Applications of nonlinear dynamics

Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures Models of such systems can be successfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence The two major book publication platforms of the Springer Complexity program are the monograph series “Understanding Complex Systems” focusing on the various applications of complexity, and the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foundations In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works Editorial and Programme Advisory Board ´ P´eter Erdi Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of Sciences, Budapest, Hungary Karl Friston Institute of Cognitive Neuroscience, University College London, London, UK Hermann Haken Center of Synergetics, University of Stuttgart, Stuttgart, Germany Janusz Kacprzyk System Research, Polish Academy of Sciences, Warsaw, Poland Scott Kelso Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA J¨urgen Kurths Potsdam Institute for Climate Impact Research (PIK), Potsdam, Germany Linda Reichl Center for Complex Quantum Systems, University of Texas, Austin, USA Peter Schuster Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria Frank Schweitzer System Design, ETH Z¨urich, Z¨urich, Switzerland Didier Sornette Entrepreneurial Risk, ETH Z¨urich, Z¨urich, Switzerland Understanding Complex Systems Founding Editor: J.A Scott Kelso Future scientific and technological developments in many fields will necessarily depend upon coming to grips with complex systems Such systems are complex in both their composition – typically many different kinds of components interacting simultaneously and nonlinearly with each other and their environments on multiple levels – and in the rich diversity of behavior of which they are capable The Springer Series in Understanding Complex Systems series (UCS) promotes new strategies and paradigms for understanding and realizing applications of complex systems research in a wide variety of fields and endeavors UCS is explicitly transdisciplinary It has three main goals: First, to elaborate the concepts, methods and tools of complex systems at all levels of description and in all scientific fields, especially newly emerging areas within the life, social, behavioral, economic, neuroand cognitive sciences (and derivatives thereof); second, to encourage novel applications of these ideas in various fields of engineering and computation such as robotics, nano-technology and informatics; third, to provide a single forum within which commonalities and differences in the workings of complex systems may be discerned, hence leading to deeper insight and understanding UCS will publish monographs, lecture notes and selected edited contributions aimed at communicating new findings to a large multidisciplinary audience Applications of Nonlinear Dynamics Model and Design of Complex Systems Edited by Visarath In SPAWAR, San Diego Patrick Longhini SPAWAR, San Diego and Antonio Palacios SDSU 123 Visarath In Space and Naval Warfare Systems Center Code 2373 53560 Hull Street San Diego, CA 92152-5001 USA Visarath@spawar.navy.mil Visarath@euler.sdsu.edu Patrick Longhini Space and Naval Warfare Systems Center Code 2373 53560 Hull Street San Diego, CA 92152-5001 USA Antonio Palacios Department of Mathematics & Statistics San Diego State University San Diego, CA 92182-7720 USA palacios@euler.sdsu.edu ISBN: 978-3-540-85631-3 e-ISBN: 978-3-540-85632-0 DOI 10.1007/978-3-540-85632-0 Understanding Complex Systems ISSN: 1860-0832 Library of Congress Control Number: 2008936465 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: WMXDesign GmbH Printed on acid-free paper springer.com Organizers Bruno Ando, University of Catania Adi Bulsara, SPAWAR, San Diego Salvatore Baglio, University of Catania Visarath In, SPAWAR, San Diego Ljupco Kocarev, University of California, San Diego Patrick Longhini, SPAWAR, San Diego Joseph Neff, SPAWAR, San Diego Antonio Palacios, San Diego State University Toshimichi Saito, Hosei University Michael F Shlesinger, Office of Naval Research Hiroyuki Torikai, Hosei University SPONSOR: Office of Naval Research (ONR) 875 N Randolph Street, Suite 1475 Arlington, VA 22217 v Preface The field of applied nonlinear dynamics has attracted scientists and engineers across many different disciplines to develop innovative ideas and methods to study complex behavior exhibited by relatively simple systems Examples include: population dynamics, fluidization processes, applied optics, stochastic resonance, flocking and flight formations, lasers, and mechanical and electrical oscillators A common theme among these and many other examples is the underlying universal laws of nonlinear science that govern the behavior, in space and time, of a given system These laws are universal in the sense that they transcend the model-specific features of a system and so they can be readily applied to explain and predict the behavior of a wide ranging phenomena, natural and artificial ones Thus the emphasis in the past decades has been in explaining nonlinear phenomena with significantly less attention paid to exploiting the rich behavior of nonlinear systems to design and fabricate new devices that can operate more efficiently Recently, there has been a series of meetings on topics such as Experimental Chaos, Neural Coding, and Stochastic Resonance, which have brought together many researchers in the field of nonlinear dynamics to discuss, mainly, theoretical ideas that may have the potential for further implementation In contrast, the goal of the 2007 ICAND (International Conference on Applied Nonlinear Dynamics) was focused more sharply on the implementation of theoretical ideas into actual devices and systems Thus the meeting brought together scientists and engineers from all over the globe to exchange research ideas and methods that can bridge the gap between the fundamental principles of nonlinear science and the actual development of new technologies Examples of some of these new and emerging technologies include: (magnetic and electric field) sensors, reconfigurable electronic circuits, nanomechanical oscillators, chaos-based computer chips, nonlinear nano-detectors, nonlinear signal processing and filters, and signal coding The 2007 ICAND meeting was held in Hawaii, at Poipu Beach, Kauai on September 24–27, 2007 The waters off Poipu Beach are crystal clear and provided a truly beautiful atmosphere to hold a meeting of this kind The invited speakers at this seminal meeting on applied nonlinear dynamics were drawn from a rarefied mix They included a few well-established researchers in the field of nonlinear dynamics vii viii Preface as well as a “new breed” of pioneers (applied physicists, applied mathematicians, engineers, and biologists) who are attempting to apply these ideas in laboratory and, in some cases, industrial applications The discussions in the meeting cover broad topics ranging from the effects of noise on dynamical systems to symmetry mathematics in the analyses of coupled nonlinear systems to microcircuit designs in implementation of these nonlinear systems The meeting also featured, as already stated, some novel theoretical ideas that have not yet made it to the drawing board, but show great promise for the future The organizers also attempted to give some exposure to much younger researchers, such as advanced graduate students and postdocs, in the form of posters The meeting set aside singificant amount of time and provided many opportunities outside of presentation setting to promote the discussions and foster collaborations amongs the participants The organizers extend their sincerest thanks to the principal sponsors of the meeting: Office of Naval Research (Washington, DC), Office of Naval Research-Global (London), San Diego State University (College of Sciences), and SPAWAR Systems Center San Diego In particular, we wish to acknowledge Dr Michael Shlesinger from the Office of Naval Research (Washington DC) for his support and encouragement In addition, we extend our grateful thanks, in specific, to Professor Antonio Palacios and Dan Reifer at SDSU for their hardwork in making the financial transactions as smoothly as possible despite many obstacles thrown in their way We also want to thank our colleagues who chaired the sessions and to the numerous individuals who donated long hours of labor to the success of this meeting Finally, we thank Spinger-Verlag for their production of an elegant proceedings San Diego, USA May 2008 V In P Longhini A Palacios Contents Invited Speakers Construction of a Chaotic Computer Chip William L Ditto, K Murali and Sudeshna Sinha Activated Switching in a Parametrically Driven Micromechanical Torsional Oscillator 15 H.B Chan and C Stambaugh Quantum Nanomechanics 25 Pritiraj Mohanty Coupled-Core Fluxgate Magnetometer 37 Andy Kho, Visarath In, Adi Bulsara, Patrick Longhini, Antonio Palacios, Salvatore Baglio and Bruno Ando Data Assimilation in the Detection of Vortices 47 Andrea Barreiro, Shanshan Liu, N Sri Namachchivaya, Peter W Sauer and Richard B Sowers The Role of Receptor Occupancy Noise in Eukaryotic Chemotaxis 61 Wouter-Jan Rappel and Herbert Levine Applications of Forbidden Interval Theorems in Stochastic Resonance 71 Bart Kosko, Ian Lee, Sanya Mitaim, Ashok Patel and Mark M Wilde Smart Materials and Nonlinear Dynamics for Innovative Transducers 91 B And`o, A Ascia, S Baglio, N Pitrone, N Savalli, C Trigona, A.R Bulsara and V In Dynamics in Non-Uniform Coupled SQUIDs 111 Patrick Longhini, Anna Leese de Escobar, Fernando Escobar, Visarath In, Adi Bulsara and Joseph Neff ix x Contents Applications of Nonlinear and Reconfigurable Electronic Circuits 119 Joseph Neff, Visarath In, Christopher Obra and Antonio Palacios Multi-Phase Synchronization and Parallel Power Converters 133 Toshimichi Saito, Yuki Ishikawa and Yasuhide Ishige Coupled Nonlinear Oscillator Array (CNOA) Technology – Theory and Design 145 Ted Heath, Robert R Kerr and Glenn D Hopkins Nonlinear Dynamic Effects of Adaptive Filters in Narrowband Interference-Dominated Environments 163 A.A (Louis) Beex and Takeshi Ikuma Design-Oriented Bifurcation Analysis of Power Electronics Systems 175 Chi K Tse Collective Phenomena in Complex Social Networks 189 Federico Vazquez, Juan Carlos Gonz´alez-Avella, V´ıctor M Egu´ıluz and Maxi San Miguel Enhancement of Signal Response in Complex Networks Induced by Topology and Noise 201 Juan A Acebr´on, Sergi Lozano and Alex Arenas Critical Infrastructures, Scale-Free Networks, and the Hierarchical Cascade of Generalized Epidemics 211 Markus Loecher and Jim Kadtke Noisy Nonlinear Detectors 225 A Dari and L Gammaitoni Cochlear Implant Coding with Stochastic Beamforming and Suprathreshold Stochastic Resonance 237 Nigel G Stocks, Boris Shulgin, Stephen D Holmes, Alexander Nikitin and Robert P Morse Applying Stochastic Signal Quantization Theory to the Robust Digitization of Noisy Analog Signals 249 Mark D McDonnell Resonance Curves of Multidimensional Chaotic Systems 263 Glenn Foster, Alfred W H¨ubler and Karin Dahmen Learning of Digital Spiking Neuron and its Application Potentials 273 Hiroyuki Torikai Dynamics in Manipulation and Actuation of Nano-Particles 287 Takashi Hikihara Contents xi Nonlinear Buckling Instabilities of Free-Standing Mesoscopic Beams 297 S.M Carr, W.E Lawrence and M.N Wybourne Developments in Parrondo’s Paradox 307 Derek Abbott Magnetophysiology of Brain Slices Using an HTS SQUID Magnetometer System 323 Per Magnelind, Dag Winkler, Eric Hanse and Edward Tarte Dynamical Hysteresis Neural Networks for Graph Coloring Problem 331 Kenya Jin’no Semiconductor Laser Dynamics for Novel Applications 341 Jia-Ming Liu Nonlinear Prediction Intervals by the Bootstrap Resampling 355 Tohru Ikeguchi Quantum Measurements with Dynamically Bistable Systems 367 M.I Dykman Poster Session Dynamics and Noise in dc-SQUID Magnetometer Arrays 381 John L Aven, Antonio Palacios, Patrick Longhini, Visarath In and Adi Bulsara Stochastically Forced Nonlinear Oscillations: Sensitivity, Bifurcations and Control 387 Irina Bashkirtseva Simultaneous, Multi-Frequency, Multi-Beam Antennas Employing Synchronous Oscillator Arrays 395 J Cothern, T Heath, G Hopkins, R Kerr, D Lie, J Lopez and B Meadows Effects of Nonhomogeneities in Coupled, Overdamped, Bistable Systems 403 M Hernandez, V In, P Longhini, A Palacios, A Bulsara and A Kho A New Diversification Method to Solve Vehicle Routing Problems Using Chaotic Dynamics 409 Takashi Hoshino, Takayuki Kimura and Tohru Ikeguchi Self-Organized Neural Network Structure Depending on the STDP Learning Rules 413 Hideyuki Kato, Takayuki Kimura and Tohru Ikeguchi Torus Bifurcation in Uni-Directional Coupled Gyroscopes Huy Vu, Antonio Palacios, Visarath In, Adi Bulsara, Joseph Neff and Andy Kho Abstract Nowadays, the global positioning system (GPS) is popularly used by the U.S Navy in navigation systems to gain precise position, velocity, and time information One of the biggest issues for using GPS is its susceptibility to jamming and other inferences The received GPS signal is approximately 20 dB below the thermal noise level from a distance 11,000 miles away Because of these weakness and vulnerability, many other alternative navigation methods are needed to improve performance and reduce the dependency on GPS One of the main alternative methods is the Inertial Guidance System (IGS) that can operate wherever GPS signals are jammed or denied A prototypical IGS is composed of three accelerometers to measure linear movement and three angular rate sensors (gyroscopes) to gauge the rotational movement The main benefit of IGS is its low cost relatively to other methods Current MEMS (Micro-Electro-Mechanical Systems) gyroscopes are compact and inexpensive, but their performance does not meet the requirements for an inertial grade guidance system In this work, a difference approach was examined on the dynamics of coupled gyroscopes to improve performance through synchronization referred as vibratory coupled gyroscopes with drive amplitudes’ coupling One of the main discoveries from the coupled gyroscopes’ mathematical model is a Torus bifurcation, which leads to synchronized behavior in and array of three gyroscopes uni-directionally coupled H Vu (B) Nonlinear Dynamical System Group, Department of Computational Sciences, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA, e-mail: hvu@sciences.sdsu.edu V In et al (eds.), Applications of Nonlinear Dynamics, Understanding Complex Systems, DOI 10.1007/978-3-540-85632-0 45, c Springer-Verlag Berlin Heidelberg 2009 463 464 H Vu Introduction Although circuit implements have experimentally demonstrated complex behavior in arrarys of coupled simple ring uni-directionally as well as bi-directionally gyroscopes, this research concentrated on theoretical and computational analysis of forward uni-directional coupled gyroscopes with the simplest case of a ring of three gyroscopes Dynamical behaviors of these three simple uni-directional gyroscopes became plentifully complicated Surprisingly, the results obtained were different from the initial expectation Even though it was called “simplest case”, each gyroscope constituted a system of four ordinary differential equations and a total of twelve ODE’s for three coupled array gyroscopes This high dimensional system required a substantial time and effort to investigate, especially to obtain high-order numerical accuracy, even though it was assumed there was no stochastic additive noise to the system in this work The Device Vibratory gyroscopes are designed to determine the angular rate of a rotating object As is shown in Fig (1a), there are a couple of springs connected with a gyroscope One spring is the drive axis (X-axis) and the other is the sense axis (Y-axis) This model creates two orthogonal vibration modes with restoring coefficients, kx and ky The harmonic motion in the drive axis is caused by an external signal or a forcing oscillator, while the movement in the sense axis depends upon Coriolis force, FCY , through the angular velocity within the Z-axis Importantly, there is an amplitude coupling within the drive axis to induce phase synchronization [2] among gyroscopes, Fig (1b), which is a structure of uni-directional forward coupling of three gyroscopes The operating principle of these vibrating gyroscopes is based upon the transfer of energy from one vibrating mode to another by the Coriolis force This research focuses upon the modified MEMS gyroscopes with coupling drive displacements in order to achieve the synchronization between the gyroscopes and to maximize the sum of all gyroscopes’ sense displacements (a) Gyroscope Mass-Spring System Fig A gyroscopes’ system (b) Uni-directionally Coupled Gyroscopes Torus Bifurcation in Uni-Directional Coupled Gyroscopes 465 Mathematical Model and Background For a uni-directionally coupled n-gyroscopes system [1], the mathematical model that governs each gyroscope in drive- and sense-coordinate components is given as mi x¨i +Ci x˙i + ki xi + μi xi3 − 2mi Ω y˙i + λ (xi − xi+1 ) = Fdrive (t) (1) mi y¨i +Ci y˙i + ki yi + μi y3i + 2mi Ω x˙i = Ni (t) (2) where xi is the displacement of the drive component, yi is the displacement of the sense component, mi is the mass of the gyroscope, Ci is the damping coefficient of the gyroscope, ki is the restoring coefficient of the gyroscope, μi is the restoring coefficient with cubic term of the gyroscope, Ω is the angular velocity of the gyroscope, λ is the coupling constant for driving displacement (X-axis) – uni-directional between xi and xi+1 , Fdrive is the driving force function – assumed to be of the form [Ad sin( ωd t)], Ad is the maximum amplitude of the driving force, ωd is the frequency of the driving force, t is time, and Ni is the stochastic noise function, assuming to be zero In this particular research, mass-spring-dampers of the system are identical The numerical parameters investigated for the coupled gyroscopes’ system are specified as λ is varied from −12 to 12, Ω is ranged from −104 to 104 , Ad = 0.001, ωd = 51650, mi = 10−9 , ki = 5.1472×10−7 , Ci = 2.6494, and μi = 2.933 Torus Bifurcation and Analysis of Frequency Response Torus bifurcation in Fig is the only type of bifurcation found in the system These torus bifurcations are symmetric in Ω − between Ω > and Ω < - while it is not necessarily the case for λ − between λ > and λ < Generally, the examined 4 x 10 0.5 Ω Ω 0.5 −0.5 x 10 D A B −0.5 F C E A F D −1 −10 −5 10 −1 −0.1 −0.05 λ (a) λ ∈ (–12, 12) 0.05 λ (b) λ ∈ (–0.15, 0.15) Fig Two-parameter bifurcation diagram – Ω vs λ – as ωd = 51650 0.1 0.15 466 H Vu 10 10 10 10 10 10 ( FFT of X )2 10 10 1 ( FFT of X )2 10 −1 10 −2 10 −1 10 −2 10 −3 10 −3 10 −4 10 −4 −5 10 10 −6 10 −5 10 0.5 1.5 2.5 3.5 0.5 1.5 r 2.5 3.5 3.5 Frequency Ratio (Wr / Wd) Frequency Ratio (W / W ) d (b) Zone B (a) Zone A 10 10 10 10 10 −1 10 10 ( FFT of X )2 ( FFT of X )2 −2 10 10 −3 10 −4 10 −1 10 −2 10 −3 10 −4 10 −5 10 −5 10 −6 10 −6 10 −7 10 −8 10 −7 10 0.5 1.5 2.5 3.5 0.5 1.5 2.5 Frequency Ratio (Wr / Wd) Frequency Ratio (Wr / Wd) (c) Zone C (d) Zone D 4 10 10 10 10 10 10 10 0 ( FFT of X ) ( FFT of X )2 10 10 −1 10 −2 10 −1 10 −2 10 −3 10 −4 10 −3 10 −5 10 −4 10 −6 10 −5 10 −7 10 −6 10 10 −8 Frequency Ratio (Wr / Wd) (e) Zone E Fig Plots of (FFT (X1 ))2 10 0.5 1.5 2.5 3.5 Frequency Ratio (Wr / Wd) (f) Zone F vs (ωr /ωd ) region was divided into different zones – A, B, C, D, E, and F Let Rω = [ωr /ωd ] be the frequency ratio, the FFT frequency plots and torus-shape plots for the drivecomponent in Figs and respectively indicated that: (a) for zone C, one single peak occurs in the FFT plot at Rω = Thus, the frequency response for gyroscopes in the synchronized region is equal to the driving frequency – (b) for zone B, prominent peaks spread in a small range at (Rω = 1) and (Rω = 3) and a ring torus shape is formed This region is rather quasi-periodic – (c) for zone E, peaks occur at (Rω = 1) Torus Bifurcation in Uni-Directional Coupled Gyroscopes 467 are more defined than those at (Rω = 3) and its torus shape has a “loose” ring structure It indicates this region is in between quasi-periodic and chaotic areas – (d) for zone A, a number of peaks occur in a broad range at (Rω = 1) and (Rω = 3) and its corresponding torus shape is well dispersed The system enters the chaotic region – (e) for zone F, multiple peaks occur in a small range at (Rω = 1) and (Rω = 3) and its torus has multiple shapes This implies the system enters the region between quasi-periodic and chaotic areas – (f) for zone D, a few peaks occur at (Rω = 1), which more defined than those at (Rω = 3) and its torus has a ring structure Thus the region is quasi-periodic – (g) zone D and zone E are different in frequency response and torus shapes This indicates the presence of the intersection between two branches of torus bifurcations distinguishing those two zones 0.5 X3 X 0 −1 −2 2 −0.5 0.5 −2 −2 X2 X1 X2 −0.5 0.5 X (b) Zone B (a) Zone A 0.15 0.02 0.1 0.01 0.05 X3 X3 −0.01 −0.05 −0.02 0.02 −0.1 0.02 X2 0.1 −0.02 −0.02 X1 (c) Zoone C −0.1 −0.0501 −0.1 X 0.05 0.1 X (d)Zone C 0.4 0.2 0.2 0.1 X X 3 −0.1 −0.2 −0.2 0.2 −0.2 X −0.2 −0.1 0.1 0.2 −0.4 0.5 −0.5 X2 −0.4 −0.2 X1 (e) Zone E Fig Phase space plots of [X1 vs X2 vs X3 ] (f) Zone F X1 0.2 0.4 468 H Vu Acknowledgments We gratefully acknowledge support from the Office of Naval Research (Code 331) and the SPAWAR internal research funding (S&T) program This work was supported in part by National Science Foundation grants CMS-0625427 and CMS-0638814 References Brian Meadows, Joseph Neff, Visarath In, and Adi Bulsara (2005), United States Patent No US 6,880,400 B1 Arkady Pikovsky, Michael Rosenblum, and Jurgen Kurths (2001), Synchronization – A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge Index A Acoustic transducer, 227 Activated decay, 377 Adaptive equalization application, 164 narrowband interference, 166 ADC, see Analog-to-Digital Converter Agilent advanced design system, 153, 396 Allison mixtures, 307–308, 314, 316–317 Aluminum nitride, 35 AM-to-FM conversion, 351–353 Amplitude of oscillation, 16, 440 Analog-to-digital converter, 226, 250, 252, 255, 257, 259–260 Array factor, 146, 149, 152 Artificial neural network application, 226 Artificial vision and olfaction, 86 AS topology, 214 ASIC design, 395–397, 401 Atomic force microscopy (AFM), 79, 287–288, 298 Attractor landscape, 445 Auditory coding, 238 Auto-correlation function, 171, 280 Autonomous system (AS) level, 211, 213–214 AWG magnet wire, 41 Axelrod model, 192–198 co-evolving network, 194 definition, 193 square lattice, 193 B Backscatter information, 351 BA model, 217–218, 418 Bandpass filters, 243 Bandwidth-limiting effect, 348 Barabasi network, 192 Bayesian filter, 56 Beamforming matrix, 245 Bernoulli map, 268–270 Bernoulli sequence, 75, 78–79, 82 Bernoulli signal, 72–73 Bessel function, 52–53 Bias-reversal signal, 326–327 Bifurcation amplifiers, sensitivity of, 377 Bifurcation analysis, 42, 176, 179, 383–384 Bifurcation and parameter space, 449–451 Bifurcation behavior, 176, 178, 180, 184 Bifurcation frequency, 16 Bifurcation in SPICE imulations, 180 Bifurcation point, 16, 18–20, 22, 256, 368–369, 371–376, 459 metastable decay, 373–375 vicinity of, 372 Bifurcation theory, 141, 177 Biot-Savart law, 48 Bistable system, 16, 130, 202, 227–228, 230, 367–377, 403–407, 449 Bit-error-rate sense, 166 Bootstrap prediction, 358–364 bootstrap interval estimation, 360–361 Jacobian-matrix estimation, 359–360 point prediction, 358–359 replication method, 358 Bootstrap resampling, 355–366 Border collision, 178, 180 Bose-Einstein Condensation (BEC) description, 30 Bosonic channel, 80 Boundary-value problem, 391 Br¨aysy construction heuristic method, 412 Brownian (Gaussian) component, 84 Brownian motions, 49, 65, 84 469 470 Brownian ratchets thermodynamics, 313 Brusselator, 389, 393–394 Buck converter, 135–138, 178 paralleled, 138–141 WTA-based paralleled buck converters, 142 Buckling modes, 298, 306 Butterworth filters, 243 C Cauchy noise, 72, 75–77, 81 Cauchy random variable, 74 ChaoGates, 11, 12 Chaos generator, 433–437 Chaotic computer chip, construction, 3–12 Chaotic computing architectures, VLSI Implementation, 11–12 Chaotic dynamics, applications of, 344–350 chaotic optical communications, 344–346 chaotic radar, 348–349 Chaotic lidar, 341, 344, 346–348, 351 Chaotic neurodynamcis, 417–420 routing method, 418–419 Chaotic neurons, 409–411 Chaotic oscillation (CO), 342, 344, 435 Chaotic pulse, 342, 344–345 Chaotic radar, 341, 344, 348–349 Chaotic signals, 433–436 Chaotization of oscillation, 433 Chemical-vapor-deposition (CVD), 78–79 Chemotactic efficiency, 67 Clamped-clamped boundary conditions, 299 Clarion cochlear implant, 242–243 Clay-tree network, 417 CLIDAR, see Chaotic lidar CMOS image sensor, 225 CNOA, see Coupled nonlinear oscillator array Cochlear implant coding, 237–247 simulation, 243–245 stochastic beamforming, 239–242 use of noise, 238–239 Cochlear nerve fibres, 243–245 Coding DNA, 317 Coherence and spin, 29 Coherent or pre-detection integration, 250 Collective phenomena, 189–199 Combinatorial optimization problem, 409 Commercially available off-the-shelf (COTS), 153–155, 159 Communications protocol (CP), 11 Computer-aided analysis, 177 Computer simulation, 181, 243, 407, 419–420 Computer virus diffusion, 212 Index Continuous signal and discrete observations, 50–56 nonlinear filters, 55 Continuous-time nonlinear system, 6–8 Coriolis force, 100, 464 Correction converter system, 180 Coulomb functions, 53 Coupled cantilever array, 292–293 Coupled-core fluxgate magnetometer, 37–45 Coupled nano-electro-mechanical systems, 34 Coupled nonlinear oscillator array (CNOA), 145–160, 395 Coupled oscillator array beam steering, 147–148 Coupled single-electron transistor technique, 31 Coupling effect, 400 Coupling parameter, 38–39, 41, 114, 116–117, 293, 335, 383, 449–450, 452 Coupling scheme, 38, 403 Coupling topology, 40, 126 Covariance matrix, 241, 357, 388 CRADAR, see Chaotic radar Critical coupling, 38, 107, 206, 428, 430, 449–450 Cross-connection coefficient, 334–335, 339 Cross-sectional schematic of the oscillator, 17 Cross-spectral density, 244 Cryptography, 341 Cumulative distribution function, 254 Current mode control (CMC), 135, 177–180 Current spread matrix, 241–242, 245–246 CVD, see Chemical-vapor-deposition D Damping constant, 19 Data acquisition (DAQ), 79, 250, 326 Decay probability, 368, 375 Decision-directed mode, 167 Deconvolution method, 239 Delayed feedback, 436 Design-oriented bifurcation, 175–187 Deterministic time scale, 201, 227 Dictyostelium discoideum cells, 61 Differential-difference equations (DDE), 434 Digital acquisition (DAQ) card, 156 See also Data acquisition Digital averaging, 250–251, 260 Digital logic control, 134 Digital spiking neuron, 273–285 Dijkstra algorithm, 417, 419 DIMUS sonar hydrophone array, 252 Direct modulation, 350 Discrete oscillator design, 396 Index Discrete Time Markov Chain (DMTC) analysis, 309 Displacement current generator, 104 Distance constraint, 214 Dither signal, 226–227 Dithering effect, 226, 234 Dominant frequencies, 423 Drift component, 84 DSN, see Digital spiking neuron Duffing element, 38, 427 equation, 292 oscillator, 19, 34, 121 Dynamical chaotization principle, 433, 435 E E-beam lithography, 279, 298, 304, 325 E-Field sensor, 102, 105, 427–431 E-sensor, 431 Early voltage, 121 Eigenfrequency, 368 Eigenvector, 216, 266–269 Elastic theory of continuous media, 28 Electromechanical transition, 298, 304–306 Electron tunneling, 128 Electronic beam steering, 146 Electrosensory lobe (ELL), 414 Electrostatic technique, 31 Elegans, 413 Engine efficiency thermodynamics, 313 Equilibrium point, 204–206, 331–334, 336, 459 Equilibrium profile and mechanical bistability, 302 Erd´os-R´enyi network, 192 Error amplifier (EA), 181 Error-free low noise environment, 167 Eukaryotic chemotaxis, 61–69 Euler buckling theory, 297–298, 302–304 Euler equation, 48–49, 299 Evoked magnetic fields (EMFs), 323, 329 Excess growth model, 312 Excitatory post-synaptic potential (EPSP), 326–327 External modulation, 350 F Feedback coefficient matrix, 392 Ferroelectric capacitor, 102–106, 427–429, 431 Ferroelectric Materials and E-Field Sensors, 102–108 dynamic behavior, 105 elementary cell – circuital realization, 103 system description, 102 471 Ferroelectric sensor, 227 Ferrofluidic gyroscopes, 100–101 Ferrofluidic masses, 101 Ferrofluids, 92–93, 99 Ferromagnetic core, 92–94, 96, 227 Ferromagnetic material, 40–41, 92, 96 Field-effect transistor (FET), 75, 78 Field programmable gate array, 3–4, 119 Finite element method (FEM), 102, 400 First-order highpass filter, 243 FitzHugh-Nagumo (FHN) model, 85 Floating-gate transistor, 127–131 Fluctuation-free dynamic, 375 Fluid mechanics problem, 47 Flux coupling, 112, 114 Fluxgate magnetometers, 92–97, 228, 403 Fokker-Planck equation, 308–309, 313, 373, 456 Forbidden interval theorem, 71–89 levy noise diffusion, 84 quantum communcation, 79 spiking retinal neuron, 82 stochastic resonance, 71 Fourier space, 63–65 Free-running Cuk converter, 178 Free-Standing Mesoscopic Beams, 297–305 Frequency response, 352, 405–406, 465, 467 curve, 263–264 Friction coefficient, 368 Functional difference equation (FDE), 434 G Game theory, 308, 311 Gaussian noise perturbations, 54, 82, 86, 229, 238, 242, 244, 254, 257, 282, 456 Gaussian probability distribution, 204 Gaussian SQUID arrays, 113 Gaussian stochastic process, 166 Generalized epidemics, 211–223 Generalized reconfigurable array, 127–131 floating-gate basics, 128 generalized array, 129 Geomagnetic field measurement, 93 Gigahertz-frequency oscillator, 26 Global bifurcation, 449–450 Global positioning system (GPS), 463 Goodness-of-fit tests, 79 Graph Coloring Problem, 331–339 Graphene, 35 Graphical user interface (GUI), 155 Gravitational fluctuation, 74 Gunn-diode, 435 472 H Harmonic balance tool, 395 Heterodyne, 350 High frequency structure simulator (HFSS), 400 Homodyne detection, 80 Hopf bifurcation, 106, 178, 180, 182 HTS SQUID magnetometer, 323–329 Hub, 212 Hypothetical power electronics system, 184 Hysteresis neuron, 332–337 Hysteretic behavior, 104, 108 Hysteretic control scheme, 178 I Identity matrix, 169, 241, 266 Igor Pro, 326 Ikeda maps, 362 Importance sampling technique, 57 Impulsive signal detection, 86 In-phase synchronization, 334, 336, 338–339 Inclinometer, 92–93, 99–100 Independent likelihood pool, 58 Independent opinion Pool, 58 Inductive-based readout strategy, 93 Infected nodes (NINs), 213–214 Information-theoretic method, 244 Infrared imaging, 86 Integrate-and-fire model, 86, 274 Integrated circuit control bus, 11 Integration gain, 250 Inter-element spacing, 400–401 Inter-spike intervals (ISIs), 82, 415 Intrinsic localized mode (ILM), 288, 293 Inverse stochastic Hopf bifurcation, 459–461 See also Hopf bifurcation J Jacobian linearization, 226 Jacobian matrix, 265–266, 356–359 Jacobian prediction, 362–363 Jacobians computing, 179 Josephson Junction, 15, 19, 112, 367–368, 377, 382 Josephson oscillators, 377 K Kalman-Bucy filter, 47 Kalman filter, 56 KE network, 216–218 Kolmogorov equation, 51, 53, 56 Kolmogorov-Fokker-Planck (KFP) equation, 388 Kramers rate, 201, 227 Index Kramers theory, 374 Kronecker delta function, 193 Kuramoto model, 206 L LabView, 95, 130, 155 Lagrange multipliers, 265 Lagrangian meters, 49 Langevin or stochastic differential equation, 49 LEGI model, 66, 68 Lennard-Jones potential, 288–289, 293 Levy noise, 84, 86 Linear displacement, 31 Linear Euler buckling theory, 303 Linear phase gradient, 146–150 Linear stability analysis, 122, 148–150, 152 Linear variable displacement transducer, 97–98 Lipschitz continuous drift, 84 Lipschitz Levy diffusion, 87 Lipschitz measurable functions, 86 LMS algorithm, 164–166 dynamic weight behavior, 166 traditional statistical theory, 165 LMS equalizer, 167–171 Local excitation, global inhibition (LEGI) model, 66, 68 Local linear prediction, 356–357 Jacobian-matrix estimate, 356–357 nonlinear prediction by Jacobian matrix, 357 Logic gate operation, Long-term depression (LTD), 413 Long-term potentiation (LTP), 413 Low noise amplifiers (LNA), 111, 117 Low pass filter (LPF), 243, 326, 435 LVDT, see Linear variable displacement transducer Lyapunov equation, 388 Lyapunov exponent, 138, 148, 264, 268–271, 389 M Macroscopic nanoscale oscillator, 34 Macroscopic quantum system, 30 Magnetic field detection, 38–39, 117, 228 Magnetic flux modulates, 38, 111–112, 382, 404 Magneto-optical trap, 15, 22 Magneto-resistive (MR) sensor, 225 Magneto-rheological fluids, 99 Magnetomotive technique, 33 Magnetophysiology, 323–330 Markov model, 317 Markov process, 47, 50 Index Material polarization, 103 Mean first passage time, 215–216, 235 Mean-square error (MSE), 163, 165–166, 173, 254, 360 Mechanical actuation and bistability, 301–304 Mechanical actuator, 92 Mechanical sensor, 227 Memory persistence, 315–316 Mesoscopic buckling, 298–299 Mesoscopicmechanical system, bistability of, 306 Metastable decay, 368, 371, 319–374 Micro- and nanomechanical devices, 15 Micromachining technologies, 102 Microwaveactive electronic component (AEC), 435 Military sensing, 93 Mode locking, 350 Model-neuron connections, 86 Model with noise, 207–208 Model without noise, 203–206 analytical treatment, 203–205 extension to scale-free networks, 205 numerical results, 203 Modulated oscillator, 367–369, 375–376 Modulation signal, 34, 45 Molecular vibrational predissociation, 288 Monopulse, nulling and beam shaping, 148–152 Monte-Carlo sampling, 57 Monte Carlo simulation, 47, 171, 216 Morphing mechanism, Morse potential, 288 Motor tasks learning, 445–446 MRI machines, 117, 379 Multi-mode Ginzburg-Landau potential, 303 Multi-phase synchronization, 133–143 Multi-resonant impedance-transformer (MRIT), 435 Multi-resonant system, 435 Multiple scales method, 153, 395 Multiplicative learning rule, 414 Multiplicative noise, 238, 456, 459 N Nano- and micromechanical resonators, 367, 377 Nano-particle manuplation, 287–288 Nearest-neighbor coupling, 147–148 Network generalization, 311 Neural network structure, 413–414 Neural prosthetics, 86 Neurotrophins, 246 Noise cancellation application, 167 473 Noise-enabled precision measurements, 297 Noise-enhanced signal quantization, 251 Noise intensity, 19, 21, 72–73, 234, 242, 254–255, 257, 259, 459 Noise radar, 348, 433–437 Noisy analog signal, robust digitization, 249 Noisy digital signal, averaging of, 250–251 Noisy nonlinear detector, 225–235 Non-equilibrium dynamical transition, 297 Non-equilibrium phase transition, 193, 196 Non-Gaussian uncertainties, 47 Non-identical threshold, 255 Nonlinear and Dynamical Circuits, 120–123 nonlinear transconductor, 120 operational-transconductance-amplifier, 120 bistable circuit, 121 oscillator with ring topology, 123 Nonlinear and reconfigurable electronic circuit, 119–132 Nonlinear array technology, 160 Nonlinear bistable circuit, 449 Nonlinear buckling instabilities, 297 Nonlinear dynamic effects, 163 Nonlinear dynamical systems theory, 142 Nonlinear dynamics of semiconductor lasers, 341–344, 353 Nonlinear elastic field theory, 298, 302 Nonlinear filtering theory, 49 Nonlinear prediction algorithm, 365 intervals, 355–356, 365 Nonlinear system, 123–127 topological symmetry, 124 example pattern forming system, 125 Nonzero coupling, 449 NOR and NAND gates, 4, 7, 8, 9, 10, 12 schematic diagram, 9–10 See also Logic gate operation Novel sensing strategies, 102 O Ohmic dissipation, 370 1-D and 2-D array design, 153–156 Opposite-phase synchronization, 335–336, 339 Optical phase-lock loop, 350 Optical sensors, 225 Optimal control theory, 264 Optimal forcing function, 264, 266, 268–271 Ordinary differential equations (ODE), 122, 125, 434, 464 Original Parrondo games, 308 Ornstein-Uhlenbeck (OU) process, 229 474 Oscillation amplitude, 18, 20, 34 onset of, 107, 403–404 Oscillator bistability of the, 368 eigenfrequency, 369 printed circuit, prototype, 155 parametric, 15, 19–20 P Packet congestion, 417–418 Parallel power converters, 133–143 Parameter theory, 191 Parametric amplification, 15 Parametric resonance, 15, 369, 375–376 Parkinson’s disease, 421–424 DDE analysis, 422 Parrondian and Brownian ratchet phenomena, 316 Parrondo-like effects, 311 Parrondo paradox, 307–318 Allison mixture, 314–317 developments, 311 original games, 308 thermodynamics of games of chance, 313 Partial differential equations (PDE), 434 Particle filters, 56–58 data fusion, 57 Particle methods, 56 Passive components, 177 Peak-sidelobe ratio (PSL), 344, 349 Penning trap, 15, 22 Performance dynamics, 446–447 Period-doubling, 175, 178–179, 342, 389, 436, 461 bifurcation, 178–179, 389, 436, 461 Period-one oscillation, 342, 350 Periodic dynamics, applications, 349–353 all-optical AM-to-FM conversion, 351–352 dual-frequency multifunction precision lidar, 351 photonic microwave generation, 350 Periodic forcing function, 263, 449 Periodic spike position sequence, 277, 279–280 Periodic spike-train, 277, 279, 283 Perturbation schemes, 342, 349 Phase-locked loop, 157, 159 Phase model dynamical equations, 150 Phase shifters, 146, 148–149, 156, 159 Photodetectors, 346, 348 Physical quantum systems, 308 PIC microcontroller, 42 Piecewise constant (PWC), 134, 136, 139, 141–143 Index Piecewise linear (PWL), 8, 82, 134, 332 Planck constant, 369, 371, 374–375 Plasma etching, 298, 305 Point vortex models, 48–49 Poisson degree distribution, 192, 194 Poisson spike rate, 82 Poisson statistics, 20–21 Polarized state, 190 Pooling networks, 250–253, 255–258, 260 Portfolio rebalancing, 312–313 Posterior density, 54, 57 distribution, 58 Power converters, 133, 141, 177 Power-factor-correction converter, 179 Power spectral density (PSD), 37, 207 Prediction interval estimation, 363, 365 Printed circuit board (PCB), 40, 180 Projection matrix, 388–389 Proof-of-principle, 3, Proportional to absolute temperature, 399 Pseudo-periodic weight behavior, 168 Pseudoplots, 157 Pseudorandom sequence, 345 PSPICE circuit, 104–105, 177 PTAT, see Proportional to absolute temperature Q Quantization, 226–227, 249–250, 253–254, 256 error, 226 Quantum activation energy, 374–376 Quantum forbidden-interval theorem, 80 Quantum information, 297 QUantum interference device, 225, 323 Quantum kinetic equation, 369–370 Quantum measurement, 26, 367, 376–377 Quantum nanomechanics, 25–36 definitions and requirements, 27 classical and quantum regimes, 28 dimensionality, 27 requirement for the nanomechanical structure, 29 potential quantum nanomechanical system, 31 coupled nano-electro-mechanical, 34–35 multi-element oscillators, 32–34 straight-beam oscillators, 32 tunneling two-state oscillator, 34 Quantum non-demolition measurements, 368 Quantum-optical model, 80 Quasi-periodic (Q), 271, 342, 466–467 Quasi-static target signal, 92 Index R RADAR (RAdio Detection and Ranging), 146 Radio-over-fiber (RoF) system, 352 Random walk centrality (RWC), 211, 213, 215–217, 222 Re-wiring, 278–283 positions, 279, 281 theorem, 279–280, 283 learning by re-wirings, 280 Reactive ion etching (RIE), 298 Real-time oscilloscope, 346, 348 Receptor occupancy fluctuations, 61–62 application to a one-dimensional geometry, 62 numerical generation of the noise, 64 specific directional sensing model, 66 Receptor occupancy fluctuations, 61, 63 Reconfigurable chaotic logic gate (RCLG), 4, 8, 11, 12 design and construction, 8–11 Refractory effect, 410–411, 418–419 Repulsive links, 439–440, 443 Residence time approach, 228–229 Residence time detection (RTD) readout, 37 Residence times difference (RTD), 92–93, 95, 406 Residual strain, 298, 306 Resonance curve, 263–267, 271 Resonance-mode frequency, 31 Resonant forcing, 264–267, 269–271 Resonant trapping, 234 Ringer solution, 325–326 ROC curves, 424 Routing strategy, 417–420 Row-column steering, 160 RTD fluxgate magnetometer, 93 amorphous FeSiB Microwire Viscosimeter, 97 amorphous FeSiB Microwire FluxgateMagnetometer, 96 device prototypes, 99 ferrofluidic transducers, 99 ferromagnetic foil: FR4-fluxgatemagnetometer, 94 S Saddle-node bifurcation, 372 Sawyer-Tower circuit, 104 Scale-free (SF) networks, 203, 205–206, 211–212, 214, 419–420 Scale-free topological properties, 211 Scanning electron microscope (SEM), 299 Scanning tunneling microscope, 287 Schmitt Trigger, 42, 228 475 Secular perturbation theory, 263 Self feedback parameter, 333–335 Self-organized neural network, 413–416 Self pulsation, 350 Self-transition, 316 Semi-periodic dynamic behavior, 166 Semiconductor laser dynamics, 341 Sensing coil, 40–41, 100 Sensitivity control, 391–394 control and SSF, 391 control goal and choice of regulator parameters, 392 Sensitivity response, 406–407 Sensory spiking neuron models, 84, 86 Serial Peripheral Interface (SPI), 11 Shannon mutual information, 72–73 Shift-andreset dynamics, 276 Sidelobe reduction, 151 Sigmoid function, 410 Sigmoid-shaped memoryless function, 82 Signal amplitude, 72–73, 227, 231, 234–235 Signal processing model, 249 Signal response, 201–209, 384, 406 enhancement of, 201–209 Signal-to-noise ratio, 71, 207–208, 239, 255, 257–260, 264, 270, 323, 327–329, 368, 384 Simultaneous analog strategy (SAS), 243 Single electron transistor (SET), 31, 35 Single-peak correlation feature, 348 Single radiator, 400–401 Single sensor filtering theory, 57 Single-walled carbon nanotube (SWNT), 76, 78 Sliding mode control, 134 Slope compensation, 179 Smart magnetic fluids, 91 SNR, see Signal-to-noise ratio Social pressure mechanism, 190 Spatial synchronization, 206 Spectral amplification factor, 440 Spectrum control, 153 SPICE convergence problems, 399 Spike-timing dependent synaptic plasticity (STDP), 413 Spike-Train, 273, 276–281, 283–284 Spiking neuron model, 84–87, 273 Spinlike network, 439 Spring constant, 15–18, 32, 293 SQUID, see Super conductive quantum interference device SSR model, 252 Star-like networks, 205 476 Static Random Access Memory cell (SRAM), 122 Stationary probabilistic density, 456 STDP learning, 413–416 Steyskal technique, 150 Stochastic beamforming, 237–246 Stochastic equilibrium point, 456 Stochastic Hopf model, 455 Stochastic learning algorithms, 72 Stochastic limit cycle, 388, 456, 459 Stochastic Lorenz model, 394, 461 Stochastic pooling network, 250–252, 255–256, 258 Stochastic resonance (SR), 34, 71–87, 201–202, 208, 227, 229, 234, 237–239, 245, 250–251, 255, 257, 263, 297, 310, 440 carbon nanotube signal detector, 75 curve, 310 effect, 201, 227 forbidden interval theorems, 71 quantum communication, 79 spiking retinal neuron, 82 levy noise diffusion, 84 hypothesis test, 75 learning algorithm, 72 screening device, 71 Stochastic sensitivity function (SSF), 387–389, 393–394 Stochastic signal quantization theory, 249 Stochastic time scale, 201, 227 Straight-beam oscillators, 32, 35 Strange nonlinear phenomena, 177 Stroboscopic map, 178 Strong-damping limit, 374 Sum-pattern steering, 149 Super conductive quantum interference device array, 111–114, 117 electronics, 324, 326, 328–329 loop parameters, 112 magnetometer, 112, 323, 329, 381–383 nonlinearity parameter, 382 use of, 117 Superconducting Quantum Interference Filters (SQIFs), 111 Supercritical bifurcation point, 18 Supercritical pitchfork bifurcation, 375 Superspreaders, 212 Superstable periodic orbit (SSPO), 134 Support vector machine (SVM), 423 Suprathreshold stochastic resonance (SSR), 237, 239, 250–251, 255, 257 Susceptibleinfected (SI) model, 213 Swedish animal welfare agency, 325 Index Switching time bifurcation, 178 Symmetry-breaking signal, 92 Synchronization, 79, 202, 206, 263, 311, 332, 334–336, 338–339, 341, 344, 451, 464 role of, 206 Synchronized secure communications, 341 T Temporal distortions, 421–422 Temporarily annealed disorder system, 440 Tent-map, 436 Threshold neuron, 72 Threshold voltage, 6, 8, 10, 76, 79, 120, 128, 255 Time-reversibility, 313–314 Topological resonance, 202, 209 Torque restoring modulations, 17 Torsional oscillator, 15–23 Torsional rods, 16 Torus bifurcation, 463–467 tracer advection, 54 two-vortices with equal strengths, 51 Transcription errors in DNA, 311 Transverse hippocampal slices, 323, 326–327, 329 Traveling wave (TW), 124–125, 127 Triangular shift matrix, 169 Trigger zone, 86 2D-cycles, sensitivity analysis of, 389–390 2D square lattice, 193, 196 Two-time scale landscape model, 447 U Ultra-wide band (UWB), 273, 280 Universal portfolio, 312 V Valve pressure drop, 101 Van der Pol nonlinearity, 147 oscillator, 153, 395 parameters, 153–154, 395 VCO topology, 397 Vehicle navigation, 93 Vehicle routing problem, 409–412 chaotic search, 410 diversification method, 411 greedy firing, 411 fluctuating a threshold value, 411 Velet neighbors, 290 Vibrating gyroscope, 92–93, 100, 464 See also Inclinometer Villeneuve weighting, 152 Viruses in cyber-networks, 212 Index Vivaldi array, 400 VLSI chip, 11–12 Volatility pumping, 308, 312–313 Voltage controlled oscillator (VCO), 154, 156 Vortex-driven tracer dynamics, 47–49 Voter model, 190–192, 198 W Weak-resonant backward wave oscillator, 434 White gaussian noise, 54, 77, 202, 384 Wiener cascade model, 82 477 filter, 163–166 process, 384, 388–389, 391, 456 solution, 165, 168 Wigner Representation, 371–373, 375 Winner-take-all (WTA), 134 Wireless PWM control, 134 Wiring Matrix, 275–277, 279–281, 283–284 WR-BWO, see Weak-resonant backward wave oscillator X Zero-amplitude state, 18–22, 375 Zero-lag value, 78 [...]... atomistic molecular dynamics approach also becomes severely limited due to the large number of atoms The size of 100 million to 100 billion atoms requires multi-scale modeling of the elastic properties of QnM systems, which may require novel approaches to computational modeling of large systems Currently, the state -of- the-art large-scale computing power of a large cluster can handle a size of 100–200 million... from the Chua’s circuit, (f) Recovered logic output signal from V 0 The fundamental period of oscillation of this circuit is 0.33 mS 5 VLSI Implementation of Chaotic Computing Architectures – Proof of Concept Recently ChaoLogix Inc designed and fabricated a proof of concept chip that demonstrates the feasibility of constructing reconfigurable chaotic logic gates, henceforth ChaoGates, in standard CMOS... (eds.), Applications of Nonlinear Dynamics, Understanding Complex Systems, DOI 10.1007/978-3-540-85632-0 1, c Springer-Verlag Berlin Heidelberg 2009 3 4 W.L Ditto et al distinctive feature of this alternate computing paradigm was that they exploited the sensitivity and pattern formation features of chaotic systems In subsequent years, it was realized that one of the most promising direction of this... electrostatic contribution to the spring constant is modulated near twice the natural frequency of the oscillator When the parametric modulation is sufficiently strong, oscillations are induced at H.B Chan (B) Department of Physics, University of Florida, Gainesville, FL 32608, USA V In et al (eds.), Applications of Nonlinear Dynamics, Understanding Complex Systems, DOI 10.1007/978-3-540-85632-0 2, c Springer-Verlag... the thermal energy kB T The motivation behind this crude definition of the quantum regime is simple The motion of a QnM system can be described by a harmonic P Mohanty (B) Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA e-mail: mohanty@physics.bu.edu V In et al (eds.), Applications of Nonlinear Dynamics, Understanding Complex Systems, DOI 10.1007/978-3-540-85632-0... order to design the NOR or NAND gates, one has to use the knowledge of the dynamics of the nonlinear system to find the values of VC and V0 that will satisfy all the input-output associations in a consistent and robust manner Consider again the simple realization of the double-scroll chaotic Chua’s attractor represented by the set of (rescaled) 3-coupled ODEs given in Eqs (1), (2), (3) This system... W.L., Proceedings of the STATPHYS-22 Satellite conference Perspectives in Nonlinear Dynamics Special Issue of Pramana 64 (2005) 433 7 Murali, K., Sinha, S and Ditto, W.L., Int J Bif and Chaos (Letts) 13 (2003) 2669 Construction of a Chaotic Computer Chip 13 8 Murali, K., Sinha S., and I Raja Mohamed, I.R., Phys Letts A 339 (2005) 39 9 Murali, K., Sinha, S., Ditto, W.L., Proceedings of Experimental Chaos... commercialization of this technology (http://www.chaologix.com ) 1 Introduction It was proposed in 1998 that chaotic systems may be utilized to design computing devices [1] In the early years the focus was on proof -of- principle schemes that demonstrated the capability of chaotic elements to do universal computing The W.L Ditto (B) J Crayton Pruitt Family Department of Biomedical Engineering, University of Florida,... pressure of less than 1 × 10−6 torr The quality factor Q of the oscillator exceeds 7,500 The modulations of the spring constant in our torsional oscillator originate mainly from the strongly distance-dependent electrostatic interaction between the top plate and the excitation electrode The equation of motion of the oscillator is given by: θ¨ + 2γ θ˙ + ω02 θ = τ /I (1) where θ is the angular rotation of the... the excitation and ke + −C (θo )VdcVac is the effective amplitude of the modulation of spring constant Coupling of the electrostatic torque 18 H.B Chan and C Stambaugh F cos(ω t) and the nonlinear terms generates an effective modulation of spring constant However, this contribution is much smaller than the direct electrostatic modulation of the spring (ke /I) As a result, the F cos(ω t) term can be neglected ... fundamental period of oscillation of this circuit is 0.33 mS VLSI Implementation of Chaotic Computing Architectures – Proof of Concept Recently ChaoLogix Inc designed and fabricated a proof of concept... thanks to the principal sponsors of the meeting: Office of Naval Research (Washington, DC), Office of Naval Research-Global (London), San Diego State University (College of Sciences), and SPAWAR Systems... Shlesinger, Office of Naval Research Hiroyuki Torikai, Hosei University SPONSOR: Office of Naval Research (ONR) 875 N Randolph Street, Suite 1475 Arlington, VA 22217 v Preface The field of applied nonlinear