45 research outputs found

    Non linear evolution of dynamic spatial systems

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    Modeling, analysis, and control of biological oscillators

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    Modeling, analysis, and control of biological oscillators

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    Dynamical Aspects of Apoptosis

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    Enharmonic motion: Towards the global dynamics of negative delayed feedback

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    In this thesis, we establish a new method for describing the qualitative dynamics of the so-called Hopf-Smale attractors in scalar delay differential equations with symmetric negative delayed feedback. The dynamics of Hopf-Smale attractors are robust under regular perturbations. Qualitatively, the attractor consists of an equilibrium, periodic orbits, and connections between them. We describe the mechanism that produces the periodic orbits and show how their formation creates new connecting orbits via sequences of Hopf bifurcations. As a result, we obtain an enumeration of all the phase diagrams, that is, the directed graphs encoding the equilibrium and periodic orbits as vertices and the connections as edges. In particular, we have obtained a prototype, the so-called enharmonic oscillator, that realizes all Hopf-Smale phase diagrams. Besides describing the Hopf-Smale attractors, our method also sheds insight into the formation process of certain global attractors with positive delayed feedback.In dieser Arbeit wird eine neue Methode zur Beschreibung der qualitativen Dynamik der sogenannten Hopf-Smale-Attraktoren in skalaren retardierten Differentialgleichung mit symmetrischer negativer verzögerter Rückkopplung entwickelt. Die Dynamik von Hopf-Smale-Attraktoren ist robust gegenüber regelmäßigen Störungen. Qualitativ besteht der Attraktor aus einem Gleichgewicht, periodischen Orbits und Orbits zwischen diesen. Wir beschreiben den Mechanismus, der die periodischen Orbits erzeugt und zeigen, wie dieser neue verbindende Orbits über Sequenzen von Hopf-Bifurkationen erzeugt. Als Ergebnis erhalten wir eine Aufzählung aller Phasendiagramme, d.h. der gerichteten Graphen, die die Gleichgewichts- und periodischen Bahnen als Knoten und die Verbindungen als Kanten kodieren. Insbesondere haben wir einen Prototyp, den sogenannten enharmonischen Oszillator, gefunden, der alle Hopf-Smale-Phasendiagramme verwirklicht. Neben der Beschreibung der Hopf-Smale-Attraktoren gibt unsere Methode auch Aufschluss über den Entstehungsprozess bestimmter globaler Attraktoren mit positiver verzögerter Rückkopplung

    On the Application of PSpice for Localised Cloud Security

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    The work reported in this thesis commenced with a review of methods for creating random binary sequences for encoding data locally by the client before storing in the Cloud. The first method reviewed investigated evolutionary computing software which generated noise-producing functions from natural noise, a highly-speculative novel idea since noise is stochastic. Nevertheless, a function was created which generated noise to seed chaos oscillators which produced random binary sequences and this research led to a circuit-based one-time pad key chaos encoder for encrypting data. Circuit-based delay chaos oscillators, initialised with sampled electronic noise, were simulated in a linear circuit simulator called PSpice. Many simulation problems were encountered because of the nonlinear nature of chaos but were solved by creating new simulation parts, tools and simulation paradigms. Simulation data from a range of chaos sources was exported and analysed using Lyapunov analysis and identified two sources which produced one-time pad sequences with maximum entropy. This led to an encoding system which generated unlimited, infinitely-long period, unique random one-time pad encryption keys for plaintext data length matching. The keys were studied for maximum entropy and passed a suite of stringent internationally-accepted statistical tests for randomness. A prototype containing two delay chaos sources initialised by electronic noise was produced on a double-sided printed circuit board and produced more than 200 Mbits of OTPs. According to Vladimir Kotelnikov in 1941 and Claude Shannon in 1945, one-time pad sequences are theoretically-perfect and unbreakable, provided specific rules are adhered to. Two other techniques for generating random binary sequences were researched; a new circuit element, memristance was incorporated in a Chua chaos oscillator, and a fractional-order Lorenz chaos system with order less than three. Quantum computing will present many problems to cryptographic system security when existing systems are upgraded in the near future. The only existing encoding system that will resist cryptanalysis by this system is the unconditionally-secure one-time pad encryption

    Qualitative dynamics of planar and spatial Lotka-Volterra and Kolmogorov systems

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    Ordinary differential equations are an important tool for the study of many real problems. In this thesis we focus in the qualitative dynamics of some ordinary differential systems, particularly, the Lotka-Volterra and Kolmogorov systems. We accomplish the study of some Lotka-Volterra systems on dimension three, which we characterize in two families of planar Kolmogorov systems. We give the complete classification of the global phase portraits in the Poincaré disk for those families. We also analyze the limit cycles of the three-dimensional Kolmogorov systems of degree three which appear through a zero-Hopf bifurcation. Some particular systems that model real problems in the field of population dynamics are also studied

    Synthetic biology of genetic circuits

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    The combination of positive and negative feedback loops has been shown to increase the robustness of oscillations. Such breakthrough has enabled to understand the importance of that dual control in a few biological systems. The reason is that most biological systems are non-linear. One of the obstacles that must be overcome when dealing with non-linear systems, even if they are simple, is that the use of intuition to predict its behavior is no longer valid. The comprehension of the behavior of the system can only be achieved mathematical modeling and computational simulations This bachelor thesis aims to develop a mathematical model of the relaxoscillator, a gene regulatory network in which two genes with identical promoters are regulated by the same activator and repressor. At the same time, the binding of those depends on the concentrations of two inducers: arabinose and IPTG, which correspond to the control parameters of the system. The obtained model, derived from the chemical reactions, was simulated under different inducer concentrations in an attempt to comprehend the long term behavior of the system. The results show that varying these inducer concentrations allows to tune the period and the amplitude of the oscillations observed in the system. In order to analyze changes in the long term behavior of the system it will be required to include a third control parameter, the transcription repressor rate, so that the system displays different dynamic behaviors. The analysis of the simulations indicates the presence of a supercritical Hopf bifurcation for a given value of the transcription repressor rate that would explain the transition between damped oscillations and persistent oscillations. Nevertheless, due to the theoretical nature of the project, experimental studies as well as two-parameter bifurcation analysis should be performed in order to confirm such hypothesis and gain understanding of the behavior of the system as a function of inducers concentrations.Ingeniería Biomédic

    Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

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    An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u
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