The gene regulatory interaction network of Mycobacterium tuberculosis (Mtb) has been mostly reconstructed, resulting in a huge step towards revealing complicated interconnections between the network and cellular responses. Nevertheless, the temporal dynamics of gene expression in the regulatory network still remain unclear. Such information importantly provides further understandings in cellular behaviors underlying the system in the presence of external stresses such as those from the host immune system and drug treatments. Here I applied an engineering-based approach to the new context of studying the Mtb temporal dynamical system by adding a synthetic perturbation system. To do so, I developed the first synthetic gene oscillator functioning as a circadian clock in mycobacteria based on a transcriptional control circuit. The circuit encodes differentially-activated positive and negative feedback regulators, which enable autonomous, self-sustained oscillatory gene expression with an average period of 4.5 hours. I modeled robustness and tunability of oscillation period and amplitude computationally and verified them experimentally. I also used this oscillatory circuit transcriptionally to regulate one of the transcription factors (KstR), and profiled in vivo temporal dynamics of KstR and KstR-regulated gene targets by dual-color fluorescence intensities in single Mycobacterium smegmatics cells. I observed bifunctionality (activation and repression) of KstR-regulated gene expression, and different gene targets exhibited differential, characteristic response times. This constituted a proof-of-concept that one can utilize a genetic oscillator to characterize dynamical regulation of transcription factors underlying a complex regulatory system. Additionally, it presents a potential new strategy for using dynamical system modeling to guide genetic engineering of mycobacteria for future therapeutic applications.

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

This work considers the effect of symmetries on analysing bifurcations in dynamical systems. We consider an example of a laser with strong optical feedback which is modelled using coupled non-linear differential equations. A stationary point can be found in space, which can then be continued in parameter space using software such as AUTO. This software will then detect and continue bifurcations which indicate change in dynamics as parameters are varied. Due to symmetries in the equations, using AUTO may require the system of equations to be reduced in order to study periodic orbits of the original system as (relative) equilibria of the reduced system. Reasons for this are explored as well as considering how the equations can be changed or reduced to remove the symmetry. Invariant and Equivariant theory provide the tools for reducing the system of equations to the orbit space, allowing further analysis of the lasers dynamics.

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates.

Este trabalho consiste do estudo das propriedades de transporte de sistemas dinâmicos caóticos por meio do uso de técnicas de operadores de projeção. Tais sistemas podem exibir difusão determinística e relaxação para o equilíbrio. Mostramos que esse comportamento difusivo pode ser visto como uma propriedade espectral do operador de Perron-Frobenius associado. Em particular, a ressonância dominante de Policott-Ruelle é calculada analiticamente para uma classe geral de mapas que preservam área. Sua dependência do número de onda determina os coeficientes de transporte normais. Calculamos uma fórmula geral exata para o coeficiente de difusão, obtida sem qualquer aproximação de alta estocasticidade, e um novo efeito emergiu: a evolução angular pode induzir modos rápidos ou lentos de difusão mesmo no regime de alta estocasticidade. Os aspectos não-Gaussianos do transporte caótico são também investigados para esses sistemas. O estudo é realizado por meio de uma relação entre a curtose, o coeficiente de difusão e o coeficiente de Burnett de quarta ordem, os quais são calculados analiticamente. Uma escala de tempo característica que delimita os regimes Gaussiano e Markoviano para a função densidade foi estabelecida. À parte os modos acelerados, cujas propriedades cinéticas são anômalas, todo os resultados estão em excelente acordo com as simulações numéricas / This work consists in the study of the transport properties of chaotic Hamiltonian systems by using projection operator techniques. Such systems can exhibit deterministic diffusion and display an approach to equilibrium. We show that this diffusive behavior can be viewd as a spectral property of the associated Perron-Frobenius operator. In particular, the leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wavenumber dependence determines the normal transport coefficients. We calculate a general exact formula for the diffusion coefficient, derived without any high stochasticity approximation and a new effect emerges: the angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The non-Gaussian aspects of the chaotic transport are also investigated for this systems. This study is done by means of a relationship between kurtosis and diffusion coefficient and fourth order Burnett coefficient, which are calculated analytically. A characteristic time scale which delimits the Markovian and Gaussian regimes for the density function was established. Despite the accelerator modes, whose kinetics properties are anomalous, all theoretical results are in excellent agreement with the numerical simulations

Newton-Picard methods are iterative methods that work well for computing roots of nonlinear equations within a continuation framework. This project presents one of these methods and includes the results of a computation involving the Brusselator problem performed by an implementation of the method. This work was done in collaboration with Andrew Salinger at Sandia National Laboratories.

Este trabalho consiste do estudo das propriedades de transporte de sistemas dinâmicos caóticos por meio do uso de técnicas de operadores de projeção. Tais sistemas podem exibir difusão determinística e relaxação para o equilíbrio. Mostramos que esse comportamento difusivo pode ser visto como uma propriedade espectral do operador de Perron-Frobenius associado. Em particular, a ressonância dominante de Policott-Ruelle é calculada analiticamente para uma classe geral de mapas que preservam área. Sua dependência do número de onda determina os coeficientes de transporte normais. Calculamos uma fórmula geral exata para o coeficiente de difusão, obtida sem qualquer aproximação de alta estocasticidade, e um novo efeito emergiu: a evolução angular pode induzir modos rápidos ou lentos de difusão mesmo no regime de alta estocasticidade. Os aspectos não-Gaussianos do transporte caótico são também investigados para esses sistemas. O estudo é realizado por meio de uma relação entre a curtose, o coeficiente de difusão e o coeficiente de Burnett de quarta ordem, os quais são calculados analiticamente. Uma escala de tempo característica que delimita os regimes Gaussiano e Markoviano para a função densidade foi estabelecida. À parte os modos acelerados, cujas propriedades cinéticas são anômalas, todo os resultados estão em excelente acordo com as simulações numéricas / This work consists in the study of the transport properties of chaotic Hamiltonian systems by using projection operator techniques. Such systems can exhibit deterministic diffusion and display an approach to equilibrium. We show that this diffusive behavior can be viewd as a spectral property of the associated Perron-Frobenius operator. In particular, the leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wavenumber dependence determines the normal transport coefficients. We calculate a general exact formula for the diffusion coefficient, derived without any high stochasticity approximation and a new effect emerges: the angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The non-Gaussian aspects of the chaotic transport are also investigated for this systems. This study is done by means of a relationship between kurtosis and diffusion coefficient and fourth order Burnett coefficient, which are calculated analytically. A characteristic time scale which delimits the Markovian and Gaussian regimes for the density function was established. Despite the accelerator modes, whose kinetics properties are anomalous, all theoretical results are in excellent agreement with the numerical simulations

The revolution of autonomous vehicles has led to the development of robots with abundant sensors, actuators with many degrees of freedom, high-performance computing capabilities, and high-speed communication devices. These robots use a large volume of information from sensors to solve diverse problems. However, this usually leads to a significant modeling burden as well as excessive cost and computational requirements. Furthermore, in some scenarios, sophisticated sensors may not work precisely, the real-time processing power of a robot may be inadequate, the communication among robots may be impeded by natural or adversarial conditions, or the actuation control in a robot may be insubstantial. In these cases, we have to rely on simple robots with limited sensing and actuation, minimal onboard processing, moderate communication, and insufficient memory capacity. This reality motivates us to model simple robots such as bouncing and underactuated robots making use of the dynamical system techniques. In this dissertation, we propose a four-pronged approach for solving tasks in resource-constrained scenarios: 1) Combinatorial filters for bouncing robot localization; 2) Bouncing robot navigation and coverage; 3) Stochastic multi-robot patrolling; and 4) Deployment and planning of underactuated aquatic robots. First, we present a global localization method for a bouncing robot equipped with only a clock and contact sensors. Space-efficient and finite automata-based combinatorial filters are synthesized to solve the localization task by determining the robot’s pose (position and orientation) in its environment. Second, we propose a solution for navigation and coverage tasks using single or multiple bouncing robots. The proposed solution finds a navigation plan for a single bouncing robot from the robot’s initial pose to its goal pose with limited sensing. Probabilistic paths from several policies of the robot are combined artfully so that the actual coverage distribution can become as close as possible to a target coverage distribution. A joint trajectory for multiple bouncing robots to visit all the locations of an environment is incrementally generated. Third, a scalable method is proposed to find stochastic strategies for multi-robot patrolling under an adversarial and communication-constrained environment. Then, we evaluate the vulnerability of our patrolling policies by finding the probability of capturing an adversary for a location in our proposed patrolling scenarios. Finally, a data-driven deployment and planning approach is presented for the underactuated aquatic robots called drifters that creates the generalized flow pattern of the water, develops a Markov-chain based motion model, and studies the long- term behavior of a marine environment from a flow point-of-view. In a broad summary, our dynamical system approach is a unique solution to typical robotic tasks and opens a new paradigm for the modeling of simple robotics systems

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

The first part of this work deals with open dynamical systems. A natural question of how the survival probability depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole. In the second part we consider some classes of cellular automata called Deterministic Walks in Random Environments on Z^1. At first we deal with the system with constant rigidity and Markovian distribution of scatterers on Z^1. It is shown that these systems have essentially the same properties as DWRE on Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws for the dynamics of perturbations propagating in such environments with aging are obtained.