The Synthesis of Arbitrary Stable Dynamics in Non-linear Neural Networks II: Feedback and Universality

We wish to construct a realization theory of stable neural networks and use this theory to model the variety of stable dynamics apparent in natural data. Such a theory should have numerous applications to constructing specific artificial neural networks with desired dynamical behavior. The networks used in this theory should have well understood dynamics yet be as diverse as possible to capture natural diversity. In this article, I describe a parameterized family of higher order, gradient-like neural networks which have known arbitrary equilibria with unstable manifolds of known specified dimension. Moreover, any system with hyperbolic dynamics is conjugate to one of these systems in a neighborhood of the equilibrium points. Prior work on how to synthesize attractors using dynamical systems theory, optimization, or direct parametric. fits to known stable systems, is either non-constructive, lacks generality, or has unspecified attracting equilibria. More specifically, We construct a parameterized family of gradient-like neural networks with a simple feedback rule which will generate equilibrium points with a set of unstable manifolds of specified dimension. Strict Lyapunov functions and nested periodic orbits are obtained for these systems and used as a method of synthesis to generate a large family of systems with the same local dynamics. This work is applied to show how one can interpolate finite sets of data, on nested periodic orbits.

Model selection approaches for non-linear system identification: a review

The identification of non-linear systems using only observed finite datasets has become a mature research area over the last two decades. A class of linear-in-the-parameter models with universal approximation capabilities have been intensively studied and widely used due to the availability of many linear-learning algorithms and their inherent convergence conditions. This article presents a systematic overview of basic research on model selection approaches for linear-in-the-parameter models. One of the fundamental problems in non-linear system identification is to find the minimal model with the best model generalisation performance from observational data only. The important concepts in achieving good model generalisation used in various non-linear system-identification algorithms are first reviewed, including Bayesian parameter regularisation and models selective criteria based on the cross validation and experimental design. A significant advance in machine learning has been the development of the support vector machine as a means for identifying kernel models based on the structural risk minimisation principle. The developments on the convex optimisation-based model construction algorithms including the support vector regression algorithms are outlined. Input selection algorithms and on-line system identification algorithms are also included in this review. Finally, some industrial applications of non-linear models are discussed.

Determination of Non-linear Damping Coefficients of bottom-hinged Oscillating Wave Surge Converters Using Numerical Free Decay Tests

Linear wave theory models are commonly applied to predict the performance of bottom-hinged oscillating wave surge converters (OWSC) in operational sea states. To account for non-linear effects, the additional input of coefficients not included in the model itself becomes necessary. In ocean engineering it is
common practice to obtain damping coefficients of floating structures from free decay tests. This paper presents results obtained from experimental tank tests and numerical computational fluid dynamics simulations of OWSC’s. Agreement between numerical and experimental methods is found to be very good, with CFD providing more data points at small amplitude rotations.
Analysis of the obtained data reveals that linear quadratic-damping, as commonly used in time domain models, is not able to accurately model the occurring damping over the whole regime of rotation amplitudes. The authors
conclude that a hyperbolic function is most suitable to express the instantaneous damping ratio over the rotation amplitude and would be the best choice to be used in coefficient based time domain models.

Non-linear finite element analysis of thin composite structures

A non-conforming three-node triangular finite element with 18 degree of freedom, is used in conjugation with the Kirchhoff theory for the non-linear analysis of thin composite plate-shell structure. The formulation of the geometrically non-linear analysis is based on an updated Lagrangian formulation associated with the Newton-Raphson iterative technique, which incorporates an automatic arc-length control procedure.

Non-linear analysis of composite structures with integrated piezoelectric sensors and actuators

This paper deals with the geometrically non linear analysis of thin plate/shell laminated structures with embedded integrated piezoelectric actuors or sensors layers and/or patches.The model is based on the Kirchhoff classical laminated theory and can be applied to plate and shell adaptive structures with arbitrary shape, general mechanical and electrical loadings. the finite element model is a nonconforming single layer triangular plate/shell element with 18 degrees of fredom for the generalized displacements and one eçlectrical potential degree of freedom for each piezoelectric layer or patch. An updated Lagrangian formulation associated to Newton-Raphson technique is used to solve incrementally and iteratively the equilibrium equation.The model is applied in the solution of four illustrative cases, and the results are compared and discussedwith alternative solutions when available.

Some non- linear problems in theoretical physics

An immense variety of problems in theoretical physics are of the non-linear type. Non~linear partial differential equations (NPDE) have almost become the rule rather than an exception in diverse branches of physics such as fluid mechanics, field theory, particle physics, statistical physics and optics, and the construction of exact solutions of these equations constitutes one of the most vigorous activities in theoretical physics today. The thesis entitled ‘Some Non-linear Problems in Theoretical Physics’ addresses various aspects of this problem at the classical level. For obtaining exact solutions we have used mathematical tools like the bilinear operator method, base equation technique and similarity method with emphasis on its group theoretical aspects. The thesis deals with certain methods of finding exact solutions of a number of non-linear partial differential equations of importance to theoretical physics. Some of these new solutions are of relevance from the applications point of view in diverse branches such as elementary particle physics, field theory, solid state physics and non-linear optics and give some insight into the stable or unstable behavior of dynamical Systems The thesis consists of six chapters.

Relevance of sampling schemes in light of Ruelle's linear response theory

We reconsider the theory of the linear response of non-equilibrium steady states to perturbations. We �rst show that by using a general functional decomposition for space-time dependent forcings, we can de�ne elementary susceptibilities that allow to construct the response of the system to general perturbations. Starting from the de�nition of SRB measure, we then study the consequence of taking di�erent sampling schemes for analysing the response of the system. We show that only a speci�c choice of the time horizon for evaluating the response of the system to a general time-dependent perturbation allows to obtain the formula �rst presented by Ruelle. We also discuss the special case of periodic perturbations, showing that when they are taken into consideration the sampling can be �ne-tuned to make the de�nition of the correct time horizon immaterial. Finally, we discuss the implications of our results in terms of strategies for analyzing the outputs of numerical experiments by providing a critical review of a formula proposed by Reick.

Fractional order and non-linear system identification algorithms for biomedical applications

We discuss the modelling of dielectric responses of amorphous biological samples. Such samples are commonly encountered in impedance spectroscopy studies as well as in UV, IR, optical and THz transient spectroscopy experiments and in pump-probe studies. In many occasions, the samples may display quenched absorption bands. A systems identification framework may be developed to provide parsimonious representations of such responses. To achieve this, it is appropriate to augment the standard models found in the identification literature to incorporate fractional order dynamics. Extensions of models using the forward shift operator, state space models as well as their non-linear Hammerstein-Wiener counterpart models are highlighted. We also discuss the need to extend the theory of electromagnetically excited networks which can account for fractional order behaviour in the non-linear regime by incorporating nonlinear elements to account for the observed non-linearities. The proposed approach leads to the development of a range of new chemometrics tools for biomedical data analysis and classification.

A state space based approach in non-linear hemodynamic response modeling with fMRI data

In this paper we use the modified and integrated version of the balloon model in the analysis of fMRI data. We propose a new state space model realization for this balloon model and represent it with the standard A,B,C and D matrices widely used in system theory. A second order Padé approximation with equal numerator and denominator degree is used for the time delay approximation in the modeling of the cerebral blood flow. The results obtained through numerical solutions showed that the new state space model realization is in close agreement to the actual modified and integrated version of the balloon model. This new system theoretic formulation is likely to open doors to a novel way of analyzing fMRI data with real time robust estimators. With further development and validation, the new model has the potential to devise a generalized measure to make a significant contribution to improve the diagnosis and treatment of clinical scenarios where the brain functioning get altered. Concepts from system theory can readily be used in the analysis of fMRI data and the subsequent synthesis of filters and estimators.

On the performance of a dual-mode non-linear vibration energy harvesting device

The research trend for harvesting energy from the ambient vibration sources has moved from using a linear resonant generator to a non-linear generator in order to improve on the performance of a linear generator; for example, the relatively small bandwidth, intolerance to mistune and the suitability of the device for low-frequency applications. This article presents experimental results to illustrate the dynamic behaviour of a dual-mode non-linear energy-harvesting device operating in hardening and bi-stable modes under harmonic excitation. The device is able to change from one mode to another by altering the negative magnetic stiffness by adjusting the separation gap between the magnets and the iron core. Results for the device operating in both modes are presented. They show that there is a larger bandwidth for the device operating in the hardening mode compared to the equivalent linear device. However, the maximum power transfer theory is less applicable for the hardening mode due to occurrence of the maximum power at different frequencies, which depends on the non-linearity and the damping in the system. The results for the bi-stable mode show that the device is insensitive to a range of excitation frequencies depending upon the input level, damping and non-linearity.

On non-linear dynamics and control designs applied to the ideal and non-ideal variants of the Fitzhugh-Nagumo (FN) mathematical model

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Non-linear boundary element analysis of floor slabs reinforced with rectangular beams

In this work, a numerical model to perform non-linear analysis of building floor structures is proposed. The presented model is derived from the Kirchhoff-s plate bending formulation of the boundary element method (BENI) for zoned domains, in which the plate stiffness is modified by the presence of membrane effects. In this model, no approximation of the generalized forces along the interface is required and the compatibility and equilibrium conditions along interfaces are imposed at the integral equation level. In order to reduce the number of degrees of freedom, the Navier Bernoulli hypothesis is assumed to simplify the strain field for the thin sub-regions (rectangular beams). The non-linear formulation is obtained from the linear formulation by incorporating initial internal force fields, which are approximated by using the well-known cell sub-division. Then, the non-linear solution of algebraic equations is obtained by using the concept of the consistent tangent operator. The Von Mises criterion is adopted to govern the elasto-plastic material behaviour checked at points along the plate thickness and along the rectangular beam element axes. The numerical representations are accurately obtained by either computing analytically the element integrals or performing the numerical integration accurately using an appropriate sub-elementation scheme. (C) 2007 Elsevier Ltd. All rights reserved.

Non-linear analysis of reinforced concrete plates by the boundary element method and continuum damage mechanics

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)