Proponents of mechanistic explanation have argued that dynamical models are mere phenomenal models, in that they describe rather than explain the scientific phenomena produced by complex systems. I argue instead that dynamical models can, in fact, be explanatory. Using an example from neuroscientific research on epilepsy, I show that dynamical models can meet the explanatory demands met by mechanistic models, and as such occupy their own unique place within the space of explanatory scientific models.

In recent years, technological advances in animal tracking have renewed interests in collective animal behavior, and in particular, locust swarms. These swarms pose a major threat to agriculture in northern Africa, the Middle East, and other regions. In their early life stages, locusts move in hopper bands, which are huge aggregations traveling on the ground. Our main goal is to understand the underlying mechanisms for the emergence and organization of these bands. We construct an agent-based model that tracks individual locusts and a continuum model that tracks the evolution of locust density. Both these models are motivated by experimental observations of individuals’ behavior. The macroscopic emergent behavior of the group is studied through numerical simulation of these models.

The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and Poincaré. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the Bishop-Jones formula for strongly discrete groups, equating the Hausdorff dimension of the radial limit set with the Poincaré exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on the limit set.

Despite the high-dimensional nature of the nervous system, humans produce low-dimensional cognitive and behavioral dynamics. How high-dimensional networks with complex connectivity give rise to functionally meaningful dynamics is not well understood. How does a neural network encode function? How can functional dynamics be systematically obtained from networks? There exist few frameworks in the current literature that answer these questions satisfactorily. In this dissertation I propose a general theoretical framework entitled 'Structured Flows on Manifolds' and its underlying mathematical basis. The framework is based on the principles of non-linear dynamical systems and Synergetics and can be used to understand how high-dimensional systems that exhibit multiple time-scale behavior can produce low-dimensional dynamics. Low-dimensional functional dynamics arises as a result of the timescale separation of the systems component's dynamics. The low-dimensional space in which the functi onal dynamics occurs is regarded as a manifold onto which the entire systems dynamics collapses. For the duration of the function the system will stay on the manifold and evolve along the manifold. From a network perspective the manifold is viewed as the product of the interactions of the network nodes. The subsequent flows on the manifold are a result of the asymmetries of network's interactions. A distributed functional architecture based on this perspective is presented. Within this distributed functional architecture, issues related to networks such as flexibility, redundancy and robustness of the network's dynamics are addressed. Flexibility in networks is demonstrated by showing how the same network can produce different types of dynamics as a function of the asymmetrical coupling between nodes. Redundancy can be achieved by systematically creating different networks that exhibit the same dynamics. The framework is also used to systematically probe the effects of lesion / (removal of nodes) on network dynamics. It is also shown how low-dimensional functional dynamics can be obtained from firing-rate neuron models by placing biologically realistic constraints on the coupling. Finally the theoretical framework is applied to real data. Using the structured flows on manifolds approach we quantify team performance and team coordination and develop objective measures of team performance based on skill level. / by Ajay S. Pillai. / Thesis (Ph.D.)--Florida Atlantic University, 2008. / Includes bibliography. / Electronic reproduction. Boca Raton, FL : 2008 Mode of access: World Wide Web.

This thesis examines the nonlinear behaviour of thermoacoustic systems by using approaches from the field of nonlinear dynamics. The underlying behaviour of a nonlinear system is determined by two things: first, by the type and form of the attractors in phase space, and second, by the mechanism that the system transitions from one attractor to another. For a thermoacoustic system, both of these things must be understood in order to define a safe operating region in parameter space, where no high-amplitude oscillations exist. Triggering in thermoacoustics is examined in a simple model of a horizontal Rijke tube. A triggering mechanism is presented whereby the system transitions from a stable fixed point to a stable limit cycle, via an unstable limit cycle. The practical stability of the Rijke tube was investigated when the system is forced by stochastic noise. Low levels of noise result in triggering much before the linear stability limit. Stochastic stability maps are introduced to visualise the practical stability of a thermoacoustic system. The triggering mechanism and stochastic dependence of the Rijke tube match extremely well with results from an experimental combustor. The most common attractors in thermoacoustic systems are fixed points and limit cycles. In order to define the nonlinear behaviour of a thermoacoustic system, it is therefore important to find the regions of parameter space where limit cycles exist. Two methods of finding limit cycles in large thermoacoustic sytems are presented: matrix-free continuation methods and gradient methods. Continuation methods find limit cycles numerically in the time domain, with no additional assumptions other than those used to form the governing equations. Once the limit cycles are found, these continuation methods track them as the operating condition of the system changes. Most continuation methods are impractical for finding limit cycles in large thermoacoustic systems because the methods require too much computational time and memory. In the literature, there are therefore only a few applications of continuation methods to thermoacoustics, all with low-order models. Matrix-free shooting methods efficiently calculate the limit cycles of dissipative systems and have been demonstrated recently in fluid dynamics, but are as yet unused in thermoacoustics. These matrix-free methods are shown to converge quickly to limit cycles by implicitly using a 'reduced order model' property. This is because the methods preferentially use the influential bulk motions of the system, whilst ignoring the features that are quickly dissipated in time. The matrix-free methods are demonstrated on a model of a ducted 2D diffusion flame, and the safe operating region is calculated as a function of the Peclet number and the heat release parameter. Both subcritical and supercritical Hopf bifurcations are found. Physical information about the flame-acoustic interaction is found from the limit cycles and Floquet modes. Invariant subspace preconditioning, higher order prediction techniques, and multiple shooting techniques are all shown to reduce the time required to generate bifurcation surfaces. Two types of shooting are compared, and two types of matrix-free evaluation are compared. The matrix-free methods are also demonstrated on a model of a ducted axisymmetric premixed flame, using a kinematic G-equation solver. The methods find limit cycles, period-2 limit cycles, fold bifurcations, period-doubling bifurcations and Neimark-Sacker bifurcations as a function of two parameters: the location of the flame in the duct, and the aspect ratio of the steady flame. The model is seen to display rich nonlinear behaviour and regions of multistability are found. Gradient methods can also efficiently calculate the limit cycles of large systems. A scalar cost function is defined that describes the proximity of a state to a limit cycle. The gradient of the cost function is used in an optimisation routine to iteratively converge to a limit cycle (or fixed point). The gradient of the cost function is found with a forwards-backwards process: first, the direct equations are marched forwards in time, second, the adjoint equations are marched backwards in time. The adjoint equations are derived by partially differentiating the direct governing equations. The gradient method is demonstrated on a model of a horizontal Rijke tube. This thesis describes novel nonlinear analysis techniques that can be applied to coupled systems with both advanced acoustic models and advanced flame models. The techniques can characterise the rich nonlinear behaviour of thermoacoustic models with a level of detail that was not previously possible.

abstract: I investigate two models interacting agent systems: the first is motivated by the flocking and swarming behaviors in biological systems, while the second models opinion formation in social networks. In each setting, I define natural notions of convergence (to a ``flock" and to a ``consensus'', respectively), and study the convergence properties of each in the limit as $t \rightarrow \infty$. Specifically, I provide sufficient conditions for the convergence of both of the models, and conduct numerical experiments to study the resulting solutions. / Dissertation/Thesis / Masters Thesis Mathematics 2018

Dynamical systems theory represents a prominent theoretical framework for the investigation of movement coordination and control in complex neurobiological systems. Central to this theory is the investigation of pattern formation in biological movement through application of tools from nonlinear dynamics. Movement patterns are regarded as attractors and changes in movement coordination can be described as phase transitions. Phase transitions typically exhibit certain key indicators like critical fluctuations, critical slowing down and hysteresis, which enable the formulation of hypotheses and experimental testing. An extensive body of literature exists which tested these characteristics and robustly supports the tenets of dynamical systems theory in the movement sciences. However, the majority of studies have tended to use a limited range of movement models for experimentation, mainly bimanual rhythmical movements, and at present it is not clear to what extent the results can be transferred to other domains such as discrete movements and/or multi-articular actions. The present work investigated coordination and control of discrete, multi-articular actions as exemplified by a movement model from the sports domain: the basketball hook shot. Accordingly, the aims of the research programme were three-fold. First, identification of an appropriate movement model. Second, development of an analytical apparatus to enable the application of dynamical systems theory to new movement models. Third, to relate key principles of dynamical systems theory to investigations of this new movement model. A summary of four related studies that were undertaken is as follows: 1. Based on a biomechanical analysis, the kinematics of the basketball hook shot in four participants of different skill levels were investigated. Participants were asked to throw from different shooting distances, which were varied in a systematic manner between 2m and 9m in two different conditions (with and without a defender present). There was a common significant trend for increasing throwing velocity paired with increasing wrist trajectory radii as shooting distance increased. Continuous angle kinematics showed high levels of inter- and intra-individual variability particularly related to throwing distance. Comparison of the kinematics when throwing with and without a defender present indicated differences for a novice performer, but not for more skilled individuals. In summary, the basketball hook shot is a suitable movement model for investigating the application of dynamical systems theory to a discrete, multi-articular movement model where throwing distance resembles a candidate control parameter. 2. Experimentation under the dynamical systems theoretical paradigm usually entails the systematic variation of a candidate control parameter in a scaling procedure. However there is no consensus regarding a suitable analysis procedure for discrete, multiarticular actions. Extending upon previous approaches, a cluster analysis method was developed which made the systematic identification of different movement patterns possible. The validity of the analysis method was demonstrated using distinct movement models: 1) bimanual, wrist movement, 2) three different basketball shots, 3) a basketball hook shot scaling experiment. In study 1, the results obtained from the cluster analysis approach matched results obtained by a traditional analysis using discrete relative phase. In study 2, the results from the method matched the a-priori known distinction into three different basketball techniques. Study 3 was designed specifically to facilitate a bimodal throwing pattern due to laboratory restrictions in throwing height. The cluster analysis again was able to identify the a-priori known distribution. Additionally, a hysteresis effect for throwing distance was identified further strengthening the validity of the chosen movement model. 3. Using eight participants, hook shot throwing distance was varied between 2m and 9m in both directions. Some distinct inter-individual differences were found in regards to movement patterning. For two subjects clear transitions between qualitatively distinct different patterns could be established. However, no qualitative differences were apparent for the remaining participants where it was suggested that a single movement pattern was continually scaled according to the throwing distance. The data supported the concept of degeneracy in that once additional movement degrees of freedom are made available these can be exploited by actors. The underlying attractor dynamics for the basketball hook shot were quite distinct from the bistable regime typically observed in rhythmical bimanual movement models. 4. To provide further evidence in support of the view that observed changes in movement patterning during a hook shot represented a phase transition, a perturbation experiment with five participants was performed. Throwing distance was once again varied in a scaling manner between 2m and 9m. The participants wore a wristband which could be attached to a weight which served as a mechanical perturbation to the throwing movement. Investigation of relaxation time-scales did not provide any evidence for critical slowing down. The movements showed high variation between all subsequent trials and no systematic variation in relation to either the mechanical perturbation or the successive jumps in throwing distance was indicated by the data. In summary, the results of the research programme highlighted some important differences between discrete multi-articular and bimanual rhythmical movement models. Based on these differences many of the findings ubiquitous in the domain of rhythmical movements may be specific to these and accordingly may not be readily generalized to movement models from other domains. This highlights the need for more research focussing on various movement models in order to broaden the scope of the dynamical systems framework and enhance further insight into movement coordination and control in complex neurobiological systems.

Let X be a locally compact Hausdorff space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A(X, σ). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy. In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U (n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group.

Let X be a locally compact Hausdorff space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A(X, σ). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy. In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U (n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group.

Thesis (Ph. D.)--Harvard University, 1968. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 1-5 (last group)).