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Mechanical systems always operate under some degree of uncertainty, which can be due to the inherent properties of the system parameters, to random inputs or external excitations, to poorly known parameters in the interface between different systems, or to inadequate knowledge of the dynamic process. Also, mechanical systems are large and highly nonlinear, while the magnitude of uncertainties may be very large. This dissertation addresses the critical need for understanding of the stochastic nature of mechanical system, especially vehicle and terramechanics systems, and need for developing efficient computational tools to model mechanical systems in the presence of parametric and external uncertainty. This dissertation investigates the influence of parametric and external uncertainties on vehicle dynamics and terramechanics. The uncertainties studied include parametric uncertainties, stochastic external excitations, and random variables between vehicle-terrain and vehicle-soil/snow interface. The methodology developed has been illustrated on a stochastic vehicle-terrain interaction model, a stochastic vehicle-soil interaction model, two stochastic tire-snow interaction models, and two stochastic tire-force relations. The uncertainties are quantified and propagated through vehicle and terramechanics systems using a polynomial chaos approach. Algorithms which can predict the geometry of the contact patch and the interfacial forces and torques on the vehicle-soil interfaces are developed. All stochastic models and algorithms are simulated for various scenarios and maneuvers. Numerical results are analyzed from the computational effort point of view, or from the angle of vehicle dynamics and terramechanics, and provide a deeper understanding of the evolution of stochastic vehicle and terramechanics systems. They can also be used in guiding vehicle design and development. This dissertation represents a pioneer study on stochastic vehicle dynamics and terramechanics. Moreover, the methodology developed is not limited to such systems. Any mechanical system with uncertainties can be treated using the polynomial chaos approach presented, considering their specific characteristics. / Ph. D.

The main limitations in performing uncertainty analysis of CFD models using conventional methods are associated with cost and effort. For these reasons, there is a need for the development and implementation of efficient stochastic CFD tools for performing uncertainty analysis. One of the main contributions of this research is the development and implementation of Intrusive and Non-Intrusive methods using polynomial chaos for uncertainty representation and propagation. In addition, a methodology was developed to address and quantify turbulence model uncertainty. In this methodology, a complex perturbation is applied to the incoming turbulence and closure coefficients of a turbulence model to obtain the sensitivity derivatives, which are used in concert with the polynomial chaos method for uncertainty propagation of the turbulence model outputs. / Ph. D.

In typical attempts to control spatio-temporal chaos, spatially extended systems were subjected to protocols that perturbed them as a whole, often overlooking the potential stabilizing interaction between adjacent regions. We have shown that through the application of a time-delayed feedback scheme to a specific localized region of a system periodic patterns can be generated that are distinct from those observed when controlling the whole system. In this thesis, we present the results of two interconnected studies: 1) Spatio-temporal patterns emerging from spatially localized time-delayed feedback perturbations within transient chaotic states of the Gray-Scott reaction-diffusion system 2) Spatio-temporal patterns emerging from spatially localized time-delayed feedback perturbations within chaotic states of the cubic complex Ginzburg-Landau equation We present an investigation of two model systems: the Gray-Scott reaction-diffusion equation and the complex Ginzburg-Landau equation. Specifically we numerically study two models characterized by exhibiting various chaotic regimes. We first consider a comprehensive study of the Gray-Scott model highlighting key details about different parameter space regimes and their relative proximity to the chaotic regime. Through a systematic investigation of the effects of the model control parameters, time-delayed feedback control strength parameters, perturbed region widths, and other quantities, we show that novel patterns can be formed through the appropriate choice of perturbation region and strength. For the second study we use spatially localized time-delayed feedback on the one-dimensional complex Ginzburg-Landau equation and demonstrate, through the numerical integration of the resulting real and imaginary equations, the stabilization of novel periodic patterns within three distinct chaotic regimes. In these studies we have shown that selectively applying a time-delayed feedback scheme to a specific spatially localized region of a chaotic system can bring forth periodic patterns that are distinct from those observed when applying a perturbation to the whole system. Depending on the protocol used, these new patterns can emerge either in the perturbed or the unperturbed region. The mechanism underlying the observed pattern generation is related to the interplay between diffusion across the interfaces separating the different regions and time-delayed feedback. Research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF-17-1-0156. / Doctor of Philosophy / In typical attempts to control spatio-temporal chaos, spatially extended systems were subjected to protocols that perturbed them as a whole, often overlooking the potential stabilizing interaction between adjacent regions. We have shown that through the application of a time-delayed feedback scheme to a specific localized region of a system periodic patterns can be generated that are distinct from those observed when controlling the whole system. We present an investigation of two model systems: the Gray-Scott reaction-diffusion equation and the complex Ginzburg-Landau equation. We first consider a comprehensive study of the Gray-Scott model highlighting key details about different parameter space regimes and their relative proximity to the chaotic regime. Through a systematic investigation of the effects of the model control parameters, time-delayed feedback control strength parameters, perturbed region widths, and other quantities, we show that novel patterns can be formed through the appropriate choice of perturbation region and strength. For the second study we use spatially localized time-delayed feedback on the one-dimensional complex Ginzburg-Landau equation and demonstrate, through the numerical integration of the resulting real and imaginary equations, the stabilization of novel periodic patterns within chaotic regimes. In these studies we have shown that selectively applying a time-delayed feedback scheme to a specific region of a chaotic system can generate periodic patterns that are distinct from those observed when controlling the whole system. Depending on the protocol used, these new patterns can emerge either in the perturbed or the unperturbed region. The mechanism underlying the observed pattern generation is related to the interplay between diffusion across the interfaces separating the different regions and time-delayed feedback. Research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF-17-1-0156.

Rivers are amongst the most dynamic ecosystems on earth. River ecosystems are highly disturbed environments, where riparian vegetation, water and sediments, are interconnected by positive and negative feedbacks, driven by a set of interactions. In the last two decades, it has been widely recognized that these eco-morphodynamic feedbacks play a crucial role in governing the equilibrium and dynamics of river ecosystem. However, the incomplete understanding and quantification of these feedbacks limit the comprehension of river behavior and the development of efficient predictive models. Thus, in this research, fundamental intrinsic feedbacks between riparian vegetation and hydro-morphodynamic disturbance are modeled, where the disturbance is generated bymthe vegetation itself. The aim is to investigate how these intrinsic feedbacks govern themequilibrium and dynamics of a simplified river ecosystem.mTo this end, numerical simulations were conducted using both a 0D model (non-spatial)mand a 1D model (spatial) coupling hydro-morphodynamics with vegetation dynamics. The case study is a straight channel where vegetation can grow only in the central patch, while upstream and downstream there are bare soil regions. The system is perturbed periodically by a succession of floods of constant amplitude. Vegetation growth occurs in between of two consecutive floods, during low flood periods. Vegetation consists of two components, the above-ground biomass (canopy) and below-ground biomass (root depth). In both models, the canopy increases the roughness, reducing flow velocity. Variations in the flow field and the reduction of bottom shear stress modify sediment transport, leading to a greater imbalance between the vegetated and bare areas and thus, inducing erosion. Erosion increases the probability of vegetation uprooting, and when scour reaches root depth, uprooting occurs. The overall feedback loop is negative: higher vegetation biomass causes greater sediment flux imbalance and more erosion, ultimately resulting in less vegetation. However, root growth may inhibit the negative feedback loop, promoting positive feedbacks. Indeed, this interplay between hydro-morphodynamic disturbance (erosion) and the vegetation resistance (root depth), governs the predominance of either a positive or a negative feedback overall balance. Model results demonstrate that when the positive feedback overall balance prevails, the system always reaches a stable configuration. Furthermore, the system can exhibit hysteresis, meaning that, depending on the initial condition, it can achieve a stable configuration in two alternative states, the fully vegetated condition or bare soil. In the presence of the vegetated patch, the system can also exhibit a more complex multi-stable behavior, with infinite equilibria between the two alternative states. This also implies that spatial interactions smooth out critical transitions and tipping points, by facilitating smoother shifts that occur gradually through multiple smaller intermediate steps. Indeed, the resilience of the system, which is the ability of the system to still maintain its fundamental structure and functions after being subject to the ecological disturbance, increases due to spatial interactions. In contrast, when the negative feedback overall balance prevails, the system never reaches a steady state but exhibits dynamic oscillations. The oscillations can be either (i) periodic or (ii) aperiodic, strongly dependent on initial conditions, and with a positive Maximum Lyapunov Exponent, indicating chaotic behavior. The study also reveals that the route to chaos is a period-doubling bifurcation, and the calculation of time scale of predictability shows that the system is predictable only for a few growth-flood cycles. These results suggest that altering the ratio between hydro-morphodynamic disturbance and vegetation resistance, such as through anthropogenic pressure and climate change, may shift the system from a positive to a negative feedback overall balance. This shift could lead from a stable state to periodic oscillations or unpredictable chaotic behavior, limiting long-term predictions of river trajectories. Additionally, understanding how positive and negative eco-morphodynamic feedbacks govern river dynamics can contribute to develop efficient predictive models. Models are essential tools for implementing efficient river management and facilitate effective communication with stakeholders.

This thesis details the investigation into the Whirling Planar Pendulum system. The WPP is a pendulum that is spun around a vertical spin axis at a controllable horizontal offset. This dynamical system exhibits both local and global bifurcations. The local pitchfork bifurcation leads to the splitting of a single stable equilibrium point into three (two stable and one unstable), as the spin rate is increased. The global bifurcations lead to two independent types of chaotic oscillations which are induced by sinusoidal excitations. The types of chaos are each associated with one of two homoclinic orbits in the system's phase portraits. The onset of each type of chaos is investigated through Melnikov's Method applied to the system's Hamiltonian, to find parameters at which the stable and unstable manifolds intersect transversely, indicating the onset of chaotic motion. These results are compared to simulation results, which suggest chaotic motion through the appearance of strange attractors in the Poincaré maps. Additionally, evidence of the WPP system experiencing both types of chaos simultaneously was found, resulting in a merger of two distinct types of strange attractor. / Master of Science / This report details the investigation into the Whirling Planar Pendulum system. The WPP is a pendulum that is spun around a vertical spin axis at a controllable horizontal offset. This system can be used to investigate what are known as local and global bifurcations. A local bifurcation occurs when the single equilibrium state (corresponding to the pendulum hanging straight down) when spun at low speeds, bifurcates into three equilibria when the spin rate is increased beyond a certain value. The global bifurcations occur when the system experiences sinusoidal forcing near certain equilibrium conditions. The resulting chaotic oscillations are investigated using Melnikov’s method, which determines when the sinusoidal forcing results in chaotic motion. This chaotic motion comes in two types, which cause the system to behave in different ways. Melnikov’s method, and results from a simulation were used to determine the parameter values in which the pendulum experiences each type of chaos. It was seen that at certain parameter values, the WPP experiences both types of chaos, supporting the observation that these types of chaos are not necessarily independent of each other, but can merge and interact.

The transport of a species in complex flow fields is an important phenomenon related to many areas in science and engineering. There has been significant progress theoretically and experimentally in understanding active transport in steady, periodic flows such as a chain of vortices but many open questions remain for transport in complex and chaotic flows. This thesis investigates the active transport in a three-dimensional, time-dependent flow field characterized by a spatiotemporally chaotic state of Rayleigh-Be?nard convection. A nonlinear Fischer-Kolmogorov-Petrovskii-Piskunov reaction is selected to study the transport within these flows. A highly efficient, parallel spectral element approach is employed to solve the Boussinesq and the reaction-advection-diffusion equations in a spatially-extended cylindrical domain with experimentally relevant boundary conditions. The transport is quantified using statistics of spreading and in terms of active transport characteristics like front speed and geometry and are compared with those results for transport in steady flows found in the literature. The results of the simulations indicate an anomalous diffusion process with a power law 2 < ? < 5/2 a result that deviates from other superdiffusive processes in simpler flows, and reveals that the presence of spiral defect chaos induces strongly anomalous transport. Additionally, transport was found to most likely occur in a direction perpendicular to a convection roll in the flow field. The presence of the spiral defect chaos state of the fluid convection is found to enhance the front perimeter by t^3/2 and by a perimeter enhancement ratio r(p) = 2.3. / Master of Science

The term spatiotemporal chaos refers to physical phenomena that exhibit irregular oscillations in both space and time. Examples of such phenomena range from cardiac dynamics to fluid turbulence, where the motion is described by nonlinear partial differential equations. It is well known from the studies of low dimensional chaotic systems that the state space, the space of solutions to the governing dynamical equations, is shaped by the invariant sets such as equilibria, periodic orbits, and invariant tori. State space of partial differential equations is infinite dimensional, nevertheless, recent computational advancements allow us to find their invariant solutions (exact coherent structures) numerically. In this thesis, we try to elucidate the chaotic dynamics of nonlinear partial differential equations by studying their exact coherent structures and invariant manifolds. Specifically, we investigate the Kuramoto-Sivashinsky equation, which describes the velocity of a flame front, and the Navier-Stokes equation for an incompressible fluid in a circular pipe. We argue with examples that this approach can lead to a theory of turbulence with predictive power.

One of the phenomenal results emerging from string theory is the AdS/CFT correspondence or gauge-gravity duality: In certain cases a theory of gravity is equivalent to a "dual" gauge theory, very similar to the one describing non-gravitational interactions of fundamental subatomic particles. A difficult problem on one side can be mapped to a simpler and solvable problem on the other side using this correspondence. Thus one of the theories can be understood better using the other. The mapping between theories of gravity and gauge theories has led to new approaches to building models of particle physics from string theory. One of the important features to model is the phenomenon of confinement present in strong interaction of particle physics. This feature is not present in the gauge theory arising in the simplest of the examples of the duality. However this N = 4 supersymmetric Yang-Mills gauge theory enjoys the property of being integrable, i.e. it can be exactly solved in terms of conserved charges. It is expected that if a more realistic theory turns out to be integrable, solvability of the theory would lead to simple analytical expressions for quantities like masses of the hadrons in the theory. In this thesis we show that the existing models of confinement are all nonintegrable--such simple analytic expressions cannot be obtained. We moreover show that these nonintegrable systems also exhibit features of chaotic dynamical systems, namely, sensitivity to initial conditions and a typical route of transition to chaos. We proceed to study the quantum mechanics of these systems and check whether their properties match those of chaotic quantum systems. Interestingly, the distribution of the spacing of meson excitations measured in the laboratory have been found to match with level-spacing distribution of typical quantum chaotic systems. We find agreement of this distribution with models of confining strong interactions, conforming these as viable models of particle physics arising from string theory.

Dans cette thèse, nous étudions un système à temps discret de dimension N, dont les paramètres varient périodiquement. Le système de dimension N est construit à partir de n sous-systèmes de dimension un couplés symétriquement. Dans un premier temps, nous donnons les propriétés générales du système de dimension N. Dans un second temps, nous étudions le cas particulier où le sous-système de dimension un est défini à l’aide d’une transformation logistique. Nous nous intéressons plus particulièrement à la structure des bifurcations lorsque N=1 ou 2. Des zones échangeurs centrées sur des points cuspidaux sont obtenues dans le cas de courbes de bifurcation de type fold (noeud-col).Ensuite, nous nous intéressons au comportement de circuits de type Chua couplés lorsqu’un paramètre varie lui aussi périodiquement, la période étant celle d’une des variables d’état interne au système. A partir de l’étude des bifurcations du système, la non existence de cycles d’ordre impair et la coexistence de plusieurs attracteurs est mise en évidence. D’autre part, on peut mettre en évidence la coexistence de différents attracteurs pour lesquels les états de synchronisation sont distincts. Le cas continu est comparé avec le cas discret. Des phénomènes tout à fait similaires sont obtenus. Il est important de noter que l’étude d’un système à temps discret est plus facile et plus rapide que celle d’un système à temps continu. L’étude du premier système permet donc d’avoir des informations sur ce qui peut se produire dans le cas continu. Pour terminer, nous analysons le comportement d’un autre système couplé à temps continu, basé lui aussi sur le circuit de Chua, mais pour lequel la commutation qui contrôle la variation du paramètre s’effectue différemment du premier système. Ce type de commutation génère une augmentation du nombre d’attracteurs / In this thesis, we propose a N-dimensional coupled discrete-time system whose parameters are forced into periodic varying, the N-dimensional system being constructed of n same one-dimensional subsystems with mutually influencing coupling and also coupled continuous-time system including periodically parameter varying which correspond to the periodic varying in the discrete-time system.Firstly, we introduce the N-dimensional coupled parametrically forced discrete-time system and its general properties. Then, when logistic maps is used as the one-dimensional subsystem constructing the system, bifurcations in the one or two-dimensional parametrically forced logistic map are investigated. Crossroad area centered at fold cusp points regarding several order cycles are confirmed.Next, we investigated behaviors of the coupled Chua's circuit whose parameter is forced into periodic varying associated with the period of an internal state value. From the investigation of bifurcations in the system, non-existence of odd order cycles and coexistence of different attractors are observed. From the investigation of synchronizations coexisting of many attractors whose synchronizations states are different are observed. Observed phenomena in the system is compared with the parametrically forced discrete-time system. Similar phenomena are confirmed between the parametrically forced discrete-time system and the parametrically forced Chua's circuit. It is worth noting that this facilitates to analyze parametrically forced continuous-time systems, because to analyze discrete-time systems is easier than continuous-time systems. Finally, we investigated behaviors of another coupled continuous-time system in which Chua's circuit is used, while, the motion of the switch controlling the parametric varying is different from the above system. Coexisting of many attractors whose synchronizations states are different are observed. Comparing with theabove system, the number of coexisting stable state is increased by the effect of the different switching motion