Abstract In this paper, several three dimensional (3-D) four-wing smooth quadratic autonomous chaotic systems are analyzed. It is shown that these systems have a number of similar features. A new 3-D continuous autonomous system is proposed based on these features. The new system can generate a four-wing chaotic attractor with less terms in the system equations. Several basic properties of the new system is analyzed by means of Lyapunov exponents, bifurcation diagrams and Poincare maps. Phase diagrams show that the equilibria are related to the existence of multiple wings.

The work of this thesis concerns how cortical memories are stored and retrieved. In particular, large-scale simulations are used to investigate the extent to which associative attractor theory is compliant with known physiology and in vivo dynamics. The first question we ask is whether dynamical attractors can be stored in a network with realistic connectivity and activity levels. Using estimates of biological connectivity we demonstrated that attractor memories can be stored and retrieved in biologically realistic networks, operating on psychophysical timescales and displaying firing rate patterns similar to in vivo layer 2/3 cells. This was achieved in the presence of additional complexity such as synaptic depression and cellular adaptation. Fast transitions into attractor memory states were related to the self-balancing inhibitory and excitatory currents in the network. In order to obtain realistic firing rates in the network, strong feedback inhibition was used, dynamically maintaining balance for a wide range of excitation levels. The balanced currents also led to high spike train variability commonly observed in vivo. The feedback inhibition in addition resulted in emergent gamma oscillations associated with attractor retrieval. This is congruent with the view of gamma as accompanying active cortical processing. While dynamics during retrieval of attractor memories did not depend on the size of the simulated network, above a certain size the model displayed the presence of an emergent attractor state, not coding for any memory but active as a default state of the network. This default state was accompanied by oscillations in the alpha frequency band. Such alpha oscillations are correlated with idling and cortical inhibition in vivo and have similar functional correlates in the model. Both inhibitory and excitatory, as well as phase effects of ongoing alpha observed in vivo was reproduced in the model in a simulated threshold-stimulus detection task. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper8: In press.</p>

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-10-10T17:42:51Z No. of bitstreams: 1 DissMBB.pdf: 961528 bytes, checksum: 881a8b78aa15e0ec1fc94c8c87e7e3a3 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-20T19:59:42Z (GMT) No. of bitstreams: 1 DissMBB.pdf: 961528 bytes, checksum: 881a8b78aa15e0ec1fc94c8c87e7e3a3 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-20T19:59:47Z (GMT) No. of bitstreams: 1 DissMBB.pdf: 961528 bytes, checksum: 881a8b78aa15e0ec1fc94c8c87e7e3a3 (MD5) / Made available in DSpace on 2016-10-20T19:59:53Z (GMT). No. of bitstreams: 1 DissMBB.pdf: 961528 bytes, checksum: 881a8b78aa15e0ec1fc94c8c87e7e3a3 (MD5) Previous issue date: 2016-07-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The mainly purpose of this paper is to study the asymptotic behaviour of abstract evolution equations. The first part of this work is dedicated to the attraction theory for univoque autonomous and nonautonomous problems and for multivoque autonomous problems. After that, we analyse the existence of the appropriate type of attractor for a reaction-diffusion equation (autonomous and with uniqueness property), a variation of the previous equation (which makes it no longer possible to ensure the uniqueness property) and a delayed differential equation (non-autonomous). For the two lasting equations, we also investigate the upper-semicontinuity of the families of the corresponding attractors. / O principal objetivo desta dissertação é estudar o comportamento assintótico de equações de evolução abstratas. A primeira parte do trabalho apresenta e compara, quando possível, a teoria de atração para problemas autônomos e não autônomos unívocos e problemas autônomos multívocos. Após apresentados os resultados, analisamos a existência dos atratores apropriados para uma equação de reação-difusão (autônoma e com unicidade de solucão), uma variação da equação anterior (fazendo com que o problema não tenha mais unicidade de solução) e uma equação diferencial com retardo (não autônoma). Nos dois últimos, investigamos também a semicontinuidade superior para as famílias de atratores correspondentes.

Feedforward neural networks have been widely studied and used in many applications in science and engineering. The training of this type of networks is mainly undertaken using the well-known backpropagation based learning algorithms. One major problem with this type of algorithms is the slow training convergence speed, which hinders their applications. In order to improve the training convergence speed of this type of algorithms, many researchers have developed different improvements and enhancements. However, the slow convergence problem has not been fully addressed. This thesis makes several contributions by proposing new backpropagation learning algorithms based on the terminal attractor concept to improve the existing backpropagation learning algorithms such as the gradient descent and Levenberg-Marquardt algorithms. These new algorithms enable fast convergence both at a distance from and in a close range of the ideal weights. In particular, a new fast convergence mechanism is proposed which is based on the fast terminal attractor concept. Comprehensive simulation studies are undertaken to demonstrate the effectiveness of the proposed backpropagataion algorithms with terminal attractors. Finally, three practical application cases of time series forecasting, character recognition and image interpolation are chosen to show the practicality and usefulness of the proposed learning algorithms with comprehensive comparative studies with existing algorithms.

Dynamical systems possess an interesting and complex behaviour that have attracted a number of researchers across different fields, such as Biology, Economics and most importantly in Engineering. The complex and unpredictability of nonlinear customary behaviour or the chaotic behaviour, makes it strange to analyse them. This thesis presents the analysis of the system of nonlinear differential equations of the so--called Lu--Chen--Cheng system. The system has similar dynamical behaviour with the famous Lorenz system. The nature of equilibrium points and stability of the system is presented in the thesis. Examples of chaotic dynamical systems are presented in the theory. The thesis shows the dynamical structure of the Lu--Chen--Cheng system depending on the particular values of the system parameters and routes to chaos. This is done by both the qualitative and numerical techniques. The bifurcation diagrams of the Lu--Chen--Cheng system that indicate limit cycles and chaos as one parameter is varied are shown with the help of the largest Lyapunov exponent, which also confirms chaos in the system. It is found out that most of the system's equilibria are unstable especially for positive values of the parameters $a, b$. It is observed that the system is highly sensitive to initial conditions. This study is very important because, it supports the previous findings on chaotic behaviours of different dynamical systems.

Este trabalho contém resultados sobre a existência, unicidade e comportamento assintótico de soluções para uma equação de viga não linear do tipo Kirchhoff, \'u IND. tt\' \'+ \'DELTA\' POT. 2\' u - M(\'INT.IND. OMEGA\' | \'NABLA\' u| 2 dx) \'DELTA\' u+ f (\'u IND. t\' ) +g(u) = h em × R +, onde \'R POT. N\' é um domínio limitado com fronteira regular \\GAMA. Essa equação é um modelo matemático para pequenas vibrações transversais de vigas ou placas extensíveis. O termo não local M(\'INT.IND. OMEGA\' | \\NABLA u |2 dx) u está relacionado à variação de tensão na viga devida à sua extensibilidade. O termo f (\'u IND. t\' ) representa uma dissipação para o sistema e g(u) representa a força exercida pelo meio. A função h representa uma força externa adicional. Consideramos o problema com as condições de fronteira u|×R + = \'INT. u SUP. \'INT. v\' | \\\'GAMA\' ×\'R +\' = 0, que corresponde ao modelo de vigas fixadas pelo bordo \\\'GAMA\'. Discutiremos o caso em que a dissipação é linear e o caso em que é não linear. Mostraremos que em ambos os casos o sistema dinâmico associado ao problema possui um atrator global. Entretanto, para o caso em que a dissipação é linear, obtemos num espaço de fase mais regular, a existência de um conjunto inércia de dimensão finita, que atrai exponencialmente todos os limitados deste espaço / This work contains some results on the existence, uniqueness and asymptotic behavior of solutions for a nonlinear beam equation of Kirchhoff type, \'u IND. tt\' + \' DELTA POT. 2\' u+ M(\'INT. IND.\' |u| 2 dx) u + g(\'u IND. t\') + f (u) = h; where \'R POT. N\' is a bounded domain with smooth boundary . This equation is a model for small vibrations of extensible beams. The nonlocal term M(\' INT. IND.\' |u| 2 dx) u is related to the variation of tensions in the beam due to its extensibility. The term f (\'u IND. t\') represents a damping mechanism for the system and g(u) represents the force exerted by the foundation. The function h represents an additional external force. We consider the problem with boundary condition u|×R+ = \' u SUP. \' |×R+ = 0, which corresponds to the model of clamped beams. We discuss the cases where the dissipation is linear and the case nonlinear. We show that in both cases, the dynamical system associated to the problem has a global attractor. However, when the dissipation is linear, we obtain, in a more regular space, the existence of an inertial set of finite dimension, which attracts exponentially all bounded sets of this space

Este trabalho contém resultados sobre a existência, unicidade e comportamento assintótico de soluções para uma equação de viga não linear do tipo Kirchhoff, \'u IND. tt\' \'+ \'DELTA\' POT. 2\' u - M(\'INT.IND. OMEGA\' | \'NABLA\' u| 2 dx) \'DELTA\' u+ f (\'u IND. t\' ) +g(u) = h em × R +, onde \'R POT. N\' é um domínio limitado com fronteira regular \\GAMA. Essa equação é um modelo matemático para pequenas vibrações transversais de vigas ou placas extensíveis. O termo não local M(\'INT.IND. OMEGA\' | \\NABLA u |2 dx) u está relacionado à variação de tensão na viga devida à sua extensibilidade. O termo f (\'u IND. t\' ) representa uma dissipação para o sistema e g(u) representa a força exercida pelo meio. A função h representa uma força externa adicional. Consideramos o problema com as condições de fronteira u|×R + = \'INT. u SUP. \'INT. v\' | \\\'GAMA\' ×\'R +\' = 0, que corresponde ao modelo de vigas fixadas pelo bordo \\\'GAMA\'. Discutiremos o caso em que a dissipação é linear e o caso em que é não linear. Mostraremos que em ambos os casos o sistema dinâmico associado ao problema possui um atrator global. Entretanto, para o caso em que a dissipação é linear, obtemos num espaço de fase mais regular, a existência de um conjunto inércia de dimensão finita, que atrai exponencialmente todos os limitados deste espaço / This work contains some results on the existence, uniqueness and asymptotic behavior of solutions for a nonlinear beam equation of Kirchhoff type, \'u IND. tt\' + \' DELTA POT. 2\' u+ M(\'INT. IND.\' |u| 2 dx) u + g(\'u IND. t\') + f (u) = h; where \'R POT. N\' is a bounded domain with smooth boundary . This equation is a model for small vibrations of extensible beams. The nonlocal term M(\' INT. IND.\' |u| 2 dx) u is related to the variation of tensions in the beam due to its extensibility. The term f (\'u IND. t\') represents a damping mechanism for the system and g(u) represents the force exerted by the foundation. The function h represents an additional external force. We consider the problem with boundary condition u|×R+ = \' u SUP. \' |×R+ = 0, which corresponds to the model of clamped beams. We discuss the cases where the dissipation is linear and the case nonlinear. We show that in both cases, the dynamical system associated to the problem has a global attractor. However, when the dissipation is linear, we obtain, in a more regular space, the existence of an inertial set of finite dimension, which attracts exponentially all bounded sets of this space

The dissertation studies about the existence of three different types of attractors of three multi-component reaction-diffusion systems. These reaction-diffusion systems play important role in both chemical kinetics and biological pattern formation in the fast-growing area of mathematical biology. In Chapter 2, we prove the existence of a global attractor and an exponential attractor for the solution semiflow of a reaction-diffusion system called Boissonade equations in the L2 phase space. We show that the global attractor is an (H, E) global attractor with the L∞ and H2 regularity and that the Hausdorff dimension and the fractal dimension of the global attractor are finite. The existence of exponential attractor is also shown. The upper-semicontinuity of the global attractors with respect to the reverse reaction rate coefficient is proved. In Chapter 3, the existence of a pullback attractor for non-autonomous reversible Selkov equations in the product L2 phase space is proved. The method of grouping and rescaling estimation is used to prove that the L4-norm and L6-norm of solution trajectories are asymptotic bounded. The new feature of pinpointing a middle time in the process turns out to be crucial to deal with the challenge in proving pullback asymptotic compactness of this typical non-autonomous reaction-diffusion system. In Chapter 4, asymptotical dynamics of stochastic Brusselator equations with multiplicative noise is investigated. The existence of a random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimation than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the L2 to H1 attracting regularity by the flattening method.

I study sets of attractors and non-attractors of finite iterated function systems. I provide examples of compact sets which are attractors of iterated function systems as well as compact sets which are not attractors of any iterated function system. I show that the set of all attractors is a dense Fs set and the space of all non-attractors is a dense Gd set it the space of all non-empty compact subsets of a space X. I also investigate the small trans-finite inductive dimension of the space of all attractors of iterated function systems generated by similarity maps on [0,1].

No description available.