京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18768号 / 理博第4026号 / 新制||理||1580(附属図書館) / 31719 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 上田 哲生, 教授 加藤 毅 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The objective of this dissertation is to develop a mechanical model of a nonlinear piezoelectric energy harvesting system induced by Duffing oscillator and predict the periodic motions of such a nonlinear dynamical system under different excitation frequency. In this dissertation, analytical distributed-parameter electromechanical modeling of a piezoelectric energy harvester will be presented. The electromechanically coupled circuit equation excited by infinitely many vibration modes is derived. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates and eventually an infinite set of algebraic equations is obtained for the complex modal vibration response and the complex voltage response of the energy harvester beam. One single vibration mode is chosen and discussed. The periodic motions are obtained through an implicit mapping method with high accuracy, stability and bifurcations of periodic motions are determined by the eigenvalue analysis. Frequency-amplitude characteristics of periodic motions are achieved by the Fourier transform

The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle configurations. We also investigate Hopf bifurcation surfaces of various dynamical systems.

In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized Pochhammer-Chree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions of both models here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves for each model, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. For the microstructure equation, the new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including the dynamics of each family of solitary waves using exponential asymptotics techniques, are also mentioned.

Rich dynamics have been demonstrated when a discrete time delay is introduced in a simple predator-prey system. For example, Hopf bifurcations and a sequence of period doubling bifurcations that appear to lead to chaotic dynamics have been observed. In this thesis we consider two different predator-prey models: the classical Gause-type predator-prey model and the chemostat predator-prey model. In both cases, we explore how different ways of modeling the time between the first contact of the predator with the prey and its eventual conversion to predator biomass affects the possible range of dynamics predicted by the models. The models we explore are systems of integro-differential equations with delay kernels from various distributions including the gamma distribution of different orders, the uniform distribution, and the Dirac delta distribution. We study the models using bifurcation theory taking the mean delay as the main bifurcation parameter. We use both an analytical approach and a computational approach using the numerical continuation software XPPAUT and DDE-BIFTOOL. First, general results common to all the models are established. Then, the differences due to the selection of particular delay kernels are considered. In particular, the differences in regions of stability of the coexistence equilibrium are investigated. Finally, the effects on the predicted range of dynamics between the classical Gause-type and the chemostat predator-prey models are compared. / Thesis / Doctor of Philosophy (PhD)

Many important engineering phenomena such as turbulent flow, fluid-structure interactions, and climate diagnostics are chaotic and sensitivity analysis of such systems is a challenging problem. Computational methods have been proposed to accurately and efficiently estimate the sensitivity analysis of these systems which is of great scientific and engineering interest. In this thesis, a new approach is applied to compute the direct and adjoint sensitivities of time-averaged quantities defined from the chaotic response of the Lorenz system and the double pendulum system. A stabilized time-integrator with adaptive time-step control is used to maintain stability of the sensitivity calculations. A study of convergence of a quantity of interest and its square is presented. Results show that the approach computes accurate sensitivity values with a computational cost that is multiple orders-of-magnitude lower than competing approaches based on least-squares-shadowing approach.

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Particle adhesion to the vasculature depends critically upon particle/cell properties (size, receptors), scale/geometric features of vasculature (diameter, bifurcation, etc.) and local hemodynamic factors (stress, torque, etc.) Current investigations using in vitro parallel-plate flow chambers suffer from several limitations including (a) idealized constructs, (b) lack of critical morphological features (bifurcations, network), (c) inability to distinguish between healthy vs. diseased vasculature, (d) large volumes and (e) non-disposability. To overcome these limitations, microvascular networks, obtained from digitization of in vivo topology were prototyped using soft-lithography techniques to generate Synthetic Microvascular Networks (SMN). CFD-ACE+, a finite volume based Computational Fluid Dynamics (CFD) software, was used to develop a computational model of the digitized networks. Dye perfusion patterns predicted by the simulations matched well with experimental observations indicating presence of well perfused as well as stagnant regions. Studies using functionalized microparticles showed non-uniform particle adhesion, with preferential adhesion at a distance of 2 vessel diameters or less from the nearest bifurcation which was validated with in vivo data. Bifurcation adhesion ratio (BAR) was found to be significantly higher for experiments (49% and 36%) and simulations (67% and 52%) compared to expected values of 24% and 21%. A single experimental run in SMN generated the entire shear adhesion map highlighting the benefits of the SMN assay. Green Fluorescent Protein (GFP) gene delivery studies with a nanopolymeric based gene delivery system showed preferential GFP expression in the vicinity of bends and bifurcation of the microvascular networks. The developed SMN based microfluidic device will have critical applications both in basic research, where it can be used to characterize and develop next generation delivery vehicles, and in drug discovery, where it can be used to study the efficacy of the drug in these realistic microvascular networks. / Mechanical Engineering

<p> A model of two predators competing for the same prey also involving predation interaction between the two predators is considered. Coexistence in forms of equilibria and periodic orbits is obtained by using bifurcation and dynamical systems theory. Global dynamics is obtained by studying the survival functions and persistence is obtained by using a theorem of Freedman and Waltman. Finally, numerical results for a specific example demonstrate the above. A Hopf bifurcation at the interior equilibrium and its unstable periodic orbit are observed.</p> / Thesis / Master of Science (MSc)

Circadian rhythms in mammals are self-sustained repeating activities driven by the circadian gene expression in cells, which is regulated at both transcriptional and posttranscriptional stages. In this work, we first used mathematical modeling to investigate the transcriptional regulation of circadian gene expression, with a focus on the mechanisms of robust genetic oscillations in the mammalian circadian core clock. Secondly, we built a coarse-grained model to study the post-transcriptional regulation of the rhythmicities of poly(A) tail length observed in hundreds of mRNAs in mouse liver. Lastly, we examined the application of Sobol indices, which is a global sensitivity analysis method, to mathematical models of biological oscillation systems, and proposed two methods tailored for the calculation of circular Sobol indices. In the first project, we modified the core negative feedback loop in a mathematical model of the mammalian genetic oscillator so that the unrealistic tight binding between the repressor PER and the activator BMAL1 is relaxed for robust oscillations. By studying the modified extended models, we found that the auxiliary positive feedback loop, rather than the auxiliary negative feedback loop, makes the oscillations more robust, yet they are similar when accounting for circadian rhythms (~24h period). In the second project, we investigated the regulation of rhythmicities in poly(A) tail length by four coupled rhythmic processes, which are transcription, deadenylation, polyadenylation, and degradation. We found that rhythmic deadenylation is the strongest contributor to the rhythmicity in poly(A) tail length and the rhythmicity in the abundance of the mRNA subpopulation with long poly(A) tails. In line with this finding, the model further showed that the experimentally observed distinct peak phases in the expression of deadenylases, regardless of other rhythmic controls, can robustly cluster the rhythmic mRNAs by their peak phases in poly(A) tail length and abundance of the long-tailed subpopulation. In the last project, we reviewed the theoretical basis of Sobol indices and identified potential problems when it is applied to mathematical models of biological oscillation systems. Based on circular statistics, we proposed two methods for the calculation of circular Sobol indices and compared their performance with the original Sobol indices in several models. We found that though the relative rankings of the contribution from parameters are the same across three methods, circular Sobol indices can better quantitatively distinguish the contribution of individual parameters. Through this work, we showed that mathematical modeling combined with sensitivity analysis can help us understand the mechanisms underlying the circadian gene expression in mammalian cells. Also, testable predictions are made for future experiments and new ideas are provided that can enable potential chronopharmacology research. / Doctor of Philosophy / Circadian rhythms are repeating biological activities with ~24h period observed in most living organisms. Disruption of circadian rhythms in humans has been found to be promote cancer, metabolic diseases, cognitive degeneration etc. In this work, we first used mathematical modeling to study the mechanisms of robust oscillations in the mammalian circadian core clock, which is a molecular regulatory network that drives circadian gene expression at transcriptional stage. Secondly, we built a coarse-grained model to investigate the post-transcriptional regulation of the rhythmicities in poly(A) tail length, which are observed in hundreds of mRNAs in mouse liver. Lastly, we examined the application of Sobol indices, which is a global sensitivity analysis method, to mathematical models of biological oscillation systems, and proposed two methods tailored for the calculation of circular Sobol indices. In the first project, we modified a previous mathematical model of the mammalian genetic oscillator so that it sustains robust oscillation with more realistic parameter values. Our analysis of the model further showed that the auxiliary positive feedback loop, rather than the auxiliary negative feedback loop, makes the oscillations more robust. In the second project, we found that rhythmic deadenylation, among the coupled transcription, polyadenylation, and degradation processes, mostly controls the rhythmicity of poly(A) tail length and mRNA subpopulation with long poly(A) tails. Lastly, we reviewed the theoretical basis of Sobol indices and found potential problems when it is applied to mathematical models of biological oscillation systems. Based on circular statistics, we proposed two circular Sobol indices, which can better distinguish the contribution of individual parameters to model outputs than the original Sobol indices. Altogether, we used mathematical modeling and sensitivity analysis to investigate the regulation of circadian gene expression in mammalian cells, providing testable predictions and new ideas for future experiments and chronopharmacology research.