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On Selecting Models for Nonlinear Time Series
 Physica D
, 1995
"... Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintainin ..."
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Cited by 49 (14 self)
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Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...
Modeling Chaotic Motions of a String From Experimental Data
 Physica D
, 1996
"... Experimental measurements of nonlinear vibrations of a string are analyzed using new techniques of nonlinear modeling. Previous theoretical and numerical work suggested that the motions of a string can be chaotic and a Shil'nikov mechanism is responsible. We show that the experimental data is c ..."
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Cited by 5 (5 self)
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Experimental measurements of nonlinear vibrations of a string are analyzed using new techniques of nonlinear modeling. Previous theoretical and numerical work suggested that the motions of a string can be chaotic and a Shil'nikov mechanism is responsible. We show that the experimental data is consistent with a Shil'nikov mechanism. We also reveal a period doubling cascade with a period three window which is not immediately observable because there is sufficient noise, probably of a dynamical origin, to mask the perioddoubling bifurcation and the period three window. 1 Introduction Bajaj and Johnson [2] have conducted an analysis of weakly nonlinear partial differential equations describing the forced vibrations of stretched uniform strings. The equations take into account motions transverse to the plan of forcing, which are induced by a coupling with longitudinal displacements, and changes in tension that occur in large amplitude motions. The averaged equations of a resonant system c...
Wavelet Reconstruction Of Nonlinear Dynamics
, 1998
"... this paper investigates whether or not these properties translate well to be used in modeling embedded timeseries from dynamical systems. We outline two strategies for using wavelets for such modeling. ..."
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Cited by 2 (0 self)
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this paper investigates whether or not these properties translate well to be used in modeling embedded timeseries from dynamical systems. We outline two strategies for using wavelets for such modeling.
Fuzzy differential inclusions in atmospheric and medical cybernetics
 IEEE Trans. Syst., Man, Cybern. B, Cybern
, 2004
"... Abstract—Uncertainty management in dynamical systems is receiving attention in artificial intelligence, particularly in the fields of qualitative and model based reasoning. Fuzzy dynamical systems occupy a very important position in the class of uncertain systems. It is well established that the f ..."
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Cited by 1 (0 self)
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Abstract—Uncertainty management in dynamical systems is receiving attention in artificial intelligence, particularly in the fields of qualitative and model based reasoning. Fuzzy dynamical systems occupy a very important position in the class of uncertain systems. It is well established that the fuzzy dynamical systems represented by a set of fuzzy differential inclusions (FDI) are very convenient tools for modeling and simulation of various uncertain systems. In this paper, we discuss about the mathematical modeling of two very complex natural phenomena by means of FDIs. One of them belongs to the atmospheric cybernetics (the term has been used in a broad sense) of the genesis of a cyclonic storm (cyclogenesis), and the other belongs to the biomedical cybernetics of the evolution of tumor in a human body. Since a discussion of the former already appears in a previous paper by the first author, here, we present very briefly a theoretical formalism of cyclone formation. On the other hand, we treat the latter system more elaborately. We solve the FDIs with the help of an algorithm developed in this paper to numerically simulate the mathematical models. From the simulation results thus obtained, we have drawn a number of interesting conclusions, which have been verified, and this vindicates the validity of our models. Index Terms—Carcinogenesis, cybernetics, cyclogenesis, fuzzy differential inclusions, fuzzy dynamical systems. I.
EMPIRICAL PSEUDORANDOM NUMBER GENERATORS
"... The most common pseudorandom number generator or PRNG, the linear congruential generator or LCG, belongs to a whole class of rational congruential generators. These generators work by multiplicative congruential method for integers, which implements a ”growandcut procedure”. We extend this concept ..."
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The most common pseudorandom number generator or PRNG, the linear congruential generator or LCG, belongs to a whole class of rational congruential generators. These generators work by multiplicative congruential method for integers, which implements a ”growandcut procedure”. We extend this concept to real numbers and call this the real congruence, which produces another class of random number generators called real congruential generators or RCG. The method in RCG inherits the procedure in LCG. Let m be a positive integer and I the interval (0,1). Consider a mapping fm:I → [m, m] and generate a sequence (xn) as follows: (1) Let x0ɛI be the seed; (2) For every positive integer n, xn+1 = [[fm (xn)]], where [[x]] = x [x], [ ] is the greatest integer function. In this paper, we investigate the sequence generated by fm (x) = msin(πx). It turns out that the finite nature of computer numbers contributes to the ”randomness ” of the sequence. Simple applications and empirical testing were implemented. 1