[Truncated abstract. Please see the pdf version for the complete text. Also, formulae and special characters can only be approximated here. Please see the pdf version of this abstract for an accurate reproduction.] This thesis treats two aspects of intelligent control: The first part is about long-term optimization by approximating dynamic programming and in the second part a specific class of a fast neural network, related to support vector machines (SVMs), is considered. The first part relates to approximate dynamic programming, especially in the framework of adaptive critic designs (ACDs). Dynamic programming can be used to find an optimal decision or control policy over a long-term period. However, in practice it is difficult, and often impossible, to calculate a dynamic programming solution, due to the 'curse of dimensionality'. The adaptive critic design framework addresses this issue and tries to find a good solution by approximating the dynamic programming process for a stationary environment. In an adaptive critic design there are three modules, the plant or environment to be controlled, a critic to estimate the long-term cost and an action or controller module to produce the decision or control strategy. Even though there have been many publications on the subject over the past two decades, there are some points that have had less attention. While most of the publications address the training of the critic, one of the points that has not received systematic attention is training of the action module.¹ Normally, training starts with an arbitrary, hopefully stable, decision policy and its long-term cost is then estimated by the critic. Often the critic is a neural network that has to be trained, using a temporal difference and Bellman's principle of optimality. Once the critic network has converged, a policy improvement step is carried out by gradient descent to adjust the parameters of the controller network. Then the critic is retrained again to give the new long-term cost estimate. However, it would be preferable to focus more on extremal policies earlier in the training. Therefore, the Calculus of Variations is investigated to discard the idea of using the Euler equations to train the actor. However, an adaptive critic formulation for a continuous plant with a short-term cost as an integral cost density is made and the chain rule is applied to calculate the total derivative of the short-term cost with respect to the actor weights. This is different from the discrete systems, usually used in adaptive critics, which are used in conjunction with total ordered derivatives. This idea is then extended to second order derivatives such that Newton's method can be applied to speed up convergence. Based on this, an almost concurrent actor and critic training was proposed. The equations are developed for any non-linear system and short-term cost density function and these were tested on a linear quadratic regulator (LQR) setup. With this approach the solution to the actor and critic weights can be achieved in only a few actor-critic training cycles. Some other, more minor issues, in the adaptive critic framework are investigated, such as the influence of the discounting factor in the Bellman equation on total ordered derivatives, the target interpretation in backpropagation through time as moving and fixed targets, the relation between simultaneous recurrent networks and dynamic programming is stated and a reinterpretation of the recurrent generalized multilayer perceptron (GMLP) as a recurrent generalized finite impulse MLP (GFIR-MLP) is made. Another subject in this area that is investigated, is that of a hybrid dynamical system, characterized as a continuous plant and a set of basic feedback controllers, which are used to control the plant by finding a switching sequence to select one basic controller at a time. The special but important case is considered when the plant is linear but with some uncertainty in the state space and in the observation vector, and a quadratic cost function. This is a form of robust control, where a dynamic programming solution has to be calculated. &sup1Werbos comments that most treatment of action nets or policies either assume enumerative maximization, which is good only for small problems, except for the games of Backgammon or Go [1], or, gradient-based training. The latter is prone to difficulties with local minima due to the non-convex nature of the cost-to-go function. With incremental methods, such as backpropagation through time, calculus of variations and model-predictive control, the dangers of non-convexity of the cost-to-go function with respect to the control is much less than the with respect to the critic parameters, when the sampling times are small. Therefore, getting the critic right has priority. But with larger sampling times, when the control represents a more complex plan, non-convexity becomes more serious.

Un défi important pour la gestion participative est de gérer des ressources renouvelables et la biodiversité. Dans le cadre de l'aide la la décision pour la gestion participative des territoires protégés, nous proposons dans cette thèse un agent expert qui fournit des informations sur la viabilité des solutions envisagées pour aider les joueurs à prendre une décision. C'est une nouvelle application de la théorie de la viabilité qui propose des concepts et méthodes pour contrôler un système dynamique afin de le maintenir dans un ensemble de contraintes de viabilité. Tout d'abord, nous avons étudié en quoi la théorie de la viabilité est particulièrement adaptée la la problématique de la gestion durable. L'analyse de viabilité prend en compte à la fois les comportements asymptotiques et transitoires qui doivent tous deux être considérés dans le cadre de la gestion de systèmes environnementaux. C'est l'intérêt de cette étude qui motive l'introduction d'un agent "viabilité" dans le cadre d'aide à la gestion participative. Nous décrivons ensuite un algorithme d'approximation du noyau de viabilité qui se concentre sur la réduction du temps de calcul. Cet algorithme utilise une procédure de classification du type "plus proche voisin", qui satisfait les conditions nécessaires pour garantir la convergence asymptotique de l'approximation. Nous comparons les résultats obtenus pour des problèmes de viabilité pour lesquels les noyaux théoriques ont été calculés. La fidélité de cet algorithme est toujours supérieure à 90 %. Le temps de calcul, relativement rapide, est très intéressant dans l'objectif d'intégrer ces calculs de noyaux dans des outils d'aide à la décision. Nous développons également un support logiciel qui est facile à utiliser par les joueurs pour les aider à analyser le problème de viabilité. Les joueurs définissent leur problème de viabilité grâce à une influence interface graphique utilisateur, puis lancent les simulations. Le logiciel permet aussi de visualiser les noyaux de viabilité et d'observer les trajectoires en 2D et 3D. Enfin nous étudions un agent expert viabilité dans un jeu de SimParc pour observer ses influences. Nous présentons des résultats fournis par l'agent expert viabilité en utilisant un modèle abstrait qui est axé sur la biodiversité. Puis nous présentons une expérimentation avec et sans agent expert viabilité. / An important challenge for participatory management is to manage renewable resources and biodiversity. In the framework of the decision support for participatory management of protected areas, we propose in this thesis an expert agent who provides information on the viability of the proposed solutions to help players to make a decision. This is a new application of the viability theory that provides concepts and methods for controlling a dynamic system in order to maintain the system in a set of viability constraints. Firstly, we study how the viability theory is particularly adapted to the problem of sustainable management. Sustainability analysis takes into account both the asymptotical dynamics and transient behaviors that have both to be considered in the context of environmental management systems. This study motivates the introduction of agent "viability" in the context of support participatory management. Then, we describe an algorithm of approximation of the viability kernel, which focuses on the reduction of computing time. This algorithm uses the "nearest neighbors" as a classification method, which satisfies the conditions guaranteeing the asymptotic convergence of the approximation. We compare the results obtained in cases where the theoretic viability kernel could be calculated. The accuracy of this algorithm is always higher than 90 percent. The computing time is relatively short, which is very interesting in the aim of integrating these calculations of viability kernel in tools of decision support. We also develop a software that is easy to use by players to help analyze the problem of sustainability. Players define their sustainability problem through a graphical user interface and launch simulations. The software also allows users to visualize the viability kernels and to observe the trajectories in 2D and 3D. Finally we study a viability expert agent in a game SimParc to observe these influences. We present the results provided by the viability expert agent using an abstract model which focuses on biodiversity. Then we present an experiment with and without viability expert agent.

Submitted by HERACLIO LEDGAR LÓPEZ LÁZARO null (herack_11@hotmail.com) on 2016-03-21T12:48:28Z No. of bitstreams: 1 Heracliodissertação.pdf: 1074830 bytes, checksum: eacc291c2e8f474bef30477ea2c47a2f (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-03-22T14:20:35Z (GMT) No. of bitstreams: 1 lazaro_hll_me_sjrp.pdf: 1074830 bytes, checksum: eacc291c2e8f474bef30477ea2c47a2f (MD5) / Made available in DSpace on 2016-03-22T14:20:35Z (GMT). No. of bitstreams: 1 lazaro_hll_me_sjrp.pdf: 1074830 bytes, checksum: eacc291c2e8f474bef30477ea2c47a2f (MD5) Previous issue date: 2016-03-07 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / The problem that we are going to study in this work, is motivated by the dynamics of differential equations nonautonomous. We will establish the existence and uniqueness of solution for a class of parabolic semilineares equations with Dirichlet boundary condition, in a family of domains that varies with time. In addition, certain hypotheses about the non-linearity, we will show the existence of a family of attractors pullback. / O problema que vamos estudar neste trabalho é motivado pela dinâmica de equações diferenciais não autônomas. Vamos estabelecer a existência e unicidade de solução para uma classe de equaçõoes parabólicas semilineares com condição de fronteira de Dirichlet, em uma família de domínios que varia com o tempo. Além disso, sob certas hipóteses sobre a não linearidade, mostraremos a existência de uma família de atratores pullback.

Cette thèse vise à démêler les principaux mécanismes impliqués dans les écoulements transitoires et turbulents. L’idée centrale est d'utiliser une technique d’optimisation non linéaire pour étudier l’origine et le rôle des structures cohérentes habituellement observées dans ces écoulements. Cette méthode a été utilisée dans trois contextes différents. Tout d’abord, un écoulement laminaire linéairement stable a été considéré et l'optimisation a été utilisée pour calculer les perturbations les plus amplifiées parmi toutes les perturbations capables de déclencher une transition vers la turbulence. Une fois que la turbulence est bien établie, une optimisation non linéaire entièrement 3D maximisant l'énergie cinétique turbulente est utilisée pour étudier les structures cohérentes qui peuplent l’écoulement turbulent et les mécanismes responsables de la croissance et de l’échange d’énergie (optimale) sont étudiés. Ensuite, une approche de type système dynamique est appliquée aux équations du mouvement. La géométrie de l’espace des phases est étudiée en utilisant la théorie de la croissance transitoire pour évaluer l’importance des variétés stable et instable dans la dynamique. Dans le même cadre, un algorithme de minimisation non linéaire est utilisé pour calculer les connexions hétérocliniques parmi les solutions invariantes des équations de Navier-Stokes. / This thesis aims at unraveling the main mechanisms involved in transitional and turbulent flows. The central idea is that of using a nonlinear optimization technique to investigate the origin and role of coherent structures usually observed in these flows. This method has been used in three different contexts. First, a linearly stable laminar flow has been considered and the optimization has been used to compute the most amplified perturbations among all disturbances able to trigger transition to turbulence. Once turbulence is well established, a fully 3D nonlinear optimization maximizing the turbulent kinetic energy is used to study coherent structures populating turbulent shear flow as well as investigate the mechanisms responsible for the energy (optimally) growth and exchange. Then, a dynamical system approach is applied to fluid flow equations. The geometry of the state space is investigated by using transient growth theory to reveal the importance of the stable and unstable manifold. In the same framework, a nonlinear minimization algorithm is used to compute heteroclinic connections among invariant solutions of the Navier-Stokes equations.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A new cryptographic method based on cellular automata (CA) has been investigated. In this method, the ciphering process is performed by preimages computation while deciphering is performed by CA temporal evolution. The preimages calculus is able to be used in a cryptographic system only if any arbitrary CA lattice has at least one preimage. In a previous method proposed by Gutowitz, this guarantee of pre-image was obtained due to the toggle property of the rules used as keys and some additional bits used in the calculus. Here a new method is proposed in which two distinct types of toggle rules are used without the need of additional bits. As a consequence, the cipher text and the plain text have the same size. One of the rules has the toggle property and it is responsible for CA dynamical behavior. The other rule is responsible to guarantee preimage existence, without need of additional bits. This method also differs from other cryptographic models based on additive rules previously published. The model is resistant against an attack known as differential cryptanalysis. Besides, it is efficient considering hardware implementation due to the intrinsic parallelism of the model. / O presente trabalho investiga um novo método criptográfico baseado em autômatos celulares (ACs). Neste método o processo de cifragem é realizado através do cálculo de préimagens enquanto a decifragem é realizada através da evolução temporal dos ACs. Para que o cálculo de pré-imagem possa ser utilizado em um sistema criptográfico, é necessário que todos os reticulados possíveis de um AC possuam pelo menos uma pré-imagem. Em um método anterior, proposto por Gutowitz, essa garantia de existência de pré-imagem foi conseguida graças à propriedade de sensitividade das regras empregadas como chaves e à utilização de bits adicionais. Nessa dissertação, um novo método é proposto onde dois tipos distintos de regras fazem com que não seja necessário acrescentar bits adicionais, tornando o texto cifrado e texto original do mesmo tamanho. Uma das regras utilizadas também possui a propriedade de sensitividade e é responsável pelo comportamento dinâmico médio do AC. A outra regra é responsável por garantir que sempre exista uma pré-imagem, sem a necessidade dos bits adicionais. Esse método também difere de outros modelos criptográficos publicados anteriormente que utilizam ACs com regras aditivas. Uma das características do modelo aqui proposto é a sua resistência contra um tipo de ataque conhecido como criptoanálise diferencial, além da possibilidade de implementação eficiente em hardware, usufruindo do paralelismo do modelo. / Mestre em Ciência da Computação

Neste trabalho, são estudados os problemas de sincronismo de fase nas redes mestre-escravo de via única (OWMS), nas topologias Estrela Simples, Cadeia Simples e mista, através da Teoria Qualitativa de Equações Diferenciais, com ênfase no Teorema da Variedade Central. Através da Teoria das Bifurcações, analisa-se o comportamento dinâmico das malhas de sincronismo de fase (PLL) de segunda ordem que compõem cada rede, frente às variações nos seus parâmetros constitutivos. São utilizadas duas funções de excitação muito comuns na prática: o degrau e a rampa de fase, aplicadas pelo nó mestre. Em cada caso, discute-se a existência e a estabilidade do estado síncrono. A existência de pontos de equilíbrio não-hiperbólicos, não permite uma aproximação linear, e nesses casos é aplicado o Teorema da Variedade Central. Através dessa rigorosa técnica de simplificação de sistemas dinâmicos é possível fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases. Desse modo, é possível determinar, localmente, suas estabilidades. / This work presents stability analysis of the syncronous state for three types of one-way master-slave time distribution network topologies: single star, single chain and both of them, mixed. Using bifurcation theory, the dynamical behavior of second-order phase-locked loops employed to extract the syncronous state in each node is analyzed in function of the constitutive parameters. Two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node and, in each case, the existence and stability of the syncronous state are studied. For parameter combinations resulting in non hyperbolic synchronous states, the linear approximation does not provide any information, even about the local behaviour of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behaviour of the original system in the neighborhood of these points. Thus, the local stability can be determined.

Este trabalho apresenta um estudo qualitativo das equações diferenciais nãolineares que descrevem o sincronismo de fase nos PLLs de 3ª ordem que compõem redes OWMS de topologia mista, Estrela Simples e Cadeia Simples. O objetivo é determinar, através da Teoria de Bifurcações, os valores ou relações entre os parâmetros constitutivos da rede que permitam a existência e a estabilidade do estado síncrono, quando são aplicadas, no oscilador mestre, duas funções de excitação muito comuns na prática: o degrau e a rampa de fase. Na determinação da estabilidade dos pontos de equilíbrio, sob o ponto de vista de Lyapunov, a existência de pontos de equilíbrio não-hiperbólicos não permite uma aproximação linear e, nesses casos, é aplicado o Teorema da Variedade Central. Essa técnica de simplificação de sistemas dinâmicos permite fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases e possibilitando determinar localmente suas estabilidades. / This work presents a qualitative study of the non-linear differential equations that describe the synchronous state in 3rd order PLLs that compose One-way masterslave time distribution networks with Single Star and Single Chain topologies. Using bifurcation theory, the dynamical behavior of third-order phase-locked loops employed to extract the syncronous state in each node is analyzed depending on constitutive node parameters when two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node. When parameter combinations result in non hyperbolic synchronous states, from Lyapunov point of view, the linear approximation does not provide any information about the local behavior of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behavior of the original system in the neighborhood of these points. Thus, the local stability can be determined.

Made available in DSpace on 2016-03-15T19:38:09Z (GMT). No. of bitstreams: 1 Marcos Kleber Soares Kunyosi.pdf: 2418852 bytes, checksum: ab6795f8d39445430da1eca23e865c56 (MD5) Previous issue date: 2006-09-11 / Instituto Presbiteriano Mackenzie / Recognition of patterns can be performed by using neural networks built with oscillators, like phase-locked loops (PLLs). These networks are modeled with differential equation systems and can be studied by using Dynamical System Theory, which is used in this work in order to investigate the dynamical behavior related to a synaptic configuration of a neural network. As a result of such an investigation, two methods (Brute Force and Algebric) that help to build neural networks formed by PLLs are presented. These methods aim to relate the synaptic configuration of the network to the corresponding basin of attraction of fixed points, which represent the stored patterns on the network. Also general properties of synaptic configuration are presented in order to generate other useful configurations. Then a model of an image recognition machine able to store in its memory a monochromatic image and able to determine if other image is similar to the memorized one is proposed. / Reconhecimento de padrões pode ser feito usando redes neurais construídas com osciladores, como malhas de sincronismo de fase (PLLs). Essas redes são modeladas por sistemas de equações diferenciais e podem ser estudas pela Teoria de Sistemas Dinâmicos, que é usada neste trabalho para investigar o comportamento dinâmico associado a uma configuração sináptica de uma rede neural. Como resultado dessa investigação, são apresentados dois métodos (Força Bruta e Algébrico) que auxiliam na construção de redes neurais formadas por PLLs. Esses métodos têm como objetivo relacionar a configuração sináptica da rede às respectivas bacias de atração de pontos atratores, os quais representam os padrões memorizados na rede. Também são apresentadas propriedades gerais da configuração sináptica que podem ser usadas para compor outras configurações de interesse. Por fim, é proposto um modelo de máquina reconhecedora de imagem capaz de armazenar em sua memória uma figura monocromática e determinar se uma imagem qualquer apresentada a ela é semelhante à memorizada.

Made available in DSpace on 2016-04-18T21:39:48Z (GMT). No. of bitstreams: 2 Angelo Schranko de Oliveira1.pdf: 925871 bytes, checksum: 812a592f67dbda8b36f5168fbd5f2598 (MD5) Angelo Schranko de Oliveira2.pdf: 2918106 bytes, checksum: 0969a0bf28b426ce84fe4595d80a73c2 (MD5) Previous issue date: 2009-01-28 / Wolfram Research, Inc. / Cellular automata (CAs) can be defined as discrete dynamical systems over n-dimensional networks of locally connected components, whose evolution occur in a discrete, synchronous and homogeneous fashion. Among their several applications, they have been used as a tool for complex systems modeling governed by fundamental laws of conservation (number-conserving cellular automata) or reversibility (reversible cellular automata). Another fundamental property that can be observed in CAs is regarding to their linearity (linear cellular automata) or nonlinearity. Usually, linear phenomena present low dynamic complexity, however, nonlinear phenoma can present complex behaviours like sensitive dependence on initial conditions and routes to chaos. This work focuses on investigating properties of cellular automata belonging to the intersection of those four classes, namely, reversible, number-conserving, and linear or nonlinear cellular automata. After presenting basic definitions, the notions of number-conserving cellular automata, conservation degree and reversibility are reviewed. Following, a dynamical characterisation parameter which relates the reversibility property of a onedimensional cellular automaton and the pre-images of their basic blocks is introduced, and some proofs of its general properties are given. Empirical observations herein suggest that a cellular automaton is reversible and number-conserving if, and only if, its local transition function is a composition of the local transition functions of the reversible, number-conserving cellular automata with neighbourhood size n=2; such an observation was drawn for neighbourhood sizes n∈{2, 3, 4, 5, 6} and number of states q=2; n∈{2, 3} and q=3; n∈{2, 3} and q=4. A proof for such a conjecture would allow the enumeration between neighbourhood lengths and the quantity of reversible, numberconserving cellular automata in the corresponding space, which can be easily identified by working out the compositions of the local transition functions with n=2. Finally, some relationships between reversible, number-conserving, linear and nonlinear CA rules, their spatio-temporal diagrams and basin of attraction fields are presented. / Autômatos celulares (ACs) podem ser definidos como sistemas dinâmicos sobre redes ndimensionais de componentes localmente conectados, cuja evolução ocorre de forma discreta, síncrona e homogênea. Dentre suas diversas aplicações, têm sido utilizados como ferramenta para modelagem de sistemas complexos regidos por leis fundamentais de conservação (autômatos celulares conservativos) ou reversibilidade (autômatos celulares reversíveis). Outra propriedade fundamental que pode ser observada nos ACs diz respeito à sua linearidade (autômatos celulares lineares) ou nãolinearidade. Fenômenos lineares normalmente apresentam menor complexidade dinâmica, enquanto fenômenos não-lineares podem apresentar propriedades tais como sensibilidade às condições iniciais e rotas para caos. O presente trabalho concentra-se na investigação de propriedades de autômatos celulares unidimensionais pertencentes à interseção dessas quatro classes, isto é, autômatos celulares unidimensionais conservativos, reversíveis, e lineares ou não-lineares. Após definições básicas, são revisitados os conceitos de conservabilidade e reversibilidade. Em seguida, introduz-se um parâmetro de caracterização dinâmica que relaciona a distribuição do número de pré-imagens dos blocos básicos à reversibilidade de autômatos celulares unidimensionais e apresentam-se algumas demonstrações decsuas propriedades gerais. Observações empíricas aqui realizadas sugerem que um autômato celular unidimensional é conservativo e reversível se, e somente se, sua função local de transição de estados é uma composição das funções locais de transição de estado dos autômatos celulares conservativos e reversíveis de vizinhança de comprimento n=2; tal observação foi constatada para vizinhanças de comprimento n∈{2, 3, 4, 5, 6} e quantidade de estados q=2; n∈{2, 3} e q=3; n∈{2, 3} e q=4. Uma demonstração para tal conjectura permitiria estabelecer uma enumeração entre os comprimentos das vizinhanças e a quantidade de autômatos celulares unidimensionais conservativos e reversíveis no espaço correspondente, os quais podem ser facilmente identificados através do cálculo das composições das funções locais de transição de estados com n=2. Por fim, apresentam-se relações entre as classes dos ACs conservativos, reversíveis, lineares e não-lineares, suas dinâmicas espaçotemporais e campos de bacias de atração.

Abstract This study fills an urgent need for qualitative analyses of relationships resulting in human change. It is a result of sixteen years of independent study by the author. It combines postgraduate study of nonlinear methodology, applied research of children’s pretend play, experience in educational psychology and Gestalt-counselling, as well as the practical training of graduate students at the Karelian State Pedagogical University (Petrozavodsk, Russia), and the Kajaani Department of Teacher Education (Kajaani, Finland). In this thesis, an attempt is made to reveal the fundamental reality of relationships between human beings. Using theories of helping relationships and data from developmental psychology, a qualitative nonlinear dynamical model of human relationships is elaborated. The scientific findings of Kurt Lewin and the Gestalt-therapy theory are widely used. To illustrate the explanatory potential of the proposed relationship model and the possibility of qualitative analyses, children’s pretend play is analyzed. In the first chapter, the basic connectedness between humans is studied. The author is focused on theories of relationships and their application to the organizing of relationships’ flow. The second chapter is devoted to detailed analyses of dynamic features of these theories and Kurt Lewin’s conception of tension system. The ontological philosophy of relationships is briefly reviewed. This helps to formulate the main problem of the research – how is a nonlinear phenomenological model of human relationships possible? In the third chapter, a new nonlinear dynamic model of human relationships is elaborated. Several conceptions from Lewin’s dynamic psychology and Gestalt-therapy are further developed in the model. A number of examples are analyzed. Video-data on children’s pretend play is analyzed in the fourth chapter. In the subsequent discussions some advantages and shortcomings of the suggested dynamic nonlinear model are examined. / Tiivistelmä Tämä tutkimus pyrkii vastaamaan kysymykseen miten inhimilliset suhteet voivat johtaa laadullisiin muutoksiin. Työssä paneudutaan ihmisten välisten suhteiden psykologisiin perusteisiin. Siinä kehitellään ihmisten välisten suhteiden ei-lineaarinen dynaaminen malli käyttäen kehityspsykologian ja auttamissuhteiden teorioita. Analyysi pohjautuu Kurt Lewinin ja hahmoterapian teoreettisiin oivalluksiin. Kehitellyn mallin selitysvoiman ja laadullisen analyysin mahdollisuuksien osoittamiseksi mallia sovelletaan lasten juonellisen roolileikin erittelyyn. Ensimmäisessä luvussa pohditaan esimerkkien avulla ihmisten välisten kontaktien perusluonnetta. Erityisesti keskitytään suhteiden teorioihin ja niiden sovelluksiin suhteiden jatkumon rakentamiseksi. Toinen luku paneutuu näiden teorioiden kuvaamien suhteiden dynaamisten piirteiden yksityiskohtaiseen tarkasteluun ja Kurt Lewinin ”tension system” käsitteeseen. Siinä esitellään myöskin lyhyesti suhteiden yksilökehityksen filosofiaa. Tältä pohjalta muotoillaan tutkimuksen pääongelma: Kuinka inhimillisten suhteiden ei-lineaarinen fenomenologinen malli on mahdollinen? Kolmannessa luvussa kehitellään uusi ei-lineaarinen inhimillisten suhteiden malli. Mallissa on kehitelty ja annettu uusi tulkinta useille Lewinin dynaamisen psykologian ja hahmoterapian käsitteille. Kehittelyä on tuettu käytännön esimerkein. Neljännessä luvussa on analysoitu lasten juonellisen roolileikin videotallenteita mallia käyttäen. Pohdinta tuo esille joitakin uuden mallin etuja ja jatkokehittelyn tarpeita.