Phase Drift in Networks of Coupled Colpitts Oscillators

Coupled oscillator networks (CONs) occur in various domains such as biology and electronics. The nonlinear phenomenon of “injection locking” leads to frequency synchronization among the oscillators in a CON. In this chapter, we first obtain simplified tools to analyze the phenomenon of injection locking and present the Generalized Adler’s equation. The Generalized Adler’s equation is applicable for analyzing the injection locking in any type of oscillator. We demonstrate the use of Generalized Adler’s equation for injection locking analysis on a ring oscillator and a Hodgkin-Huxley neuronal oscillator. Next, we describe efficient system-level simulation techniques (synchronized steady state and transient) for coupled oscillator networks using the Perturbation Projection Vector (PPV) phase macromodel of the oscillator. The coupled oscillator simulation techniques are generic in nature and can be applied for the simulation of any coupled oscillator network, like the ones found in biological or electronic systems. We apply our system-level techniques on a −G m LC (harmonic) and a Hodgkin-Huxley (non-harmonic) CON.

Effect of bioactive protein supplementation on equine gait kinematics

A high performance Space-Time Reference in orbit could be realized using a\nstable atomic clock in a precisely defined orbit and linking that to high\naccuracy atomic clocks on the ground using a laser based time-transfer link.\nThis would enhance performance of existing systems and provide unique\ncapabilities in navigation, precise timing, earth sciences, geodesy and the\nsame approach could provide a platform for testing fundamental physics in\nspace. Precise laser time- and frequency-transfer from the ground to an\norbiting satellite would make it possible to improve upon the current state of\nthe art in timing (about 1 to 30 ns achieved with GPS) by roughly a factor of\n1000 to the 1 ps level.\n

A critical component of many Inertial Navigation Systems (INS) is the gyroscope — a device used for detecting rotation rates and orientation. Gyroscopes are subject to material imperfections and manufacturing defects, potentially reducing their ability to detect signals and to minimize phase drift. This manuscript explores the idea of performance enhancement by combining the input–output response of an ensemble of gyroscopes coupled bidirectionally, through the driving- and sensing-modes, in a ring fashion. Numerical simulations and bifurcation analysis via perturbation methods show the existence of a stable synchronization state in which all gyroscopes vibrate with the same phase and amplitude. In this state, the sum output of the sensing modes is significantly larger than any individual sensing mode, which leads to significant reductions in phase drift caused by material imperfections, fabrication variations, and system noise. Similarities and differences with respect to a system of gyroscopes coupled only along the driving-axis are discussed in great detail as well.

Coupled oscillator networks occur in various domains such as biology, astrophysics and electronics. In this paper, we present a comprehensive procedure for rapid and accurate simulation of large coupled oscillator networks using widely accepted, fully-nonlinear perturbation projection vector (PPV) phase macromodels. We validate our method against full simulation of 20times20 coupled network of Brusselator biochemical oscillator and obtain computational speedups of 170x over full simulation. Furthermore, we apply the method to study self-organization phenomenon of Brusselator under asymmetric coupling and time period variations.

The dynamical tolerance of coupled oscillator networks against local failures is studied. As the fraction of failed oscillator nodes gradually increases, the mean oscillation amplitude in the entire network decreases and then suddenly vanishes at a critical fraction as a phase transition. This critical fraction, widely used as a measure of the network robustness, was analytically derived for random failures but not for targeted attacks so far. Here we derive the general formula for the critical fraction, which can be applied to both random failures and targeted attacks. We consider the effects of targeting oscillator nodes based on their degrees. First we deal with coupled identical oscillators with homogeneous edge weights. Then our theory is applied to networks with heterogeneous edge weights and to those with nonidentical oscillators. The analytical results are validated by numerical experiments. Our results reveal the key factors governing the robustness and fragility of oscillator networks.

A set of N identical oscillators (clocks) are connected to form a mutually synchronous network. General matrix expressions are derived for the phase error spectrum at each oscillator when the network operates in the linear regime. For certain topologies of practical interest the phase error variance and the correlation between the errors at different oscillators is computed and compared in their behavior for large N . The results presented assume no thermal noise and no delay between the oscillators. However, the theory can be easily extended to include these two effects.

In this chapter we describe phenomenon of ragged synchronizability. First we give a theoretical introduction about detecting of the complete synchronization in the network of coupled systems. As an example we choose a network of non-diagonally coupled forced van der Pol oscillators.We present construction of the single electrical van der Pol oscillator and array of such systems. We show numerical and experimental results of the existence of the complete synchronization in the array and we discuss it in the very detailed way. We also consider the synchronization in clusters, experimentally in the small array and numerically in the larger network. Our numerical simulations show good agreement with the experimental observations.

Recent evidence suggests that electrotonic coupling is an important mechanism for neuronal synchronisation in the mammalian cortex and hippocampus. Various types of network oscillations have been shown to depend on, or be sharpened by, gap junctions between inhibitory interneurones or excitatory projection cells. Here we made use of a targeted disruption of the gene coding for Cx36, a recently discovered neuronal gap junction subunit, to analyse its role in hippocampal network behaviour. Mice lacking Cx36 are viable and lack obvious morphological or behavioural abnormalities. Stimulation of afferent and efferent fibre pathways in hippocampal slices revealed a largely normal function of the synaptic circuitry, including tetanically evoked network oscillations. Spontaneous sharp waves and ripple (approximately 200 Hz) oscillations, however, occurred less frequently in slices from Cx36 -/- mice, and ripples were slightly slower than in littermate controls. Moreover, epileptiform discharges elicited by 4-aminopyridine were attenuated in slices from Cx36 -/- mice. Our findings indicate that Cx36 plays a role in the generation of certain forms of network synchronisation in the hippocampus, namely sharp wave-ripple complexes and hypersynchronous epileptiform discharges.

The “youngest” of the major satellite applications is the field of satellite navigation. This field of measurement and ranging through the use of satellite positioning systems first started in the context of scientific research. This initial use of satellites was simply for positioning and location. These activities, that were first based on using Doppler frequency shifts as a satellite orbited above, were largely scientific and not strategic. These types of activities included geodetics (i.e., such as the measurement of continental drift over time) or the collection of scientific information and data from atmospheric landor ocean-based sensors where the specific locations of the sensors were important. The real strides in the development of satellite navigation, however, came when space systems were developed for the specific purpose of precise targeting of missiles and various other types of weapon systems. The Soviet Union/Russian J.N. Pelton (*) International Space University, Arlington, VA, USA e-mail: joepelton@verizon.net S. Camacho-Lara Centro Regional de Ensenanza de Ciencia y Tecnologia del Espacio para America Latina y el Caribe (CRECTEALC), Santa Maria Tonantzintla, Puebla, Mexico e-mail: sergio.camacho@inaoep.mx # Springer Science+Business Media New York 2016 J.N. Pelton et al. (eds.), Handbook of Satellite Applications, DOI 10.1007/978-1-4614-6423-5_10-3 1 Federation satellite navigation system, known as GLONASS, and the US-based Navstar system –with its Global Positioning Satellite (GPS) network – developed space-based systems that provided unprecedented capabilities to determine locations on the ground or oceans with great accuracy. Early systems such as Argos that relied on Doppler shift technology were only accurate within a precision of hundreds of meters. Today’s advanced satellite navigation systems, however, are accurate for measurements that can be indicated with a precision only a few meters and within centimeters if utilizing reference stations. The fact that GPS and GLONASS satellite signals are freely available in space for all to use has spurred the development of low-cost receivers for much more than strategic or military usage. Today, various types of civilian use of navigational and positioning satellites have become popular on a global basis. The latest development in application-specific integrated circuit (ASIC) chip technology has fueled the growth of applications based on the use of these precise spacebased navigational and positioning systems. The availability of these highly capable but increasingly low-cost chips – small enough that they could be included in handsets such as cell phones or included with the electronics available on various types of vehicles (i.e., cars, trucks, buses, trains, airplanes, ships, etc.) – represented a real breakthrough. The development of specialized computer chips to perform satellite navigation calculations has increased the number of users of this technology from a relatively small population of several thousands to tens of millions. In the coming decade, the proliferation of international satellite navigation systems in orbit (i.e., USA, Russian Federation, China, India, Europe, and Japan) plus the continued development of ever lower cost of satellite navigation chips for receivers will continue the popular expansion of these increasingly “easy to use” and versatile systems. Wide and easy access to low-cost consumer receivers for satellite navigation services strongly suggests that this trend of expanded use will continue to surge. Thus, within a decade, use of these devices will increase to the hundreds of millions if not billions of people. Atomic clocks with incredible temporal precision today allow satellite navigation systems to be used by military organizations, governmental geospatial scientists, and scientists to determine locations with great accuracy. But these applications are now just a small fraction of total usage. The deployment of the new international satellite navigational systems and smaller and lower cost ground receivers will support an ever-increasing civilian consumer market for an ever-expanding range of everyday applications. These consumer applications include driving a vehicle to a desired location, safely sailing a boat, going mountain climbing or hiking without getting lost, or simply finding out where you are within a city. The precise time keeping ability of today’s satellite navigation systems also means that these spacecrafts can also serve as a global timekeeper for computers and scientific experiments. The time stamp from a satellite navigation satellite can also be used not only for scientific or regulatory purposes but for other applications such as security and banking systems as well. These satellites are also used to support the synchronization needs of communications satellites. In short, 2 J.N. Pelton and S. Camacho-Lara

The Global Positioning System (GPS), in addition to providing precise navigation and positioning information, produces precise time and frequency measurements. These measurements result from the atomic clocks in the GPS satellites, which are closely coupled to Universal Coordinated Time as maintained by the U. S. Naval Observatory [UTC(USNO)]. The application of these measurements to timekeeping and other systems requiring precise time and frequency is distinct from the navigation/positioning mission. This article will describe the differences in application of GPS to time and frequency uses. These uses will be described in the major areas of timekeeping; stationary uses, such as communications networking; and mobile use for aircraft and shipboard applications. The major considerations in application and operation with precise and less-capable oscilators will be described. Examples and data will be presented to illustrate the applications. © 1999 John Wiley & Sons, Inc.

Heretofore, all precision clocks and oscillators have required one or more ovens to control the temperature of frequency dependent components. Ovens always add significantly to the power budget and limit the upper operating temperature. In the 1970s, Jack Kusters, et al. developed the dual-mode oscillator, which allows one mode to be used to measure temperature [1]. Then the other mode is made as frequency independent with temperature as can reasonably be done. The modes used are the B- and C-modes. With knowledge of the temperature behavior for these two modes, the B-mode measurements can be used to compensate the much smaller frequency dependent changes in the C-mode so that the output is temperature independent [2]. In 1968, David W. Allan developed the time-scale algorithm for the atomic-clock ensemble providing time for the USA at the National Bureau of Standards in Boulder, Colorado. German and Allan have modified that algorithm to operate with a quartz-crystal oscillator ensemble. In addition, they have developed an algorithm that uses the ensemble to determine the temperature coefficients for each of the oscillators (clocks) in the ensemble — called automatic temperature compensation (ATC). The output of the ATC feeds the time-scale ensemble algorithm, which has the property of providing an output better than the best clock, and even the worst clock enhances the performance. With this configuration, flicker-floors of parts in 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sup> have been obtained with only a gradual deterioration in stability in the long-term. The ensemble output also compensates for long-term frequency drift. To date, the EQUATE timing stability has demonstrated timing errors of less than 100 microseconds for prediction intervals out to 10 days. The ensemble operates without an oven, with micro-degree temperature measurement precision, has been tested over a temperature range of −40 to +85°C and can reasonably operate at even higher temperatures. This R&D system is now in its bread-board phase, and efforts are in progress to turn it into a small, low-powered, timing product [3]. This quartz-crystal oscillator timing ensemble is an essential part of the EQUATE technology package, which provides both precise timing as well as six-axis inertial sensing. We have demonstrated excellent frequency stability during large dynamic motion and during large temperature excursions. The whole timing and navigation package should be small and require very little power — well suited for a GPS denied environment.

We investigate a minimal network model consisting of a 2D linear (non-oscillatory) resonator and a 1D linear cell, mutually inhibited with piecewise-linear graded synapses. We demonstrate that this network can produce oscillations in certain parameter regimes and the corresponding limit gradually transition from regular oscillations (of non-relaxation type) to relaxation oscillations as the levels of mutual inhibition increase.

Based on analogy with the oscillator networks, the auto-associative Hopfield network's behavior is investigated for the effect of reducing the connections number. It is shown that the exclusion of connections, with weights modules strictly less than the maximum for a given neuron, significantly improves the network performance. In this case, the allowed share of the distorted input vector elements increases with the network dimension.

We present here a dynamic formalism which allow to compute analitically the stationary properties of networks of neural oscillators. This technique, derived originally to study situations away from equilibrium, is an alternative to standard methods developed to analyze the behaviour of attractor neural networks.