Analysis of nonlinear oscillators in the frequency domain using volterra series Part II : identifying and modelling jump Phenomenon

In this the second part of the paper, a common and severe nonlinear phenomenon called jump, a behaviour associated with the Duffing oscillator and the multi-valued properties of the response solution, is investigated. The new frequency domain criterion of establishing the upper limits of the nonlinear oscillators, developed in Part I of this paper, is applied to predict the onset point of the jump, and the Volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques

A comparison of polynomial and wavelet expansions for the identification of chaotic coupled map lattices

A comparison between polynomial and wavelet expansions for the identification of coupled map lattice (CML) models for deterministic spatio-temporal dynamical systems is presented in this paper. The pattern dynamics generated by smooth and non-smooth nonlinear maps in a well-known 2-dimensional CML structure are analysed. By using an orthogonal feedforward regression algorithm (OFR), polynomial and wavelet models are identified for the CML’s in chaotic regimes. The quantitative dynamical invariants such as the largest Lyapunov exponents and correlation dimensions are estimated and used to evaluate the performance of the identified models

Piecewise Volterra modelling of the Duffing oscillator in the frequency domain

When analysing the nonlinear Duffing oscillator, the weak nonlinearity is basically dependent on the amplitude range of the input excitation. The nonlinear differential equation models of such nonlinear oscillators, which can be transformed into the frequency domain, can generally only provide Volterra modelling and analysis in the frequency-domain over a fraction of the entire framework of weak nonlinearity. This paper discusses the problem of using a new non-parametric routine to extend the capability of Volterra analysis, in the frequency domain, to weakly nonlinear Duffing systems at a wider range of excitation amplitude range which the current underlying nonlinear differential equation models fail to address

The identification of cellular automata

Although cellular automata have been widely studied as a class of the spatio temporal systems, very few investigators have studied how to identify the CA rules given observations of the patterns. A solution using a polynomial realization to describe the CA rule is reviewed in the present study based on the application of an orthogonal least squares algorithm. Three new neighbourhood detection methods are then reviewed as important preliminary analysis procedures to reduce the complexity of the estimation. The identification of excitable media is discussed using simulation examples and real data sets and a new method for the identification of hybrid CA is introduced

Identification of binary cellular automata from spatiotemporal binary patterns using a fourier representation

The identification of binary cellular automata from spatio-temporal binary patterns is investigated in this paper. Instead of using the usual Boolean or multilinear polynomial representation, the Fourier transform representation of Boolean functions is employed in terms of a Fourier basis. In this way, the orthogonal forward regression least-squares algorithm can be applied directly to detect the significant terms and to estimate the associated parameters. Compared with conventional methods, the new approach is much more robust to noise. Examples are provided to illustrate the effectiveness of the proposed approach

Feature subset selection and ranking for data dimensionality reduction

A new unsupervised forward orthogonal search (FOS) algorithm is introduced for feature selection and ranking. In the new algorithm, features are selected in a stepwise way, one at a time, by estimating the capability of each specified candidate feature subset to represent the overall features in the measurement space. A squared correlation function is employed as the criterion to measure the dependency between features and this makes the new algorithm easy to implement. The forward orthogonalization strategy, which combines good effectiveness with high efficiency, enables the new algorithm to produce efficient feature subsets with a clear physical interpretation

Identification of the neighborhood and CA rules from spatio-temporal CA patterns

Extracting the rules from spatio-temporal patterns generated by the evolution of cellular automata (CA) usually produces a CA rule table without providing a clear understanding of the structure of the neighborhood or the CA rule. In this paper, a new identification method based on using a modified orthogonal least squares or CA-OLS algorithm to detect the neighborhood structure and the underlying polynomial form of the CA rules is proposed. The Quine-McCluskey method is then applied to extract minimum Boolean expressions from the polynomials. Spatio-temporal patterns produced by the evolution of 1D, 2D, and higher dimensional binary CAs are used to illustrate the new algorithm, and simulation results show that the CA-OLS algorithm can quickly select both the correct neighborhood structure and the corresponding rule

Neighborhood detection and rule selection from cellular automata patterns

Using genetic algorithms (GAs) to search for cellular automation (CA) rules from spatio-temporal patterns produced in CA evolution is usually complicated and time-consuming when both, the neighborhood structure and the local rule are searched simultaneously. The complexity of this problem motivates the development of a new search which separates the neighborhood detection from the GA search. In the paper, the neighborhood is determined by independently selecting terms from a large term set on the basis of the contribution each term makes to the next state of the cell to be updated. The GA search is then started with a considerably smaller set of candidate rules pre-defined by the detected neighhorhood. This approach is tested over a large set of one-dimensional (1-D) and two-dimensional (2-D) CA rules. Simulation results illustrate the efficiency of the new algorith

Sensitivity study of generalised frequency response functions

The dependence and independence of input signal amplitudes for Generalised Frequency Response Functions(GFRF’s) are discussed based on parametric modelling

Characterising linear spatio-temporal dynamical systems in the frequency domain

A new concept, called the spatio-temporal transfer function (STTF), is introduced to characterise a class of linear time-invariant (LTI) spatio-temporal dynamical systems. The spatio-temporal transfer function is a natural extension of the ordinary transfer function for classical linear time-invariant control systems. As in the case of the classical transfer function, the spatio-temporal transfer function can be used to characterise, in the frequency domain, the inherent dynamics of linear time-invariant spatio-temporal systems. The introduction of the spatio-temporal transfer function should also facilitate the analysis of the dynamical stability of discrete-time spatio-temporal systems