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.

An approach to determine the local boundary of small disturbance voltage stability region in a cut-set power space:基于微扰的割集电压稳定域局部边界求解方法

A new approach to determine the local boundary of voltage stability region in a cut-set power space (CVSR) is presented. Power flow tracing is first used to determine the generator-load pair most sensitive to each branch in the interface. The generator-load pairs are then used to realize accurate small disturbances by controlling the branch power flow in increasing and decreasing directions to obtain new equilibrium points around the initial equilibrium point. And, continuous power flow is used starting from such new points to get the corresponding critical points around the initial critical point on the CVSR boundary. Then a hyperplane cross the initial critical point can be calculated by solving a set of linear algebraic equations. Finally, the presented method is validated by some systems, including New England 39-bus system, IEEE 118-bus system, and EPRI-1000 bus system. It can be revealed that the method is computationally more efficient and has less approximation error. It provides a useful approach for power system online voltage stability monitoring and assessment. This work is supported by National Natural Science Foundation of China (No. 50707019), Special Fund of the National Basic Research Program of China (No. 2009CB219701), Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 200439), Tianjin Municipal Science and Technology Development Program (No. 09JCZDJC25000), National Major Project of Scientific and Technical Supporting Programs of China During the 11th Five-year Plan Period (No. 2006BAJ03A06). ©2009 State Grid Electric Power Research Institute Press.

Optimal control and numerical optimization applied to epidemiological models

A relação entre a epidemiologia, a modelação matemática e as ferramentas computacionais permite construir e testar teorias sobre o desenvolvimento e combate de uma doença. Esta tese tem como motivação o estudo de modelos epidemiológicos aplicados a doenças infeciosas numa perspetiva de Controlo Ótimo, dando particular relevância ao Dengue. Sendo uma doença tropical e subtropical transmitida por mosquitos, afecta cerca de 100 milhões de pessoas por ano, e é considerada pela Organização Mundial de Saúde como uma grande preocupação para a saúde pública. Os modelos matemáticos desenvolvidos e testados neste trabalho, baseiam-se em equações diferenciais ordinárias que descrevem a dinâmica subjacente à doença nomeadamente a interação entre humanos e mosquitos. É feito um estudo analítico dos mesmos relativamente aos pontos de equilíbrio, sua estabilidade e número básico de reprodução. A propagação do Dengue pode ser atenuada através de medidas de controlo do vetor transmissor, tais como o uso de inseticidas específicos e campanhas educacionais. Como o desenvolvimento de uma potencial vacina tem sido uma aposta mundial recente, são propostos modelos baseados na simulação de um hipotético processo de vacinação numa população. Tendo por base a teoria de Controlo Ótimo, são analisadas as estratégias ótimas para o uso destes controlos e respetivas repercussões na redução/erradicação da doença aquando de um surto na população, considerando uma abordagem bioeconómica. Os problemas formulados são resolvidos numericamente usando métodos diretos e indiretos. Os primeiros discretizam o problema reformulando-o num problema de optimização não linear. Os métodos indiretos usam o Princípio do Máximo de Pontryagin como condição necessária para encontrar a curva ótima para o respetivo controlo. Nestas duas estratégias utilizam-se vários pacotes de software numérico. Ao longo deste trabalho, houve sempre um compromisso entre o realismo dos modelos epidemiológicos e a sua tratabilidade em termos matemáticos.

Hopf bifurcation in parallel polarized Nd:YAG laser

Dynamics of Nd:YAG laser with intracavity KTP crystal operating in two parallel polarized modes is investigated analytically and numerically. System equilibrium points were found out and the stability of each of them was checked using Routh–Hurwitz criteria and also by calculating the eigen values of the Jacobian. It is found that the system possesses three equilibrium points for (Ij, Gj), where j = 1, 2. One of these equilibrium points undergoes Hopf bifurcation in output dynamics as the control parameter is increased. The other two remain unstable throughout the entire region of the parameter space. Our numerical analysis of the Hopf bifurcation phenomena is found to be in good agreement with the analytical results. Nature of energy transfer between the two modes is also studied numerically.

Global attractivity in a recursive sequence

In this paper, we Study the invariant intervals, the globally attractivity of the two equilibrium points, and the oscillatory behavior of tile solutions of the difference equation x(n =) ax(n-1) - bx(n-2)/c + x(n-2), n = 1,2,......, where a, b. c > 0. (C) 2003 Elsevier Inc. All rights reserved.

On the recursive sequence Xn = axn-1+bxn-2/c+dxn-1xn-2

In this Paper, we study the invariant intervals, the global attractivity of the equilibrium points, and the asymptotic behavior of the solutions of the difference equation x(n) = ax(n-1) + bx(n-2) / c + dx(n-1)x(n-2), n =1, 2, ..., where a greater than or equal to 0, b, c, d > 0. (C) 2004 Elsevier Inc. All rights reserved.

Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua`s circuit behavior

This paper presents an experimental characterization of the behavior of an analogous version of the Chua`s circuit. The electronic circuit signals are captured using a data acquisition board (DAQ) and processed using LabVIEW environment. The following aspects of the time series analysis are analyzed: time waveforms, phase portraits, frequency spectra, Poincar, sections, and bifurcation diagram. The circuit behavior is experimentally mapped with the parameter variations, where are identified equilibrium points, periodic and chaotic attractors, and bifurcations. These analysis techniques are performed in real-time and can be applied to characterize, with precision, several nonlinear systems.

Reaction-diffusion systems coupled at the boundary and the Morse-Smale property

We study an one-dimensional nonlinear reaction-diffusion system coupled on the boundary. Such system comes from modeling problems of temperature distribution on two bars of same length, jointed together, with different diffusion coefficients. We prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic. To this end, we write the system as an equation with noncontinuous diffusion coefficient. We then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the Sturm-Liouville properties of the solutions of a linear elliptic problem. (C) 2008 Elsevier Inc. All rights reserved.

O uso de modelos compartimentais do tipo hospedeiro-macroparasita na dinâmica de doenças infecciosas causadas por helmintos diretamente transmitidos

Neste trabalho, nos propomos a estudar o desenvolvimento teórico de alguns modelos matemáticos básicos de doenças infecciosas causadas por macroparasitas, bem como as dificuldades neles envolvidas. Os modelos de transmissão, que descrevemos, referem-se ao grupo de parasitas com transmissão direta: os helmintos. O comportamento reprodutivo peculiar do helminto dentro do hospedeiro definitivo, no intuito de produzir estágios que serão infectivos para outros hospedeiros, faz com que a epidemiologia de infecções por helmintos seja fundamentalmente diferente de todos os outros agentes infecciosos. Uma característica importante nestes modelos é a forma sob a qual supõe-se que os parasitas estejam distribuídos nos seus hospedeiros. O tamanho da carga de parasitas (intensidade da infecção) em um hospedeiro é o determinante central da dinâmica de transmissão de helmintos, bem como da morbidade causada por estes parasitas. Estudamos a dinâmica de parasitas helmintos de ciclo de vida direto para parasitas monóicos (hermafroditas) e também para parasitas dióicos (machos-fêmeas) poligâmicos, levando em consideração uma função acasalamento apropriada, sempre distribuídos de forma binomial negativa. Através de abordagens analítica e numérica, apresentamos a análise de estabilidade dos pontos de equilíbrio do sistema. Cálculos de prevalências, bem como de efeitos da aplicação de agentes quimioterápicos e da vacinação, no controle da transmissão e da morbidade de parasitas helmintos de ciclo de vida direto, também são apresentados neste trabalho.

HOPF BIFURCATION FROM LINES of EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A novel approach based on recurrent neural networks applied to nonlinear systems optimization

This paper presents an efficient approach based on recurrent neural network for solving nonlinear optimization. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid subspace technique. These parameters guarantee the convergence of the network to the equilibrium points that represent an optimal feasible solution. The main advantage of the developed network is that it treats optimization and constraint terms in different stages with no interference with each other. Moreover, the proposed approach does not require specification of penalty and weighting parameters for its initialization. A study of the modified Hopfield model is also developed to analyze its stability and convergence. Simulation results are provided to demonstrate the performance of the proposed neural network. (c) 2005 Elsevier B.V. All rights reserved.

A neural network approach for robust nonlinear parameter estimation in presence of unknown-but-bounded errors

Systems based on artificial neural networks have high computational rates due to the use of a massive number of simple processing elements and the high degree of connectivity between these elements. This paper presents a novel approach to solve robust parameter estimation problem for nonlinear model with unknown-but-bounded errors and uncertainties. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the network convergence to the equilibrium points. A solution for the robust estimation problem with unknown-but-bounded error corresponds to an equilibrium point of the network. Simulation results are presented as an illustration of the proposed approach. Copyright (C) 2000 IFAC.

Stability and convergence analysis of a neural model applied in nonlinear systems optimization

A neural model for solving nonlinear optimization problems is presented in this paper. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the convergence of the network to the equilibrium points that represent an optimal feasible solution. The network is shown to be completely stable and globally convergent to the solutions of nonlinear optimization problems. A study of the modified Hopfield model is also developed to analyze its stability and convergence. Simulation results are presented to validate the developed methodology.

A modified Hopfield model for solving the N-Queens problem

A neural network model for solving the N-Queens problem is presented in this paper. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the convergence of the network to the equilibrium points. The network is shown to be completely stable and globally convergent to the solutions of the N-Queens problem. Simulation results are presented to validate the proposed approach.