Synchronization of complex dynamical networks with fractional order

Complex dynamical networks (CDN) can be applied to many areas in real world, from medicine, biology, Internet to sociology. Study on CDNs has drawn great attention in recent years. Nodes in a CDN can be modelled as systems represented by differential equations. Study has shown that fractional order differential equations (DF) can better represent some real world systems than integer-order DFs. This research work focuses on synchronization in fractional CDNs.  A literature review on CDNs with fractional order has summarized the latest works in this area.  Fractional chaotic systems are studied in our initial investigation.  Fractional calculus is introduced and the relevant fundamentals to model, describe and analyse dynamical networks are presented. It is shown that the structure and topological characteristics of a network can have a big impact on its synchronizability. Synchronizability and its various interpretations in dynamical networks are studied. To synchronize a CDN efficiently, controllers are generally needed. Controller design is one of the main tasks in this research. Our first design is a new sliding mode control to synchronize a dynamical network with two nodes. Its stability has been proven and verified by simulations.  Its convergence speed outperforms Vaidyanathan's scheme, a well-recognized scheme in this area. The design can be generalized to CDNs with more nodes.  As many applications can be modelled as CDNs with node clustering, a different sliding mode control is designed for cluster synchronization of a CDN with fractional order. Its stability is proven by using Lyapunov method. Its convergence and efficiency is shown in a simulation. Besides these nonlinear methods mentioned, linear control is also studied intensively for the synchronization.  A novel linear method for synchronization of fractional CDNs using a new fractional Proportional-Integral (PI) pinning control is proposed.  Its stability is proven and the synchronization criteria are obtained. The criteria have been simplified using two corollaries so the right value for the variables can be easily assigned. The proposed method is compared with the conventional linear method which uses Proportional (P) controller. In the comparison, the mean squared error function is used. The function measures the average of the squared errors and it is an instant indicator of the synchronization efficiency. A numerical simulation is repeated 100 times to obtain the averages over these runs. Each simulation has different random initial values for both controllers. The average of the errors in all the 100 simulations is obtained and the area under the function curve is defined as an overall performance index (OPI), which indicates the controller's overall performance. In control, small overshoot is always desired. In our work, the error variation is also used as a measure.  The maximum variation from the average of 100 simulations is calculated and compared for both methods. With all the statistical comparisons, it is clear that with the same power consumption, the proposed method outperforms the conventional one and achieves faster and smoother synchronization. Communication constraints exist in most real world CDNs. Communication constraints and their impact on control and synchronization of CDNs with fractional order are investigated in our study. A new adaptive method for synchronizing fractional CDN with disturbance and uncertainty is designed. Its stability is proven and its synchronization criteria are obtained for both fractional CDN with known and unknown parameters. Random disturbance is also included in both cases. Our results show that the new method is efficient in synchronizing CDNs with presence of both disturbance and uncertainty

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Invariant manifold theory for impulsive functional differential equations with applications

The primary contribution of this thesis is a development of invariant manifold theory for impulsive functional differential equations. We begin with an in-depth analysis of linear systems, immersed in a nonautonomous dynamical systems framework. We prove a variation-of-constants formula, introduce appropriate generalizations of stable, centre and unstable subspaces, and develop a Floquet theory for periodic systems. Using the Lyapunov-Perron method, we prove the existence of local centre manifolds at a nonhyperbolic equilibrium of nonlinear impulsive functional differential equations. Using a formal differentiation procedure in conjunction with machinery from functional analysis -- specifically, contraction mappings on scales of Banach spaces -- we prove that the centre manifold is smooth in the state space. By introducing a coordinate system, we are able to prove that the coefficients of any Taylor expansion of the local centre manifold are unique and sufficiently regular in the time and lag arguments that they can be computed by solving an impulsive boundary-value problem. After proving a reduction principle, this leads naturally to explorations into bifurcation theory, where we establish generalizations of the classical fold and Hopf bifurcations for impulsive delay differential equations. Aside from the centre manifold, we demonstrate the existence and smoothness of stable and unstable manifolds and prove a linearized stability theorem. One of the applications of the theory above is an analysis of a SIR model with pulsed vaccination and finite temporary immunity modeled by a discrete delay. We determine an analytical stability criteria for the disease-free equilibrium and prove the existence of a transcritical bifurcation of periodic solutions at some critical vaccination coverage level for generic system parameters. Then, using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution until a Hopf point is identifed. A cylinder bifurcation is observed; the periodic orbit expands into a cylinder in the extended phase space before eventually contracting onto a periodic orbit as the vaccination coverage vanishes. The other application is an impulsive stabilization method based on centre manifold reduction and optimization principles. Assuming a cost structure on the impulsive controller and a desired convergence rate target, we prove that under certain conditions there is always an impulsive controller that can stabilize a nonhyperbolic equilibrium with a trivial unstable subspace, robustly with respect to parameter perturbation, while guaranteeing a minimal cost. We then exploit the low-dimensionality of the centre manifold to develop a two-stage program that can be implemented to compute the optimal controller. To demonstrate the effectiveness of the two-stage program, which we call the centre probe method, we use the method to stabilize a complex network of 100 diffusively coupled nodes at a Hopf point. The cost structure is one that assigns higher cost to controlling of nodes that have more neighbours, while the jump functionals are required to be diagonal -- that is, they do not introduce further coupling. We also introduce a secondary goal, which is that the number of nodes that are controlled is minimized

Piecewise Linear Dynamical Systems: From Nodes to Networks

Piecewise linear (PWL) modelling has many useful applications in the applied sciences. Although the number of techniques for analysing nonsmooth systems has grown in recent years, this has typically focused on low dimensional systems and relatively little attention has been paid to networks. We aim to redress this balance with a focus on synchronous oscillatory network states. For networks with smooth nodal components, weak coupling theory, phase-amplitude reductions, and the master stability function are standard methodologies to assess the stability of the synchronous state. However, when network elements have some degree of nonsmoothness, these tools cannot be directly used and a more careful treatment is required. The work in this thesis addresses this challenge and shows how the use of saltation operators allows for an appropriate treatment of networks of PWL oscillators. This is used to augment all the aforementioned methods. The power of this formalism is illustrated by application to network problems ranging from mechanics to neuroscience

Room layout estimation on mobile devices

Room layout generation is the problem of generating a drawing or a digital model of an existing room from a set of measurements such as laser data or images. The generation of floor plans can find application in the building industry to assess the quality and the correctness of an ongoing construction w.r.t. the initial model, or to quickly sketch the renovation of an apartment. Real estate industry can rely on automatic generation of floor plans to ease the process of checking the livable surface and to propose virtual visits to prospective customers. As for the general public, the room layout can be integrated into mixed reality games to provide a better immersiveness experience, or used in other related augmented reality applications such room redecoration. The goal of this industrial thesis (CIFRE) is to investigate and take advantage of the state-of-the art mobile devices in order to automate the process of generating room layouts. Nowadays, modern mobile devices usually come a wide range of sensors, such as inertial motion unit (IMU), RGB cameras and, more recently, depth cameras. Moreover, tactile touchscreens offer a natural and simple way to interact with the user, thus favoring the development of interactive applications, in which the user can be part of the processing loop. This work aims at exploiting the richness of such devices to address the room layout generation problem. The thesis has three major contributions. We first show how the classic problem of detecting vanishing points in an image can benefit from an a-priori given by the IMU sensor. We propose a simple and effective algorithm for detecting vanishing points relying on the gravity vector estimated by the IMU. A new public dataset containing images and the relevant IMU data is introduced to help assessing vanishing point algorithms and foster further studies in the field. As a second contribution, we explored the state of-the-art of real-time localization and map optimization algorithms for RGB-D sensors. Real-time localization is a fundamental task to enable augmented reality applications, and thus it is a critical component when designing interactive applications. We propose an evaluation of existing algorithms for the common desktop set-up in order to be employed on a mobile device. For each considered method, we assess the accuracy of the localization as well as the computational performances when ported on a mobile device. Finally, we present a proof of concept of application able to generate the room layout relying on a Project Tango tablet equipped with an RGB-D sensor. In particular, we propose an algorithm that incrementally processes and fuses the 3D data provided by the sensor in order to obtain the layout of the room. We show how our algorithm can rely on the user interactions in order to correct the generated 3D model during the acquisition process

Abstracts on Radio Direction Finding (1899 - 1995)

The files on this record represent the various databases that originally composed the CD-ROM issue of "Abstracts on Radio Direction Finding" database, which is now part of the Dudley Knox Library's Abstracts and Selected Full Text Documents on Radio Direction Finding (1899 - 1995) Collection. (See Calhoun record https://calhoun.nps.edu/handle/10945/57364 for further information on this collection and the bibliography). Due to issues of technological obsolescence preventing current and future audiences from accessing the bibliography, DKL exported and converted into the three files on this record the various databases contained in the CD-ROM. The contents of these files are: 1) RDFA_CompleteBibliography_xls.zip [RDFA_CompleteBibliography.xls: Metadata for the complete bibliography, in Excel 97-2003 Workbook format; RDFA_Glossary.xls: Glossary of terms, in Excel 97-2003 Workbookformat; RDFA_Biographies.xls: Biographies of leading figures, in Excel 97-2003 Workbook format]; 2) RDFA_CompleteBibliography_csv.zip [RDFA_CompleteBibliography.TXT: Metadata for the complete bibliography, in CSV format; RDFA_Glossary.TXT: Glossary of terms, in CSV format; RDFA_Biographies.TXT: Biographies of leading figures, in CSV format]; 3) RDFA_CompleteBibliography.pdf: A human readable display of the bibliographic data, as a means of double-checking any possible deviations due to conversion

Engineering data compendium. Human perception and performance, volume 3

The concept underlying the Engineering Data Compendium was the product of a research and development program (Integrated Perceptual Information for Designers project) aimed at facilitating the application of basic research findings in human performance to the design of military crew systems. The principal objective was to develop a workable strategy for: (1) identifying and distilling information of potential value to system design from existing research literature, and (2) presenting this technical information in a way that would aid its accessibility, interpretability, and applicability by system designers. The present four volumes of the Engineering Data Compendium represent the first implementation of this strategy. This is Volume 3, containing sections on Human Language Processing, Operator Motion Control, Effects of Environmental Stressors, Display Interfaces, and Control Interfaces (Real/Virtual)