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This thesis takes a mathematical journey through concepts of synchronization, cascades, self-organized criticality, and flows on networks, establishing several novel emergent phenomena. It provides new analytical tools insights into the collective behaviors that can arise from the co-evolution of different dynamical processes on complex networks.Emergence entails the birth of surplus complexity in a macroscopic system, when the whole becomes more than the sum of its parts. For example, ant colonies, composed of many seemingly dull and unimpressive individual ants display surprising structure and organization on the system level. And a cup of coffee that we label as hot and wet, is composed of molecules possessing neither of these properties. Emergent properties of some systems, e.g., in thermodynamics, can be summarized with concise, elegant equations. Yet, in many systems, even the full knowledge of the fundamental rules governing each element isn't enough. The field of Chemistry, for example, relies on quantum mechanics and can in principle be fully simulated given high enough computational power, yet it remains a complex, open ended field of research with centuries of tradition and no end in sight. This is mainly due to two reasons: One is the superiority of direct analytical models for the emergent properties over a blind search with simulations. The second is the lack of remotely enough computational power required for such simulations.Complexity science studies the question of emergence in general, how the system can be more than the sum of its parts? Or why is it that ``more is different," as the esteemed Philip Anderson put it. The field of complex systems mainly focuses on the types of collective behaviors with wide applicability, including but not limited to flows, consensus formation, synchronization, cascades, self-organized criticality, and extreme events. Emergent phenomena usually rely on the interactions between subsystems, and hence network science with the primary focus on system composition through the relationships (the edges) between the subsystems (the nodes) is particularly well suited for answering such questions. In this thesis, we will embark on a mathematical journey through these concepts, heavily relying on both, complex systems' and network scientific approaches, finding new analytical insights and supporting our findings with numerical simulations.We start by coupling two quintessential models in complexity science: the Bak-Tang-Wiesenfeld (BTW) sandpile model of self-organized critical cascades and the Kuramoto model of synchronization. This fusion reveals emergent long-time oscillations not present in either model alone, punctuated by extreme dragon king events in the form of system-spanning cascades of cascades. Our analysis allows the characterization of the tipping point leading to dragon kings, frequency of oscillations, and cascade size distributions.Abstracting from this specific phenomenon, we next explore different ways through which models of self-organized criticality could give rise to dragon kings. This leads to a natural taxonomy of different types of dragon king events, and provides a so far unknown necessary condition on self-organized criticality in general.Next, we examine the interplay and co-evolution of diverse mobile agents on a network. Analogous to the social dynamics of intrinsically diverse individuals who navigate between and interact within various physical or virtual locations, agents in our model traverse a complex network of heterogeneous environments and engage with everyone they encounter. The precise nature of agents' internal dynamics and the various interactions that nodes induce are left unspecified and can be tailored to suit the requirements of individual applications. We derive effective dynamical equations for agent states which are instrumental in investigating thresholds of consensus, devising effective attack strategies to hinder coherence, and designing optimal network structures with inherent node variations in mind.Finally, we consider flows on networks under various symmetries and conservations, such as human transportation, internet packets, power grids, and mailed packages. We construct a hierarchy of field-theoretic models, capable of accurately modeling the throughput, predicting sustainability collapse under growing demand, and allowing us to explore a spectrum of routing strategies, locate flow bottlenecks, and determine the potential benefits of dissipation.

Interdisciplinary investigation has the potential to advance all fields involved.In this dissertation, three distinct fields of Quantitative Biology are discussed and advanced incrementally using the general tools of Mathematical Physics. Chapter one applies reaction-diffusion equations to explain the dispersal of cellsby localized degradation of a chemoattractant, which could explain the migration of leukocytes from the thymus and be a mechanism for morphogenesis. Chapter two investigates a particle model wherein an attractive force explains the termination of atrial fibrillation. Atrial fibrillation— the most common cardiac arrhythmia in the world with approximately 30 million patients in 2010— is associated with increased morbidity and mortality. Chapter three applies machine learning to explain social recognition in primate hippocampus, showing that cross-modal representations of identity can be achieved by at least two distinct neural mechanisms and that these representations comprise multiple social categories that reflect different relationships. Together, these chapters demonstrate the general capacity of Mathematical Physics to advance Quantitative Biology in addition to the capacity for Quantitative Biology to motivate novel analytic results and analyses within Mathematical Physics.

This dissertation is split into 3 parts. In the first part (Chapter 2) we look at shapesor manifolds on which neural activity over time lies. In a neural state space, whereeach axis represents a neuron, neural activity over time forms a point cloud. This pointcloud often occupies a small region in the space of all possible activity patterns thusrevealing structure in data. We consider point clouds from neural activities from commonpopulation codes known as “tuning curve models”. In these models, the firing rate ofeach neuron is a function of a latent variable which might be a stimulus variable or avariable related to an internal state and a tuning curve parameter which labels eachneuron. We address the question: How close are point clouds formed by such models toa linear subspace? To answer this, we define the linear dimension of the data to be thenumber of dimensions which captures a very high fraction of variance, for example 95%variance in this data. We show that the linear dimension grows exponentially with thenumber of latent variables encoded by the population. Thus the manifolds formed by theneural activities from these models are extremely non-linear. Linear dimension is not agood measure for the intrinsic dimension of the manifold on which this point cloud lies.In the second part (Chapter 3), we model connections between distant brain regionsby sparse random connections. We start by observing that such a network has a specialproperty known as the expander property. Using this property it can be shown thatinformation can be transmitted efficiently from a source region to a target region evenif the target region has fewer neurons than the source region. We also consider if thecompressed patterns in the target region can be re-coded or expanded to perform somecomputation. We show that the compressed patterns can be re-expanded by algorithmsknown as Locally Competitive Algorithms (LCA) and the re-expanded patterns can beseparated by a downstream neuron into arbitrarily defined classes. We next considerwhether long range reciprocal connections between two regions can be used to maintainpersistent activity in both the regions. Such activity is thought to be a substrate forworking memory, the ability to hold things in mind. We show that the network can indeedmaintain sparse patterns of activity through simple network dynamics. We conclude thatsparse random connections can be used to transmit information effectively and improvethe performance of certain computations compared to dense random connections.In the last part (Chapter 4), we built a computational rate model for the pre-cortexbiological neural circuit responsible for the localisation of sound in the vertical plane.Interaction of incoming sound waves with the outer ear filters out energy from specificfrequency bands in the spectrum of the incoming sound. The frequency bands with zeroor reduced power in them are known as notches. The position of the notches is a functionof the angle of elevation of the sound source. There is a dedicated set of neurons in theauditory pathway which are sensitive to the position of these notches and hence thoughtto be responsible for the localization of sound in the vertical plane. These neurons showdifferent levels of excitation or inhibition above or below their spontaneous rates for different combinations of frequencies and intensities of sound. We built a computational model to probe how this complex set of responses arise from the interaction between the various populations of neurons in the auditory pathway.

This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.

We develop methods for numerically integrating oscillatory functions and employ one of the methods in a new spectrally accurate collocation method to solve the linearized Vlasov-Poisson equation describing a collisionless plasma.In 1923, Filon provided one of the first methods for evaluating definite oscillatory integrals via a quadrature formula. Filon's method involved treating the integral as a non-oscillatory function being multiplied by an oscillatory function. An inherent difficulty to what are now known as Filon methods involves integrating a set of basis functions against the oscillatory part of the function. We present two moment free Filon method capable of calculating indefinite integrals on a finite interval for irregular oscillators. These are the first moment free Filon methods with a polynomial basis that can compute integrals for general irregular oscillators. This work is a generalization of previous work by Hasegawa and Torii for the regular oscillator case that relies on representing the non-oscillatory part of the function via shifted Chebychev polynomials. We also present a novel method for calculating the regular oscillator case.We then present a new collocation technique of computing solutions of the single-mode 1-D Vlasov-Poisson problem in which the plasma density is represented on Chebychev panels using high-order polynomials in time. We apply this method to a Volterra equation first discovered by Penrose that describes the time evolution of the plasma density. We also compute the resolvent of the Volterra integral operator using the collocation scheme. By integrating a highly oscillatory function that fits in the framework of the first half of the thesis, we are able to reconstruct the velocity distribution of the plasma at any time. The numerical simulations agree with key theoretical aspects of plasma physics such as Landau Damping. We present examples for a Maxwellian distribution and a double-hump unstable distribution.

Understanding the topological structure of phase space for dynamical systems inhigher dimensions is critical for numerous applications, including transport of objectsin the solar system, systems of uids, and charged particles in crossed magneticand electric elds. Many topological techniques have been developed tostudy maps of two-dimensional (2D) phase spaces, but extending these techniquesto higher dimensions is often a major challenge or even impossible. One such technique,homotopic lobe dynamics (HLD), has shown great success in analyzing thestable and unstable manifolds of hyperbolic xed points for area-preserving mapsin two dimensions. The output of the HLD technique is a symbolic descriptionof the minimal underlying topology of the invariant manifolds. The present workextends HLD to volume-preserving maps in three dimensions. We extend HLD tosystems that have equatorial heteroclinic intersections, pole-to-pole invariant circles,and forced pole-to-pole heteroclinic intersections. In order to extend HLD tothese cases, we went through multiple computational methodologies as well shiftour perspective of manifolds in 3D. We demonstrate the power of the HLD byapplying it to increasingly complex numerical and theoretical examples.

Free-Electron Lasers like the one at the SLAC National Accelerator Laboratory are sources of extremely bright X-rays that are useful in a variety of scientific imaging applications. Because there are only a handful of FEL facilities around the world, access to these X-rays is in high demand. Every year, hundreds of hours are spent tuning quadrupole focusing magnets to optimize the X-ray brightness. During this tuning process, the beam typically cannot be used for experiments. In this thesis, I show that by performing Bayesian optimization using a Gaussian process regression model containing prior information derived from an optical model of the accelerator in combination with historical data, we were able to significantly reduce the amount of time spent tuning the quadrupoles in comparison to previous methods.

In this work, I provide a literature review of earlier works on classification of topological phases. In doing so, we motivate the idea of topological order and phases of matter that are dependent on global geometric features of systems in consideration. I review an algebraic formalism to categorize phases of matter containing non-abelian anyons, known as Braided Tensor Categories (BTC). With the motivation to study the relationship between non-abelian anyonic and symmetry defects, I also introduce the an algebraic formalism describing the interplay of global symmetries with the anyonic degrees of freedom. This formalism is known as $G$-crossed BTC. These algebraic formalism provide observable topological quantities, such as the topological $S$-matrix, that can be probed by interference measurements. When the topological $S$-matrix of the BTC is unitary, it is said that the fusion theory is said to be modular. Hence, one calls the algebraic formalism a Modular Tensor Category (MTC). I present the results of our work showing that one can construct projective representations of the mapping class group of the punctured torus from the topological formalism given by $G$-crossed BTC. This is pivotal in constructing a $G$-crossed BTC for surfaces of higher genus, as one can build surfaces of arbitrary genus by gluing many punctured tori. Throughout my mathematical analysis, I also show that modularity of BTCs are not destroyed by the presence of symmetry defects.

This dissertation focuses on nonlinear dynamical systems with net responses. In particular, we discover a steady field induced within liquids by a sinusoidal potential, referred to as Asymmetric Rectified Electric Field (AREF). AREF helps explain several long-standing discrepancies regarding the behavior of particles and electrically induced fluid flows in response to oscillatory potentials, broadly impacts the interpretation of the experiments, and offers new avenues for research in electrokinetics. Additionally, we demonstrate that non-antiperiodic, zero-time-average, excitation of a spatially symmetric system can yield net responses. We consider an object atop a flat surface that undergoes a dual-mode horizontal vibration. Our calculations, and subsequent experimental observations, show that the object experiences a net drift if the applied frequencies are the ratio of odd and even numbers (e.g., 1 Hz and 2 Hz). As a corollary, our theory suggests that swapping the powered (non-antiperiodic potential) and the grounded parallel electrodes of an electrochemical cell alters the system behavior, a prediction verified by our experimental observations on the AREF-induced electrophoresis.

Preferential concentration describes the tendency of heavy particles to accumulate in certain regions of a turbulent flow. This process has been hypothesized to play a role in particle growth which is of crucial importance in numerous physical and engineering applications. The efficiency of preferential concentration is known to depend on the ratio of the particle stopping time to the turbulent eddy turnover time, which is called the Stokes number. In this thesis, we investigate the role of turbulence on preferential concentration of heavy particles with Stokes number less than unity. We use Direct Numerical Simulations and adopt the two-fluid formalism, where the particulate phase is treated as a continuum. In the first work, we study a two-way coupled system in the particle-induced Rayleigh-Taylor instability, and observe the striking emergence of dense, filamentary particle structures. Most notably, we find that the particle concentration enhancement primarily depends on three properties of the system: the rms fluid velocity, the Stokes number, and the assumed particle diffusivity from the two-fluid equations. Additionally, we note that when preferential concentration is dominant, the probability distribution function of the particle concentration takes on a distinctive form, characterized by an exponential tail whose slope is related to the same three properties listed above. In the second part, we further extend our study to a regime in which turbulence is externally-driven, and verify that the results found in the first study also hold. In the final work, we use a box-counting algorithm to identify and extract key features of the dense particle structures. We find in particular that these structures have a large aspect ratio. We propose an advection-diffusion model to predict their thickness, and find preliminary evidence that suggests that their long dimension depends on the Taylor microscale.

Fractional order nonlinear evolution equations have emerged in recent times as being very important model for depicting the interior behavior of nonlinear phenomena that exist in the real world. In particular, Schrodinger-type fractional nonlinear evolution equations constitute an aspect of the field of quantum mechanics. In this study, the (2 + 1)-dimensional time-fractional nonlinear Schrodinger equation and (1 + 1)-dimensional time-space fractional nonlinear Schrodinger equation are revealed as having different and novel wave structures. This is shown by constructing appropriate analytic wave solutions. A successful implementation of the advised rational ( 1 / ϕ ′ ( ξ ) ) -expansion method generates new outcomes of the considered equations, by comparing them with those already noted in the literature. On the basis of the conformable fractional derivative, a composite wave variable conversion has been used to adapt the suggested equations into the differential equations with a single independent variable before applying the scheme. Finally, the well-furnished outcomes are plotted in different 3D and 2D profiles for the purpose of illustrating various physical characteristics of wave structures. The employed technique is competent, productive and concise enough, making it feasible for future studies.

Time/space separation-based modeling methods have been widely researched for estimating lithium-ion battery (LIB) thermal dynamics. However, these methods have been developed in an offline environment and may not perform well in real-time application since the battery systems in electric vehicles (EVs) are usually subject to external disturbances. Furthermore, the onboard measurements of temperature are often corrupted by significant error. To address these problems, we present a reduced model-based observer design for online temperature distribution estimation in LIBs. First, an extreme learning machine (ELM) based offline spatiotemporal model is constructed to approximate the thermal dynamics of LIB. Second, an adaptive reduced order observer is designed based on the offline model developed in the previous step. The offline model is then updated with the estimation results of the observer. As the performance of the estimator is highly related to the placement of sensors, a genetic algorithm (GA) based integrated optimization strategy is also developed to determine the optimal sensor location for online estimation. Finally, the whole temperature distribution is estimated in real time using the observer, the measured voltage, current and the limited available temperature data. Simulation results on a battery thermal system verify the effectiveness of this developed model.

In this work, a generalized unsteady magnetohydrodynamic transport of a rate type fluid near an unbounded upright plate is analyzed under Newtonian heating and non-uniform velocity conditions. The vertical plate is suspended in a porous medium and is encountering the radiation effects. Three different fractional models for Maxwell fluid are established by using the modern definitions of Atangana–Baleanu, Caputo–Fabrizio, and Caputo fractional operators. Triple fractional analysis is conducted to reach out to the solutions of consequent flow and energy equations. Laplace transform and Stehfest’s numerical algorithm are jointly applied to solve each fractional model. Shear stress and heat transfer rate are measured at the solid–fluid interface in the form of skin friction coefficient and Nusselt number respectively. The physical significance of associated parameters in velocity and energy boundary layers is investigated and graphs are provided to discuss the relevant physical arguments. A tabular analysis is performed to demonstrate the influence of physical parameters on shear stress and heat transfer rate. An empirical comparison between fractional and classical solutions indicates that fractional operators provide a better explanation of the physical features of the model. It is also analyzed that for Newtonian heating and non-uniform velocity conditions, the Atangana–Baleanu fractional operator is the finest fractional model to describe the memory effect of velocity and energy distribution. The velocity of a Maxwell fluid is always higher for constant velocity condition as compared to the ramped velocity condition. Furthermore, the heat transfer rate declines for increasing values of the fractional parameter α and a minimum value is witnessed for the classical model but, it follows an inverse trend when the radiation parameter R d increases.

There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the gazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

Transverse galloping is a type of flow-induced vibration (FIV) that leads to critical design considerations for engineering structures. A purely nonlinear energy sink (NES) composed of a ball free to rotate in a circular track experimentally mitigated the galloping of a square in a previous study. The current study introduces a model for simulating the dynamics of the square prism coupled with a ball-in-track (BIT) NES and predicting the system behaviour at high flow speeds beyond the limits of the previously presented experiments. Numerical simulations employ the fitting of experimental data as inputs to define parameters. Wind tunnel static experiments provide the galloping force coefficient [[EQUATION]] relative to the prism angle of attack. Additionally, free rotation tests allow evaluating the ball damping coefficient [[EQUATION]] as a function of its mass and the NES track radius. The result of the rotation tests provides a critical angular speed beyond which the ball damping increases non-linearly. We point out the damping variation as an advantage of the BIT-NES; less damping at low angular velocities helps the ball start its rotation, while relatively large damping at higher speeds dissipates more energy from the vibrating system. Numerical results exhibit four response modes for the NES; oscillatory at low flow speeds, intermittent within a small range of higher flow speeds, rotational at higher flow speeds, and ineffective regime at flow speeds out of the NES effective range. Modelling the primary mass as a parametric excitation source for the NES provides an analytical estimation of the boundary between the oscillatory and intermittent regimes. Furthermore, we advance an analytical analysis of the power flow across the integrated prism-NES system to explain the NES behaviour and predict the limit of its effective range.

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

A sliding mode control (SMC) strategy is proposed in this paper for a class of perturbed nonlinear systems with unmeasurable states and state constraints to deal with the state tracking problems. First of all, a partial states observer is designed for solving the problems due to unmeasurable states. The estimation errors will approach zero in a finite time. Secondly, based on a designed barrier Lyapunov function, one designs the sliding surface function and an adaptive sliding mode tracking controller to ensure that the states have the ability to track the desired signals. Moreover, the tracking error is capable of converging to zero in a finite time without violating the given state's constraints. Perturbation estimator and adaptive mechanisms are also utilized so that there is no need to know the upper bounds of perturbations and perturbation estimation errors. Finally, a numerical example is provided to demonstrate the feasibility of the proposed control strategy.

The rotating characteristic and the stability analysis of unsymmetrical magnetically suspended motor (MSM) are investigated in the article. Firstly, the dynamic functions of unsymmetrical MSM with different moment of inertia ratios and suspension span ratios are developed. Based on the equation of radial rotation, the open-loop pole and the frequency response are analyzed, so relationships among critical whirling frequency, moment of inertia ratio and suspension span ratio are established. Moreover, relationships among rotating characteristic, moment of inertia ratio and suspension span ratio are investigated, and then the rotation stability criterion of unsymmetrical MSM based on dual-frequency bode diagram is developed. Finally, the results prove that the proposed rotation analysis method is a potential way to analyze rotating characteristic of unsymmetrical MSM.

This paper gives a condensed review of the history of solutions to the Euler-Poisson equations modeling equilibrium states of rotating stars and galaxies, leading to a recent result of Walter Strauss and the author. This result constructs a connected set of rotating star solutions for larger and larger rotation speed, so that the supports of the stars become unbounded if we assume an equation of state $p = \rho^\gamma$, $4/3

The bifurcation structure of the oscillations of ultrasound contrast agents (UCAs) was studied as a function of the driving pressure for excitation frequencies that were determined using the UCAs pressure-dependent resonances $$(f_\mathrm{s})$$ . It was shown that when excited by the $$(f_\mathrm{s})$$ , the UCA can undergo a saddle-node bifurcation (SNB) to higher amplitude oscillations. The driving pressure at which the SNB occurs is controllable and depends on the $$(f_\mathrm{s})$$ magnitude. Utilizing the appropriate $$(f_\mathrm{s})$$ , the scattering cross section of the UCAs can significantly be enhanced (e.g., $$\sim $$ ninefold) while at the same time avoiding potential UCA destruction (by limiting the radial expansion ratio $$

Under the relation of 0 0 -concordance, the set of knotted 2-spheres in S 4 S^4 forms a commutative monoid M 0 \mathcal {M}_0 with the operation of connected sum. Sunukjian [Int. Math. Res. Not. IMRN 17 (2015), pp. 7950–7978] has recently shown that M 0 \mathcal {M}_0 contains a submonoid isomorphic to Z ≥ 0 \mathbb {Z}^{\ge 0} . In this note, we show that M 0 \mathcal {M}_0 contains a submonoid isomorphic to ( Z ≥ 0 ) ∞ (\mathbb {Z}^{\ge 0})^\infty . Our argument relates the 0 0 -concordance monoid to linear independence of certain Seifert solids in the spin rational homology cobordism group.