Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

abstract: It remains unquestionable that space-based technology is an indispensable component of modern daily lives. Success or failure of space missions is largely contingent upon the complex system analysis and design methodologies exerted in converting the initial idea into an elaborate functioning enterprise. It is for this reason that this dissertation seeks to contribute towards the search for simpler, efficacious and more reliable methodologies and tools that accurately model and analyze space systems dynamics. Inopportunely, despite the inimical physical hazards, space systems must endure a perturbing dynamical environment that persistently disorients spacecraft attitude, dislodges spacecraft from their designated orbital locations and compels spacecraft to follow undesired orbital trajectories. The ensuing dynamics’ analytical models are complexly structured, consisting of parametrically excited nonlinear systems with external periodic excitations–whose analysis and control is not a trivial task. Therefore, this dissertation’s objective is to overcome the limitations of traditional approaches (averaging and perturbation, linearization) commonly used to analyze and control such dynamics; and, further obtain more accurate closed-form analytical solutions in a lucid and broadly applicable manner. This dissertation hence implements a multi-faceted methodology that relies on Floquet theory, invariant center manifold reduction and normal forms simplification. At the heart of this approach is an intuitive system state augmentation technique that transforms non-autonomous nonlinear systems into autonomous ones. Two fitting representative types of space systems dynamics are investigated; i) attitude motion of a gravity gradient stabilized spacecraft in an eccentric orbit, ii) spacecraft motion in the vicinity of irregularly shaped small bodies. This investigation demonstrates how to analyze the motion stability, chaos, periodicity and resonance. Further, versal deformation of the normal forms scrutinizes the bifurcation behavior of the gravity gradient stabilized attitude motion. Control laws developed on transformed, more tractable analytical models show that; unlike linear control laws, nonlinear control strategies such as sliding mode control and bifurcation control stabilize the intricate, unwieldy astrodynamics. The pitch attitude dynamics are stabilized; and, a regular periodic orbit realized in the vicinity of small irregularly shaped bodies. Importantly, the outcomes obtained are unconventionally realized as closed-form analytical solutions obtained via the comprehensive approach introduced by this dissertation. / Dissertation/Thesis / Doctoral Dissertation Systems Engineering 2019