
Science Fiction ◽  
State Space ◽  
Fractal Geometry ◽  
Quantum Physics ◽  
Nonlinear Dynamical Systems ◽  
Physical Reality ◽  
Invariant Set ◽  
Geometric Framework

A new law of physics is proposed, defined on the cosmological scale but with significant implications for the microscale. Motivated by nonlinear dynamical systems theory and black-hole thermodynamics, the Invariant Set Postulate proposes that cosmological states of physical reality belong to a non-computable fractal state-space geometry I , invariant under the action of some subordinate deterministic causal dynamics D I . An exploratory analysis is made of a possible causal realistic framework for quantum physics based on key properties of I . For example, sparseness is used to relate generic counterfactual states to points p ∉ I of unreality, thus providing a geometric basis for the essential contextuality of quantum physics and the role of the abstract Hilbert Space in quantum theory. Also, self-similarity, described in a symbolic setting, provides a possible realistic perspective on the essential role of complex numbers and quaternions in quantum theory. A new interpretation is given to the standard ‘mysteries’ of quantum theory: superposition, measurement, non-locality, emergence of classicality and so on. It is proposed that heterogeneities in the fractal geometry of I are manifestations of the phenomenon of gravity. Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that a successful quantum theory of gravity should unify the causal non-Euclidean geometry of space–time with the atemporal fractal geometry of state space. The task is not to make sense of the quantum axioms by heaping more structure, more definitions, more science fiction imagery on top of them, but to throw them away wholesale and start afresh. We should be relentless in asking ourselves: From what deep physical principles might we derive this exquisite structure? These principles should be crisp, they should be compelling. They should stir the soul. Chris Fuchs ( Gilder 2008 , p. 335)

Orbit of an Image Under Iterated System II

2011 ◽  
Vol 2 (4) ◽  
pp. 57-74
Author(s):  
S. L. Singh ◽  
S. N. Mishra ◽  
Sarika Jain
Keyword(s):  
Fractal Geometry ◽  
Iterative Procedure ◽  
Iterated Function System ◽  
Mathematical Structure ◽  
Program Code ◽  
Computational Mathematics ◽  
Fractal Image ◽  
Iterated Function ◽  
Iterative Procedures

An orbital picture is a mathematical structure depicting the path of an object under Iterated Function System. Orbital and V-variable orbital pictures initially developed by Barnsley (2006) have utmost importance in computer graphics, image compression, biological modeling and other areas of fractal geometry. These pictures have been generated for linear and contractive transformations using function and superior iterative procedures. In this paper, the authors introduce the role of superior iterative procedure to find the orbital picture under an IFS consisting of non-contractive or non-expansive transformations. A mild comparison of the computed figures indicates the usefulness of study in computational mathematics and fractal image processing. A modified algorithm along with program code is given to compute a 2-variable superior orbital picture.

Is there Chaos on the German Labor Market? / Ist der deutsche Arbeitsmarkt chaotisch?

1999 ◽  
Vol 218 (5-6) ◽  
Author(s):  
Michael Neugart
Keyword(s):  
Time Series ◽  
Labor Market ◽  
Chaotic Dynamics ◽  
Alternative Hypothesis ◽  
The Past ◽  
Labor Market Dynamics ◽  
Linear And Nonlinear ◽  
Bds Test ◽  
Linear And Nonlinear Systems

SummaryEvidence on the role of chaotic and nonlinear dynamics on labor markets is mixed. It is unclear whether nonlinear relationships are responsible for the dynamic patterns observed in Europe during the past decades. In this paper, we test German labor market data for the null hypothesis of an i.i.d. process with the BDS test. As several processes including chaotic, nonlinear deterministic, and stochastic linear and nonlinear systems are nested within the alternative hypothesis, time series are whitened with linear and nonlinear filters. Lyapunov exponents and correlation dimensions are applied to the residuals of the filtered time series to test for chaotic dynamics. There seems to be a nonlinear deterministic core to German labor market dynamics. Chaos does not occur.

scholarly journals Quantifying the Role of Folding in Nonautonomous Flows: The Unsteady Double-Gyre

2017 ◽  
Vol 27 (10) ◽  
pp. 1750156 ◽  
Author(s):  
K. G. D. Sulalitha Priyankara ◽  
Sanjeeva Balasuriya ◽  
Erik Bollt
Keyword(s):  
Chaotic Dynamics ◽  
Two Dimensions ◽  
Birkhoff Theorem ◽  
System A ◽  
Gyre System ◽  
Double Gyre

We analyze chaos in the well-known nonautonomous Double-Gyre system. A key focus is on folding, which is possibly the less-studied aspect of the “stretching+folding=chaos” mantra of chaotic dynamics. Despite the Double-Gyre not having the classical homoclinic structure for the usage of the Smale–Birkhoff theorem to establish chaos, we use the concept of folding to prove the existence of an embedded horseshoe map. We also show how curvature of manifolds can be used to identify fold points in the Double-Gyre. This method is applicable to general nonautonomous flows in two dimensions, defined for either finite or infinite times.

CONTROL OF THE TRANSITION TO CHAOS IN NEURAL NETWORKS WITH RANDOM CONNECTIVITY

1993 ◽  
Vol 03 (02) ◽  
pp. 279-291 ◽  
Author(s):  
B. DOYON ◽  
B. CESSAC ◽  
M. QUOY ◽  
M. SAMUELIDES
Keyword(s):  
Neural Networks ◽  
Chaotic Dynamics ◽  
Synaptic Coupling ◽  
Sustained Activity ◽  
Weak Connectivity ◽  
Fully Connected ◽  
The Brain ◽  
Theoretical Results ◽  
Random Connectivity

The occurrence of chaos in recurrent neural networks is supposed to depend on the architecture and on the synaptic coupling strength. It is studied here for a randomly diluted architecture. We produce a bifurcation parameter independent of the connectivity that allows a sustained activity and the occurrence of chaos when reaching a critical value. Even for weak connectivity and small size, we find numerical results in accordance with the theoretical ones previously established for fully connected infinite sized networks. Moreover the route towards chaos is numerically checked to be a quasiperiodic one, whatever the type of the first bifurcation is. In the discussion, we connect these results to some recent theoretical results about highly diluted networks. Hints are provided for further investigations to elicit the role of chaotic dynamics in the cognitive processes of the brain.

Role of surface roughness characterized by fractal geometry on laminar flow in microchannels

2009 ◽  
Vol 80 (2) ◽  
Author(s):  
Yongping Chen ◽  
Chengbin Zhang ◽  
Mingheng Shi ◽  
G. P. Peterson
Keyword(s):  
Surface Roughness ◽  
Laminar Flow ◽  
Fractal Geometry

scholarly journals Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

2018 ◽  
Author(s):  
Colin Grudzien ◽  
Alberto Carrassi ◽  
Marc Bocquet
Keyword(s):  
Ensemble Kalman Filter ◽  
Kalman Filters ◽  
Chaotic Dynamics ◽  
Forecast Error ◽  
Model Error ◽  
High Dimensional ◽  
Reduced Rank ◽  
Nonlinear Approximations ◽  
Complementary Subspace

Abstract. The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. A reduced rank representation of the estimated covariance, however, leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, describing the intrinsic role of covariance inflation in reduced rank, ensemble based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically.

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