Visarath In Antonio Palacios
Springer Complexity
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Henry Abarbanel, Institute for Nonlinear Science, University of California, San Diego, USA Dan Braha, New England Complex Systems, Institute and University of Massachusetts, Dartmouth, USA Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, Kalamazoo, 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 Viktor Jirsa, Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille, France Janusz Kacprzyk, System Research, Polish Academy of Sciences, Warsaw, Poland Kunihiko Kaneko, Research Center for Complex Systems Biology, The University of Tokyo, Tokyo, Japan Scott Kelso, Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry, UK Jürgen Kurths, Potsdam Institute for Climate Impact Research (PIK), Potsdam, Germany Ronaldo Menezes, Department of Computer Science, Florida Institute of Technology, Melbourne, FL, USA Andrzej Nowak, Department of Psychology, Warsaw University, Warsaw, Poland Hassan Qudrat-Ullah, School of Administrative Studies, York University, Toronto, ON, Canada 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 Zürich, Zürich, Switzerland Didier Sornette, Entrepreneurial Risk, ETH Zürich, Zürich, Switzerland Stefan Thurner, Section for Science of Complex Systems, Medical University of Vienna, Vienna, Austria
Understanding Complex Systems
Founding Editor: S. 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.
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Visarath In • Antonio Palacios
123
Visarath In Space and Naval Warfare Systems Center San Diego, CA USA
Antonio Palacios Department of Mathematics, Nonlinear Dynamical Systems Group
San Diego State University San Diego, CA USA
ISSN 1860-0832 ISSN 1860-0840 (electronic) Understanding Complex Systems ISBN 978-3-662-55543-9 ISBN 978-3-662-55545-3 (eBook) DOI 10.1007/978-3-662-55545-3
Library of Congress Control Number: 2017947684
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This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Germany The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany
To our wives, Christine and Irene, and our sons, Beredei, Reynard and Daniel, and to our beloved parents, for their love and support throughout these years.
Preface
The seminal work by Lorenz in 1963 [264], and later by May in 1976 [273–275], has led scientists and engineers to recognize that nonlinear systems can exhibit a rich variety of dynamic behavior. From simple systems, such as the evolution of single species [314], an electronic or biological oscillator [423, 424], to more complex systems, such as chemical reactions [33], climate patterns [153], bursting behavior by a single neuron cell [208], and flocking of birds [333, 393], Dynamical Systems theory provides quantitative and qualitative (geometrical) techniques to study these and many other complex systems that evolve in space and/or time. Regardless of the origins of a system, i.e., Biology, Chemistry, Engineering, Physics, or even the Social Sciences, dynamical systems theory seeks to explain the most intriguing and fundamental features of spatio-temporal phenomena.
In recent years, systems made up of individual units coupled together, either weakly or tightly, have gained considerable attention. For instance, the dynamics of arrays of Josephson junctions [18, 99, 100, 160], central pattern generators in biological systems [85, 226, 227], coupled laser systems [328, 419], synchroniza- tion of chaotic oscillators [315, 426], collective behavior of bubbles in fluidization [163], the flocking of birds [393], and synchronization among interconnected biological and electronic nonlinear oscillators. These are only a few representative examples of a new class of complex dynamical systems or complex networks. The complexity arises from the fact that individual units cannot exhibit on their own the collective behavior of the entire network. In other words, the collective behavior is the exclusive result of the mutual interaction that takes place when multiple units are interconnected in some fashion.
In most cases, three factors are normally considered when studying the collective behavior of a complex system. Mainly, the internal dynamics of each individual unit or cell, the topology of cell connections, i.e., which cells communicate with each other, and the type of coupling. More recently, a fourth factor has gained further attention—symmetry. It is well-known that symmetry alone can restrict the type of solutions of systems of ordinary– and partial differential equations, which often serve as models of complex systems. So it is reasonable to expect that certain aspects of the collective behavior of a complex system can be inferred from the
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presence of symmetry alone. In fact, the work by Golubitsky [145, 146, 149] lays down the theoretical foundations for a model-independent analysis to understand, and predict, the behavior of a dynamical system using, mainly, the underlying symmetries of the system while separating the fine details of the model. While this approach has been widely successful in explaining computer simulations and experimental observations of many different spatio-temporal phenomena, it has found limited use in the conceptualization and development of nonlinear devices even though many of those systems are, inherently, symmetric. And while many works have been dedicated to study the symmetry-preserving phenomenon of synchronization [25, 315, 317, 382, 426], significantly less is known about how one can exploit the rich variety of collective patterns that can emerge via symmetry-breaking bifurcations, such as heteroclinic cycles [63].
Over the past 17 years, we and other colleagues and students have been attempting to bridge the current gap between the theory of symmetry-based dynamics, equivariant bifurcation theory, and its application to developing non- linear devices. At the beginning, around the year 2000, we were interested in developing new methods to manipulate frequency in arrays of nonlinear oscillators for antenna devices. Collaborators from the U.S. Navy had already shown [166] that small frequency perturbations applied to the end points of a chain of nonlinear oscillators can lead to a change in the direction of the radiation pattern. That is, they demonstrated that beam steering was possible without mechanically rotating an antenna. The next puzzle that we had to solve was to manipulate the collective frequency of the array over a broad band without changing the internal frequency of each individual oscillator. But just when we were about to solve this problem, we were steered, no pun intended, into developing a new class of highly sensitive, low-power and low-cost, magnetic- and electric field sensors. Theoretical work for this new class of sensors started around 2002. The fundamental principles were twofold: to exploit coupling-induced oscillations in symmetric networks to generate self-induced oscillations, thus reducing power consumption; and to exploit symmetry-breaking effects of heteroclinic cycles to enhance sensitivity. As a starting point, we chose fluxgate magnetometers as individual units, because their behavior is governed by a one-dimensional autonomous differential equation. Consequently, based on the fundamental theory of ODEs, it follows that in the absence of any forcing term the one-dimensional dynamics of the individual units cannot produce oscillations. But when the fluxgates are coupled then the network can, under certain conditions that depend on the coupling strength, oscillate. This configuration could demonstrate to skeptics that self-induced oscillations can indeed be engineered. In practice, the network would still need, of course, a min- imum amount of energy to kick it off of its trivial equilibrium state and get the oscillations going. Overall, we were able to show that, under certain conditions, the sensitivity response of an array of weakly coupled fluxgate sensors can increase by four orders of magnitude while their cost could be simultaneously reduced to a fraction of that of an individual fluxgate sensor. This technology matured around 2005 with design, fabrication and deployment.
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In 2006, we extended the work on magnetic fields to electric field sensors. These sensors are also governed by one-dimensional, overdamped, bistable systems of equations. We conducted a complete bifurcation analysis that mirrors that of the fluxgate magnetometer and, eventually, translated the research work into a micro- circuit implementation. This microcircuit was intended to be used for measuring minute voltage or current changes that may be injected into the system. The con- ceptualization of these sensors employs the model-independent approach of Golubitsky’s theory for the study of dynamical systems with symmetry, while the development of laboratory prototypes takes into account the model-specific features of each device which, undoubtedly, may impose additional restrictions when we attempt to translate the theory into an actual experiment. For instance, a sensor device that measures magnetic flux, as oppose to electric field signals, may limit the type of coupling functions that can be realized in hardware. In other words, not every ide- alization of a network-based structure can be readily implemented in the laboratory. This and other similar restrictions need to be kept in mind while reading this book.
Around that same 2006 year, we started, in tandem, to the work on electric field sensors, theoretical studies of networks of Superconducting Quantum Interference Devices (SQUIDs). The work was suspended until 2009 when we returned to explore in greater detail the response of networks of non-uniform SQUID loops. The technology matured by 2012 with applications to antennas and communication systems. Around that same period, 2007–2009, we went back to the study of multi-frequency oscillations in arrays of nonlinear oscillators. In fact, we were able to develop a systematic way to manipulate collective frequency through cascade networks. The work matured in 2012 with the modeling, analysis, design and fabrication of the nonlinear channelizer. This is an integrated circuit made up of large parallel arrays of analog nonlinear oscillators, which, collectively, serve as a broad-spectrum analyzer with the ability to receive complex signals containing multiple frequencies and instantaneously lock-on or respond to a received signal in a few oscillation cycles. Again, the conceptualization of the nonlinear channelizer was based on the generation of internal oscillations in coupled nonlinear systems that do not normally oscillate in the absence of coupling. Between 2007–2011, we investigated various configurations of networks of coupled vibratory gyroscopes. The investigations showed that networks of vibratory gyroscopes can mitigate the negative effects of noise on phase drift. But the results were, mainly, computational and applicable only to small arrays. Finally, between 2012–2015, we developed the necessary mathematical approach to study networks of arbitrary size. This work showed the nature of the bifurcations that lead arrays of gyroscopes, connected bidirectionally, in and out of synchronization. The results were applicable to net- works of arbitrary size.
In the past few years, previous works have led us into new topics. Networks of energy harvesters, which, interestingly, are governed by ODEs that resemble those of vibratory gyroscopes. This feature highlights again the model-independent nature of the analysis of differential equations with symmetry. In the year 2011, in par- ticular, we started a new project to study the collective behavior of spin-torque nano-oscillators. The motivation for this work is a conjecture by the 2007 Nobel
Preface ix
Laureate, Prof. Albert Fert, about the possibility that synchronization of nano-oscillators could produce substantial amounts of microwave power for prac- tical applications. Determining the regions of parameter space of stable synchro- nized solutions was a very challenging problem due to the nature (non-polynomial form) of the governing equations. Finally, this year we overcame the major diffi- culties by exploiting, again, equivariant bifurcation theory. And the most recent project that we started in 2016 is about networks of coupled oscillators for improving precision timing with inexpensive oscillators, as oppose to atomic clocks.
Along the way, several patents were approved by the U.S. Patent Office for the works related to these projects, including:
2007 U.S. Patent # 7196590. Multi-Frequency Synthesis Using Symmetry Methods in Arrays of Coupled
Nonlinear Oscillators. 2008 U.S. Patent # 7420366. Coupled Nonlinear Sensor System. 2009 U.S. Patent # 7528606. Coupled Nonlinear Sensor System for Sensing a Time-Dependent Target Signal
and Method of Assembling the System. 2011 U.S. Patent # 7898250. Coupled Fluxgate Magnetometers for DC and Time-Dependent (AC) Target
Magnetic Field Detection. 2011 U.S. Patent # 8049486. Coupled Electric Field Sensors for DC Target Electric Field Detection. 2012 U.S. Patent # 8049570. Coupled Bistable Microcircuit for Ultra-Sensitive Electric and Magnetic Field
Sensing. 2012 U.S. Patent # 8212569. Coupled Bistable Circuit for Ultra-Sensitive Electric Field Sensing Utilizing
Differential Transistors Pairs. 2015 U.S. Patent # 8994461. Sensor Signal Processing Using Cascade Coupled Oscillators. 2015 U.S. Patent # 9097751. Linear Voltage Response of Non-Uniform Arrays of Bi-SQUIDS. 2016 Under review. Navy Case: 101427. Enhanced Performance in Coupled Gyroscopes and Elimination of Biasing
Signal in a Drive-free Gyroscope. 2016 Under review. Navy Case: 101950. Arrays of Superconducting Quantum Interference Devices with Self Adjusting
Transfer to Convert Electromagnetic Radiation into a Proportionate Electrical Signal to Avoid Saturation.
2016 Under review. Navy Case: 102297. 2D Arrays of Diamond Shaped Cells Having Multiple Josephson Junctions. 2016 Under review. Navy Case: 103829. Network of Coupled Crystal Oscillators for Precision Timing.
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None of these projects would have been possible without the active participation of students, joint work with collaborators, and the financial support from various sources. We would like to thank each of the students first: John Aven [21, 22], Jeremmy Banning [26], Katherine Beauvais [30], Susan Berggren [36, 159], Bernard Chan, Nathan Davies [91], Scott Gassner [132, 133], Mayra Hernandez [168], Habib Juarez, Tyler Levasseur, Patrick Longhini [261, 262], Daniel Lyons [266, 267], Antonio Matus [272], Derek Moore, Loni Olender, Steven Reeves [331] Norbert Renz [332], Richard Shaffer [359], Brian Sturgis-Jensen, James Turtle [398, 399], Huy Vu [407], Sarah Wang, Bing Zhu [433]. Special acknowledgement and thanks to Patrick Longhini, he was the first student that got involved in the work through his Master and, later on, Ph.D. thesis. He continues to be an extremely valuable asset to multiple ongoing projects. Collaborators include: Bruno Ando (Univ. of Catania), Marcio De Andrade (SPAWAR), Salvatore Baglio (Univ of Catania, Italy), Peter Blomgren (SDSU), Donald Bowling (NAWC), Pietro-Luciano Buono (Univ. of Ontario Institute of Technology, Canada), Adi Bulsara (SPAWAR), Lowell Burnett (QUASAR), Juan Carlos Chaves (HPTi), Ricardo Carretero (SDSU), Anna Leese de Escobar (SPAWAR), Jocirei Dias Ferreira (Federal Univ. of Mato Grosso, Brazil), Hugo Gonzalez-Hernadez (Instituto Tecnologico de Monterrey), Frank Gordon (SPAWAR), Takachi Hikihara (Kyoto Univ., Japan), Calvin Johnson (SDSU), Andy Kho (SPAWAR), Daniel Leung (SPAWAR), John F. Lindner (College of Wooster), Norman Liu (SPAWAR), Joseph M. Mahaffy (SDSU), LT Jerome McConnon (SPAWAR), Brian K. Meadows (SPAWAR), Oleg Mukhanov (HYPRES), Joseph Neff (SPAWAR), Suketu Naik (Weber State Univ.), Martin Nisenoff (M. Nisenoff Associates), Georgy Prokopenko (HYPRES), Wouter-Jan Rappel (UCSD), LT Sarah Rice (SPAWAR), Robert Romanofsky (NASA), Vincenzo Sacco (Univ. Catania), Benjamin Taylor (SPAWAR), Edmond Wong (SPAWAR), Yongming Zhang (QUASAR).
Many thanks to the Chaos Group at the Oak Ridge National Laboratory, Stuart Daw, Charles Finney, and Sreekanth Pannala, for very stimulating discussions. We also wish to acknowledge very fruitful interactions with John Angus and Ali Nadim who served as committee members for many of the Ph.D. theses that derived from related projects. Special thanks to Pietro-Luciano Buono with whom we have collaborated extensively in recent years to apply advance methods from equivariant bifurcation theory. One of the authors, Antonio Palacios, wishes to thank Marty Golubitsky, in particular, for his mentorship and guidance to learn from him (during a postdoctoral appointment) the principles and methods for studying dynamical systems that posses symmetry.
We also wish to acknowledge the financial support provided by several agencies to conduct the necessary research work that serves as the foundation of some of the technologies discussed in this book, including: Army Research Office, Department of Defense, Department of Energy, the National Science Foundation, the National Security Agency, the Office of Naval Research, the San Diego Foundation, and the Space and Naval Warfare Center, San Diego. We wish to acknowledge the con- tinuous support of Dr. Michael Shlesinger from ONR.
Preface xi
The book is intended for a broad audience. For engineers who might be inter- ested in applying ideas and methods from dynamical systems with symmetry and equivariant bifurcation theory to design and fabricate novel devices. For mathe- maticians and physicists who might be interested in translational research work to extrapolate fundamental research theorems into practical applications. And for scientists from many disciplines, viz. Biology, Chemistry, Computer Science, Geology, etc., who might be interested in the interplay between theory and real-life applications from the general field of nonlinear science.
The book is organized as follows. In Chap. 1 we present fundamental ideas of complex networks and bistability, which is a common feature of many sensor devices; and then we dedicate a few sections to introduce basic ideas, methods and examples in the analysis of differential equations (ODEs and PDEs) with symmetry. One particular class of solutions that rarely appears in generic versions of systems of differential equation are heteroclinic cycles. These types of solutions are, how- ever, generic features of systems with symmetry. We exploit these cycles to enhance sensitivity and, thus, we dedicate a section to explain what they are and how they can be found. The book is then organized in two parts. Part I, Chap. 2 through Chap. 6 is dedicated to translational research work that already led to mature technologies. These technologies include networks of fluxgate magne- tometers; arrays of micro-electronic electric field sensors; networks of SQUIDs; cascade arrays of nonlinear oscillators for multi-frequency generators; and a special chapter in honor of the theoretical work by Pietro-Luciano and Marty Golubitsky: a device realization of a Central Pattern Generator network of the animal gaits studied by them. Part II, Chap. 7 through Chap. 10 include, mainly, theoretical works that have not yet mature into actual device realizations. The technologies that may derive from these works are part of ongoing efforts.
San Diego, USA Visarath In 2017 Antonio Palacios
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Contents
1 A Unifying Theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Self-oscillating Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 The Role of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Symmetry-Breaking Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Coupled Cell Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7 Heteroclinic Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.8 Representative Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Coupled-Core Fluxgate Magnetometer . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Fluxgate Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Modeling Single-Core Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 Coupled Single-Domain System . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5 Sensitivity Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.6 Alternating Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.7 AC Field Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.8 Target Signal Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.9 Effects of Nonhomogeinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.10 Effects of Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.11 Laboratory Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3 Microelectric Field Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Circuit Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3 Analysis and Bifurcation Diagrams. . . . . . . . . . . . . . . . . . . . . . . 110 3.4 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . . 111 3.5 Period and Residence Times Response . . . . . . . . . . . . . . . . . . . . 118 3.6 SPICE Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
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4 Superconductive Quantum Interference Devices (SQUID) . . . . . . . . 127 4.1 History of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2 The Josephson Effect and SQUID Technology . . . . . . . . . . . . . . 131 4.3 Phase-Space Dynamics of DC SQUID . . . . . . . . . . . . . . . . . . . . 132 4.4 Chimera States in Non-locally Coupled Arrays. . . . . . . . . . . . . . 138 4.5 The DC Bi-SQUID. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.6 Serial Bi-SQUID Array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.7 Parallel Bi-SQUID Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.8 Design, Fabrication, and Evaluation . . . . . . . . . . . . . . . . . . . . . . 154
5 Frequency Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.1 Frequency Up-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.2 Experiments on Frequency Up-Conversion . . . . . . . . . . . . . . . . . 176 5.3 Frequency Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.4 Experiments on Frequency Down-Conversion . . . . . . . . . . . . . . 191 5.5 Large Frequency Downconversion Ratios. . . . . . . . . . . . . . . . . . 199 5.6 Nonlinear Channelizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.7 Experimental Setup of Nonlinear Channelizer . . . . . . . . . . . . . . 218
6 ANIBOT: Biologically-Inspired Animal Robot . . . . . . . . . . . . . . . . . 229 6.1 Central Pattern Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.2 CPG Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 6.3 Analog Fitzhugh–Nagumo Neuron Circuit . . . . . . . . . . . . . . . . . 232 6.4 Patterns and Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.5 Leg Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
7 Gyroscope Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.2 History of Navigation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.3 Evolution of Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 7.4 Vibratory Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 7.5 Bi-Directionally Coupled Ring . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.6 Unidirectionally Coupled Ring . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.7 Drive-Free Gyroscope System . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.8 Hamiltonian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8 Energy Harvesting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.2 Single Magnetostrictive Energy Harvesting Model . . . . . . . . . . . 299 8.3 Coupled Energy Harvester System . . . . . . . . . . . . . . . . . . . . . . . 302 8.4 Computational Bifurcation Results . . . . . . . . . . . . . . . . . . . . . . . 305 8.5 Hamiltonian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 8.6 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
xiv Contents
9 Spin Torque Nano Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 9.1 The Giant Magnetoresistance (GMR) Effect . . . . . . . . . . . . . . . . 317 9.2 Spin Torque Nano Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 9.3 Landau-Lifshitz-Gilbert-Slonczewski (LLGS) Model . . . . . . . . . 321 9.4 The Synchronization Challenge . . . . . . . . . . . . . . . . . . . . . . . . . 323 9.5 Series Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 9.6 Complex Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . 328 9.7 Hopf Bifurcation Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 9.8 Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 9.9 Locking into Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
10 Precision Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 10.1 History of Precision Timing Devices . . . . . . . . . . . . . . . . . . . . . 339 10.2 Crystal Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 10.3 Two-Mode Oscillator Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 10.4 Coupled Crystal Oscillator System . . . . . . . . . . . . . . . . . . . . . . . 348 10.5 Averaging and Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 10.6 Linearization and Spectrum at the Origin . . . . . . . . . . . . . . . . . . 357 10.7 Stability and Bifurcation Results. . . . . . . . . . . . . . . . . . . . . . . . . 360 10.8 Numerical Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 10.9 Phase Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 10.10 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
Contents xv
Chapter 1 A Unifying Theme
Nonlinear systems can behave very robustly, yet they can also exhibit high sensitivity to small perturbations [205, 206, 277, 404, 420]. This dichotomy is a consequence of the parameters that govern the behavior of a given system. In regionswhere changes in parameters do not yield radically different behavior the system is expected to behave robustly. However, in regions that contain a bifurcation point, high-sensitivity can ensue. Close to the onset of oscillations, in particular, a small perturbation can drasti- cally alter the behavior of a nonlinear oscillator, it can either destroy the oscillations completely or it can change their characteristics such as frequency and amplitude. For instance, temperature variations can reset a circadian oscillator, see Fig. 1.1 while noise [277] can induce an exogenous oscillator to oscillate in regions where only steady-state behavior would dominate the system dynamics absent the noise. Similar sensitivity features can be used for signal amplification in electrical, mechanical and optical systems [213]. Exploring those features is the subject of this chapter and of the entire book.
1.1 Complex Networks
Over the past seventeen years, we and other colleagues and students have been developing, through theory and experiments, a new paradigm that combines ideas and methods from the theory of Nonlinear Dynamical Systems with Symmetry in Mathematics, Physics, and Engineering, for performance enhancement of nonlinear devices. A unifying theme and goal of this paradigm is to demonstrate, theoretically and experimentally, that collective behavior, which is uniquely produced by inter- connected nonlinear devices, can be exploited to develop novel complex network based systems that can outperform the function of their individual counterparts. After all, many sensory systems [225] in animals are controlled by clusters of neu- ron cells, located in the central nervous system and physically interconnected to
© Springer-Verlag GmbH Germany 2018 V. In and A. Palacios, Symmetry in Complex Network Systems, Understanding Complex Systems, DOI 10.1007/978-3-662-55545-3_1
1
Fig. 1.1 Endogenous circadian oscillators of Drosophila and Mammals. Source Journal of Cell Science 119, 4793–4795 (2006)
produce, somehow, an optimal response, which in many cases outperforms those of humans. A dog’s sense of smell is said to be a thousand times more sensitive than that of humans, thanks to the collective behavior of more than 220 olfactory receptor cells in its nose, see Fig. 1.2(left), while the temporal resolution of the com- pound eye (Drosophila melanogaster) of a fruitfly eye is approximately ten times better than that of humans due to the collective output of 6400 photoreceptor cells, tightly bound to one another, see Fig. 1.2(right). These biologically-inspired ideas are model-independent, so we hope they can stimulate further development of other sensory systems.
The collaborative work has lead to a broad class of nonlinear devices been fabri- cated, including: highly-sensitive, low power, miniaturized magnetic- and electric- field sensors; networks of SQUID (Superconducting Quantum Interference Devices) for the development of a sensitive, low noise, significantly lower Size, Weight and Power (SWaP) antenna integrated with Low-Noise Amplifier (LNA); arrays of gyro- scopeswith reduced phase drift for improved navigation; frequency up/down convert- ers; a broad-spectrum analyzer with the ability to receive complex signals containing multiple frequencies and instantaneously lock-on or respond to a received signal in a few oscillation cycles; networks of energy harvesters with increased power output; an inexpensive precision timing device through coupled nonlinear oscillators circuits; and networks of spin-torque nano-oscillators for generating microwave signals at the nano scale. Figure1.3 shows a composite snapshot of the multiple projects outlined above. To provide readerswith a description of the translational work, involving, con-
1.1 Complex Networks 3
Fig. 1.2 Biological sensors made up of coupled cell systems. (Left) A dog’s smell is thousand times more sensitive than that of humans thanks to the collective behavior of more than 220 olfac- tory receptor cells. (Picture courtesy of http://www.faithfool.wordpress.com). (Right) The temporal resolution of a firefly is ten times better than in humans due to the collective output of approxi- mately 6400 photoreceptor cells tightly coupled to one another. Picture courtesy of “Research Penn State” [306]
Fig. 1.3 Representative projectswith a unifying theme: to exploit the collective behavior of complex network systems to model, analyze and design advanced engineering systems. From top-right, in counter-clockwise direction, coupled nonlinear oscillators for antennas; highly-sensitive coupled- core fluxgate magnetometer; networks of vibratory gyroscopes with optimal phase drift; thousands of superconductive loops for communication systems; arrays of spin-torque nano oscillators for microwave signal generation; and networks of energy harvesting systems
ceptualization, derivation of mathematical models, analysis, design and fabrication of these technologies, is the main goal of this book.
The technologies that are derived from the proposed network-based configuration can be effectively described as hybrid, discrete-continuous, nonlinear dynamical system: a discrete number of individual components, in which each unit is governed by a continuousmodel of nonlinear differential equations, physically coupled to form the network structure. These networks can also be described as complex networks in the sense that their response cannot be achieved by individual units, i.e., only by the mutual interaction of the individual components.
Although a direct relation to biological systems has yet to be demonstrated inmore detail, we show proof of concept that a network-based approach can indeed lead to significant performance enhancements for certain classes of sensors. In particular: fluxgatemagnetometers, electric-field sensors, and SQUID sensors. But the emphasis of the proposed approach onmagnetic and electric-field sensors is not exhaustive. The basic principles of cooperative behavior can be applied to enhance the performance of a wide range of nonlinear systems, such as antennas and radar systems; gyroscopes, energy harvesters; precision timing devices and microwave signal generators. By cooperative we mean patterns of collective behavior. For instance, we show that complete synchronization, in which all units oscillate in-phase with the same wave form and amplitude, can lead to significant reduction in the phase drift that affects the performance of gyroscopes. On the other hand, heteroclinic cycles are found to be ideal patterns to increase sensitivity in networks of fluxgate magnetometers. Similarly, traveling wave patterns can be exploited to improve the performance of precision timing devices. Which pattern of behavior is better suited for a particular application is a model dependent issue that has to be studied on a case-by-case basis. However, the existence of these, and many other related patterns of collective behavior are governed, mainly, by a common feature, symmetry.
In all the systems that are described in this book, our objectives include to inves- tigate the following model-independent issues:
(i) To investigate, analytically, computationally, and experimentally, the interplay between the discrete and continuous characteristics of the network dynamics.
(ii) To determine the fundamental limit of performance enhancement, e.g., reduc- tion in phase drift or increase in sensitivity, that can be achieved with a network- based device.
(iii) To develop design rules for nonlinear devices.
To accomplish these goals, we draw on methods from dynamical systems and bifurcation theory in systemswith symmetry to address someof the following,model- dependent, fundamental issues:
• What is the optimal network architecture that can produce the highest-level of performance enhancements at the lowest cost and with the lowest power con- sumption?
• How can a network be built in hardware? What are the advantages and disad- vantages of different hardware implementations, and how do they compare to simulations in software?
1.1 Complex Networks 5
• How can a network be programmed and controlled to operate in realistic (noisy) environments?
Since the majority of the technologies that are discussed in this book employ nonlinear oscillator components, we present next a brief introduction of basic ideas, concepts and principles.
1.2 Bistability
Many complex systems, natural and artificial ones, exhibit oscillatory behavior, i.e., cyclic behavior that repeats at regular intervals. Examples include: the rhythmic light pulses of fireflies [53, 111], see Fig. 1.4, the electrical activity of neuron cells that make up central pattern generators in biological systems [63, 83, 85, 148, 226, 227, 414], the patterns of lights produced by arrays of coupled lasers [328, 419], voltage variations in modern communication systems [315, 426], the growth and decay of population sizes between competing species [273–275], bubble formation and evolution in fluidization and mixing processes [163], and variations in phase and current in arrays of Josephson junctions [18, 99, 100, 160] in quantum physics.
In the absence of noise, the underlying cyclic fluctuations in a given system can arise from individual units that oscillate on their own, also known as endogenous or self-excited oscillators, or from exogenous units that oscillate only when they are externally driven or coupled together. Circadian rhythms, which regulate the daily cycle of many living organisms, plants, and animals, for instance, are endogenously
Fig. 1.4 Complex interactions among fireflies can lead them to coordinate the rhythmic flashing lights produced by each individual firefly. Collectively, the swarm can then achieve synchronization and oscillate in unison. Source National Geographic
6 1 A Unifying Theme
generated. In fact, the first endogenous circadian oscillation to be observed was the movement of the leaves of Mimosa pudica, a plant studied by the French scientist Jean-Jacques dÒetous de Mairan.1
In addition, bistability—the property that allows a system to rest in either of two states—underlies the basic oscillatory behavior of many other natural and artificial systems. Statesmay include typical invariant sets, such as equilibriumpoints, periodic and quasi-periodic solutions, and chaotic attractors. In the absence of an external stimulus, the state variable x(t) of a bistable system will relax to one of the invariant sets, and it will remain in that state unless it is switched or forced to another state. It is in this sense that the system exhibits “memory.” Which invariant set the system will relax to depends typically on the set of initial conditions. All bistable systems employ some form of energy source as the underlying principle that allows them to switch between states. The source of energy is due typically through external forcing or through the coupling mechanism.
For instance, dynamic sensors [150, 210, 252, 326, 334], operate as exoge- nous oscillators with nonlinear input-output characteristics, often corresponding to a bistable potential energy function of the form
dx
dt = −∇U (x), (1.1)
where x(t) is the state variable of the natural system or artificial device, e.g., mag- netization state, and U is the bistable potential function. Examples include: fluxgate magnetometers [45, 139], ferroelectric sensors [27], and mechanical sensors, e.g., acoustic transducers made with piezoelectric materials. Fig. 1.5(top) illustrates the case of a double-well potential function U (x) = −ax2 + bx4, whose minima are located at ±xm and the height of the potential barrier between the two minima is labeled by U0.
Without an external excitation (periodic forcing or noise), the state point x(t) of the exogenous oscillator described by Eq. (1.1) will rapidly relax to one of two stable attractors, which correspond to the minima of the potential energy function U (x). In the presence of an external periodic forcing term f (t), with frequency ω, the state variable in U (x + f (t)) can be induced to oscillate periodically (with a well- defined waveform) between its two stable attractors−xm and+xm , as is illustrated in Fig. 1.5(bottom). The forcing term is also known as biasing signal in the engineering literature.
Standard Detection Mechanism. To detect a small target signal (dc or low- frequency), typically the standard, spectral-based [323, 335, 336, 338], readout mechanism is employed. Assume ε to be the target signal. When ε = 0, the power spectral density contains only the odd harmonics of the bias frequency ω. But when ε > 0, the potential energy function U (x + f (t) + ε) is skewed, resulting in the appearance of even harmonics; the response at the second harmonic 2ω is then used
1Source: Wikipedia https://en.wikipedia.org.
1.2 Bistability 7
Fig. 1.5 (Top) Bistable Potential U (x) = −ax2 + bx4. (Bottom) Switching between wells of a potential function can be achieved by a sufficiently large biasing signals (or noise) greater than the potential barrier
x
U0
to detect and quantify the target signal, as is shown in Fig. 1.6. The standard readout mechanisms has some drawbacks, however. Chief among them is the requirement of a large onboard power to provide a high-amplitude, high-frequency, bias signal. The feedback electronics can also introduce their own noise floor into the measurement process, and finally, a high-amplitude, high-frequency, bias signals often increase the noise floor of the system.
In biological systems, bistability is a key feature for understanding and engineer- ing cellular functions such as: storing and processing information by the human brain during the decision-making process [6]; regulation of the cell cycle [368, 400]; sporu- lation, which controls the timing and dynamics of dramatic responses to stress [406]; design and construction of synthetic toggle switches [131]; and in gene regulatory networks responsible for embryonic stem cell fate decisions [78]. In chemical sys-
8 1 A Unifying Theme
0.2 0.4 0.6 0.8 1 10−8
10−6
10−4
10−2
10−6
10−4
10−2
P S
D (f
) Fig. 1.6 Power Spectrum Decomposition of the oscillations in a bistable overdamped system subject to a periodic forcing. (Left) In the absence of an external signal the PSD shows only even harmonics. (Right) In the presence of an external signal, however, the PSD exhibits odd as well as even harmonics. Typically, the first even harmonic is used as a detection mechanism
tems bistability is central to the analysis of relaxation kinetics [422]. In mechanical systems, bistablemechanisms are commonly employed in the design and fabrications ofMicro-Electro-Mechanical-Systems (MEMS) versions of relays, valves, clips, and threshold switches [175, 325]. In electronics, hysteresis and bistability are combined to design and fabricate Schmitt trigger circuits, which convert analog input signals to digital output signals [353]. In neuroscience, bistability is at the core of Hopf models [86, 178, 180], which describe the input/output response of neurons through differential equations of the form
τi dVi
dt = −Vi + g(Vi ), (1.2)
where τi is a suitable time constant that controls how quickly unit(neuron) i responds to a stimulus, Vi is the output (typically voltage) of unit i , and g is the activation function, which normally represents a saturation nonlinearity property of neurons.
Alternatively, the phenomenon of Stochastic Resonance [127, 128, 171, 204, 206, 277] can also serve as a mechanism for generating oscillations, see Fig. 1.7. Briefly speaking, stochastic resonance refers to noise-induced transitions which, due to the nonlinear nature of a system dynamics, become synchronized with the period of an external force. A distinctive feature of stochastic resonance is a rapid increase in a system’s output Signal-to-Noise Ratio (SNR) under weak coupling followed by a slower decrease in SNR for stronger coupling. At intermediate noise intensities, the system exhibits maximum SNR. This feature is inherently due to the nonlinear nature of a system’s dynamics and cannot be reproduced by linear systems.
1.3 Self-oscillating Networks 9
Fig. 1.7 (Left) Motion of a particle on a double-well potential function. In the absence of noise, the system dynamics quickly settles into an equilibrium point. Which equilibrium point is selected depends on the initial conditions. (Right) In the presence of noise, the system dynamics now lingers intermittently between the two equilibrium states of the deterministic system, independently of initial conditions
1.3 Self-oscillating Networks
A fundamental idea in the new paradigm that we have developed for performance enhancement in nonlinear systems is simple: make exogenous oscillators behave as endogenous ones. That is, we seek to minimize power consumption, and simulta- neously enhance output response, by replacing force-driven exogenous oscillators with self-oscillating networks. The paradox can be resolved through symmetry. This point deserves a little more explanation.
Many systems are known to oscillate only when they are driven by an external force. However, when they are connected in some fashion, the symmetry of the resulting topology of connections, i.e., which units are coupled with each other, and the nonlinear characteristics of each individual unit, can be exploited to induce the interconnected network to generate a collective pattern of oscillation via an appropriate coupling function, see Fig. 1.8.
From a mathematical standpoint, the choice of coupling function can be any type of function, leading to a wide range of network solutions. From an engineering standpoint, the coupling function is restricted, however, by the type of system or technology being used. This is, in other words, a model-dependent feature of the system. For instance, fluxgate magnetometers coupled through magnetic flux are restricted to unidirectional coupling since directing magnetic flux in both directions can be extremely complicated to achieve. Mechanical gyroscopes can be coupled, however, bidirectionally through a series of mass-spring systems.
An immediate advantage of the self-oscillating network approach is the cost reduc- tion that can be achieved by eliminating power-driven components which are typi- cally very expensive. A more subtle advantage is to exploit the collective pattern of oscillation to enhance output response. Consider for instance a sensor device (mag- netic or electric) whose behavior (as is described earlier) is governed by a potential
10 1 A Unifying Theme
2
1
5
−0.5
0
0.5
1
Time
0 0.2 0.4 0.6 0.8 1 10−5
100
105
Frequency
D Fig. 1.8 (Left) Representative example of self-oscillating networks. Individually, each unit cannot oscillate. (Right) Under certain coupling topologies, and the internal dynamics of each unit, the network can, collectively, oscillate
function of the form (1.1). A self-oscillating network would eliminate the need of the large onboard power that is required for a biasing signal to overcome the energy- barrier of the system and thus induce the required oscillations. Furthermore, the network oscillations can yield (under certain conditions) wave forms whose char- acteristics (frequency, amplitude, and phase) can exhibit higher sensitivity to very small target signals.
The symmetry-based approach to study nonlinear systems, while nonstandard, is not entirely new among the mathematics community. However, there is much less familiarity with the techniques of symmetry-breaking bifurcation, and applications of equivariant bifurcation theory, developed by Golubitsky and Stewart [146, 147, 149], as they apply to the engineering, design and fabrication, of complex systems. For this reason, we dedicate the next few sections to introduce, first, the mathemati- cal formalism to describe symmetry-breaking bifurcations; networks architecture of interconnected devices, through the coupled cell formalism and, then, a more com- prehensive discussion of the existence of heteroclitic cycles in nonlinear systems.
1.4 The Role of Symmetry
A general approach for the analysis of complex network systems has been to derive a detailed model of its individual parts, connect the parts and note that the system contains some sort of symmetry, then attempt to exploit this symmetry in order to simplify numerical computations. This approach can result in very complicated models that are difficult to analyze even numerically.
Our approach in this book is, however, aimed at promoting a unifying theme for the use of symmetry in a systematic way. First, determine the conditions for
1.4 The Role of Symmetry 11
the existence and stability of collective patterns of behavior of a complex system. Then perform translational work to transfer theorems and principles into prototypes devices that conform, as close as possible, to a particular model. For instance, the phenomenon of coupling-induced oscillations that regulates the dynamics of the newly postulated coupling-based sensor devices is dictated, mainly, by the groupZN , of cyclic permutations of N objects, i.e., units connected with a preferred direction or directionally. It is in this sense that the ideas and methods of our approach are device-independent so that similar principles can be readily applied to understand, and hopefully, to enhance the performance of a wide variety of sensor devices so long as the symmetry conditions are satisfied.
A few more details of how symmetry can appear in governing equations, such as in systems of differential equations, are now in order.
Definition 1 Symmetry is a geometrical concept that describes the set of transfor- mations that leave an object unchanged.
In complex systems with continuous nonlinear behavior, the objects are the gov- erning equations,which typically consist of systemsof ordinary differential equations (ODEs) or partial differential equations, and the transformations are the changes in the underlying variables that leave the equations unchanged.More formally, consider the following system of ODEs
dx
dt = f (x,λ), (1.3)
where x ∈ Rn , λ ∈ Rp is a vector of parameters and f : Rn × Rp → R is a smooth function. Let γ be a particular time-independent transformation inRn . Direct substi- tution of the transformed variable γx into (1.3) yields γ x = f (γx,λ). Consequently, for the system of ODEs (1.3) to remain unchanged, f must commute with γ. In other words, f (γx) = γ f (x). In practice, the set of all transformations that commute with f forms a group Γ . We then arrive to the following formal definition.
Definition 2 A system of ODEs such as (1.3) is said to have Γ -symmetry if
f (γx,λ) = γ f (x,λ), (1.4)
for all x ∈ Rn and for all γ ∈ Γ .
Equation (1.4) also implies that f is Γ -equivariant. But more importantly, it implies that if x is a solution of (1.3) then so is γx(t) for all γ ∈ Γ . In fact, the collection of points γx(t), for all γ ∈ Γ forms a set called the group orbit of Γ :
Γ x = {γ x : γ ∈ Γ }.
Furthermore, the concept of group orbit applies to any point x(t), not just equi- librium solutions.
12 1 A Unifying Theme
Fig. 1.9 Circuit realization of a Van der Pol oscillator. The governing equations exhibit reflectional symmetry with respect to the state variables
Example 1 As an example, consider the van der Pol circuit depicted in Fig. 1.9. IL
and IC are the currents across the inductor L and capacitor C , respectively. IR is the current across two resistors R1 and R2 located inside the rectangle labeled R in which F(V ) = −V/R1 + V 3/(3R2
2). The dynamics of the circuit shown in Fig. 1.9, after rescaling, is governed by the
following second order scalar ODE
d2V
where δ = 1/(R2C), p = R2/R1, ω = 1/ √
LC . After a change of variables, we can rewrite the model equation (1.5) as a first order system of the form
dx
(1.6)
where x(t) = V (t). We can then find two transformations that leave this system unchanged: the identity transformation γ1 = id, where γ1(x, y) → (x, y), and a second transformation, which can be described abstractly as γ2 = −1, so that γ2(x, y) → (−x,−y). The identity transformation is always a symmetry of any system, while the second transformation γ2 corresponds to a reflection through the origin in the phase space R2. Furthermore, it can be shown that γ1 and γ2 are the only transformations that leave (1.6) unchanged. Together, γ1 and γ2 form the group Z2 = {γ1, γ2} of symmetries of the Van der Pol oscillator (1.6).
But individual solutions can also exhibit symmetry. For instance, consider equi- librium solutions. A solution xe of Eq. (1.3) is an equilibrium or steady-state if and only if
f (xe) = 0.
The symmetries of equilibrium points of a Γ -equivariant ODE form a subgroup of Γ , which we define next.
1.4 The Role of Symmetry 13
Definition 3 Let xe represent an equilibrium or steady-state solution of a Γ - equivariant system of ODEs. The symmetries of xe form the isotropy subgroup Σ of Γ , which is defined by
Σxe = {γ ∈ Γ : γ · xe = xe} . (1.7)
1.5 Symmetry-Breaking Bifurcations
Equivariant systems of ODEs always posses a trivial solution x0 whose isotropy subgroup is the entire group of symmetries of the model equations [146, 147, 149]. That is,Σx0 = Γ . But as parameters are varied the system can exhibit a new solution with less symmetry. That is, Σx ⊂ Γ . It is then said that the system has undergone a symmetry-breaking bifurcation. We consider in this chapter two types of symmetry- breaking bifurcations, steady-state and Hopf bifurcations. The former case leads to new equilibrium solutions while the latter to periodic oscillations. We describe next each of these two cases.
Steady-State Bifurcations. The iconic picture in Fig. 1.10 of the milk drop coronet by the pioneering work on speed photography by Harold E. Edgerton illustrates best the phenomenon of steady-state symmetry-breaking bifurcation. The pool of milk in its unperturbed or trivial state is symmetric under arbitrary rotations and reflections on a plane, which form the orthogonal group O(2). The perturbation by the droplet breaks, however, the O(2) symmetry of the trivial solution and it induces a crown-like shape with lesser symmetry. The 24-sided polygon that appears by joining the individual clumps now has D24-symmetry, where DN is the dihedral group of symmetries of an N -gon.
Example 2 Consider now Euler’s column buckling experiment, illustrated in Fig. 1.11. It consists of an elastic beam subjected by a compressive force. Upon a critical value of the compressive force the beam deforms into one of two buck- led states, either to the right or to the left. Which state actually appears depends on model-dependent features such as material imperfections or thermal fluctuations. According to Bernoulli–Euler beam theory, a mathematical model for the angle θ(t) between the undeformed road and the tangent of the deformed rod is
E Iθ ′′ (x) + P sin θ(x) = 0, (1.8)
where x is the material coordinate, E is the elastic modulus, I is moment of inertia, P is the compressive force and L is the length of the beam. Equation (1.8) pos- sesses reflectional symmetry θ → −θ, which is described by the group Γ = Z2. The unperturbed unbuckled state is the trivial solution with Γ -symmetry. Assuming boundary conditions θ(0) = θ(L) = 0, two nontrivial solutions are ± sin(πx/L). The isotropy subgroup of these solutions is the trivial group 1. It can be shown that these two solutions emerge through a pitchfork bifurcation off of the trivial solution,
14 1 A Unifying Theme
Fig. 1.10 This iconic picture of the milk drop illustrates the phenomenon of symmetry-breaking bifurcations. An unperturbed pool of milk is invariant under arbitrary rotations and reflections on a plane, which form the orthogonal group O(2). The crown-like shape that emerges under the perturbation by the droplet is a 24-sided polygon whose symmetries are described by the dihedral group D24. Source Harold E. Edgerton, Milk Drop Coronet, 1957. 2010 Massachusetts Institute of Technology
Fig. 1.11 Euler beam experiment. An elastic beam is subjected to a compressive force. Upon reaching a certain threshold value of the compressive force, the trivial solution, unbuckled state, losses stability and a buckled state, right or left, emerges through a pitchfork bifurcation. Source Wikipedia
1.5 Symmetry-Breaking Bifurcations 15
i.e., via symmetry-breaking of Z2 symmetry. Now, if the beam were to be cylindri- cal instead of rectilinear then the group of symmetries of the experiment becomes Γ = O(2), just as in the milk drop experiment. But now the symmetry-breaking mechanism could lead, in principle, to a buckle mode with reflectional symmetry.
A critical concept in the symmetry-based analysis of differential equations is that of a representation of a group. Let Γ be a Lie group and V a vector space. A representation ofΓ is a homomorphism fromΓ to the group ofmatrices GL(V). That is, ρ : Γ → GL(V ). Thus, a group element γ ∈ Γ describes the abstract structure of the group, while ρ(γ) indicates how each group element acts on the objects, e.g., equations. It is common practice to refer to V as “the representation of γ”.
Absolute irreducibility is yet another concept that is used systematically to deter- mine the generic type of bifurcations that can occur in a symmetric system of ODEs. For instance, absolute irreducibility excludes the existence of purely imaginary eigen- values in the linearization of a model. Consequently, periodic oscillations cannot emerge via Hopf bifurcations for absolutely irreducible spaces. A brief definition follows but more details can be found in [149, 186].
Definition 4 A representation of a group Γ on a vector space V is absolutely irre- ducible if the only linear mappings on V that commute with Γ are scalar multiples of the identity.
It is also awell-known fact that symmetry forces systemsofODEs to have invariant linear subspaces. In particular, the fixed-point subspace of a solution is the invariant subspace where the isotropy subgroup acts trivially:
Definition 5 Suppose that Σ ⊂ Γ is a subgroup. Then the fixed-point subspace
Fix(Σ) = { x ∈ Rn : σx = x ∀σ ∈ Σ
}
is a flow invariant subspace [149].
Fixed point subspaces describe the regions of phase space where a particular solution resides. This suggests a model-independent strategy to find solutions of symmetric systems of ODEs. Restrict the equations to Fix(Σ) and then solve for the solutions. Since Fix(Σ) is, in general, lower dimensional that the entire space then it might be significantly easier to solve the restricted equations. A critical observation is the fact that it might be possible, under certain conditions, to predict the type of solutions of a symmetric system of ODEs without having to solve for the solutions. Details are formalized by the Equivariant Branching Lemma.
Theorem 1 (Equivariant Branching Lemma [149]) Let Γ ⊆ O(n) be a compact Lie group acting absolutely irreducibly on Rn. Let
dx
16 1 A Unifying Theme
be a Γ -equivariant bifurcation problem so that
f (0,λ) = 0 (d f )0,λ = c(λ)I.
Assume c′(0) = 0 and let Σ ⊆ Γ satisfy
dim Fix(Σ) = 1.
Then there exists a unique branch of solutions to f (x,λ) = 0 bifurcating from (0, 0), where the symmetry of the solution is Σ .
Example 3 Consider again the Euler beam experiment. Recall that the symmetries of the experiment are described by the group Γ = Z2. Furthermore, Σ = 1 is an isotropy subgroup in which Fix(Σ) = R, so that dim Fix(Σ) = 1. Thus, by the Equivariant Branching Lemma, we can predict the existence of a branch of steady- state solutions to the idealized model
x = f (x,λ),
with trivial symmetry Σ = 1. Indeed, a Lyapunov–Schmidt reduction [76] of the model Eq. (1.8) yields
x = λx − x3.
This reduced problem satisfies all conditions of the Equivariant Branching Lemma. The steady-state solutions with trivial symmetry are ±√
x . They emerge via a pitch- fork bifurcation.
Hopf Bifurcation. Symmetries of periodic solutions may arise in one of two forms. As purely spatial symmetries or as a combination of space and time symmetries. In the former case, a periodic solution x(t) is fixed at every moment in time by some γ ∈ Γ , so that γ is a purely spatial symmetry. This is similar to the symmetries of steady-states discussed above.
Now, in the latter case, the solution trajectory is fixed by a combination of the spatial action of γ ∈ Γ and a phase-shift θ ∈ S1, where S1 is the circle group of phase shifts acting on 2π periodic functions. That is,
(γ, θ) · x(t) = γx(t + θ) = x(t), ∀t.
As it was the case of steady-state solutions, both types of symmetries can be formally described through an extended version of the isotropy subgroup for periodic oscillations.
Definition 6 Let x(t) represent a periodic solution of a Γ -equivariant system of ODEs. The symmetries of x(t) form the isotropy subgroup, which is defined by
1.5 Symmetry-Breaking Bifurcations 17
} . (1.10)
Observe that the case θ = 0 corresponds to purely spatial symmetries. We will come back to discuss the issue of purely spatial symmetries in more detail a little later in the next section; within then context of coupled cell systems.
Generically, the existence of Hopf bifurcations in a symmetric system of ODEs is determined by Γ -simple irreducible representations. Formally,
Definition 7 A representation W of Γ is Γ -simple if either W is composed of two copies of an absolutely irreducible representation, so that W = V ⊕ V , or W is non-absolutely irreducible for Γ .
In either of these two cases, it can be shown [149, 186] that Γ -simple represen- tations lead to Jacobian matrices with the following structure
(d f )(0,0) = [ 0 −Im
Im 0
] , (1.11)
where m = n/2. We can now state the equivalent of the Equivariant Branching Lemma for Hopf
bifurcations with symmetry.
dx
dt = f (x,λ), x ∈ Rn, λ ∈ R (1.12)
be a Γ -equivariant bifurcation problem with Σ ⊂ Γ × S1. Assume the linearization of (1.12) satisfies Eq. (1.11) with eigenvalues σ(λ) ± w(λ)i , each of multiplicity m and also
σ′(0) = 0.
If the action of Γ is Γ -simple on Rn and Σ satisfies
dim Fix(Σ) = 2
then there exists a unique branch of periodic solutions to f (x,λ) = 0, with period near 2π, bifurcating from (0, 0), where the symmetry of the solution is Σ .
Example 4 (Premixed Flame Dynamics) A mixture of either isobutane and air, or propane and air, are burned on a circular porous plug burner in a low pressure (0.3– 0.5atm) combustion chamber. The process allowed for control of the pressure, flow rate, and fuel to oxidizer ratio to within 0.1%. The simplest cellular pattern generated by the burner is a large single cell with O(2) symmetry, as is shown in Fig. 1.12.
Changes in the experimental parameters (type of fuel, pressure, total flow, and equivalence ratio) resulted in different cellular patterns in the flame front. In
18 1 A Unifying Theme
Fig. 1.12 Combustion experiments conducted by M. Gorman et al., at the University of Houston [152] showcase cellular pattern instability. Simplest pattern that appears is a homogeneous flame front with the same O(2) symmetry as that of the circular burner
Fig. 1.13 Four snapshots from the two-cell state of the flame front that rotates clockwise. States with the opposite geometrical sense (i.e., related by reflections) rotate counter clockwise
particular, it has been shown [311] that rotating flames emerge via Hopf bifurcations that break theO(2) symmetry of the trivial or homogeneous flame front. Figure1.13 illustrates a two-cell state that rotates clockwise.
1.6 Coupled Cell Systems
Anaturalmathematical framework for the analysis of complex systems, such as those that consist of arrays of coupled nonlinear oscillators, is that of coupled cell system, Fig. 1.14 shows a representative example with N = 4 cells coupled unidirectionally (with varying coupling strength among different nodes) along a ring. By a “cell” we mean an individual component or unit that possesses its own dynamical behavior.
1.6 Coupled Cell Systems 19
Fig. 1.14 Representative example of a coupled cell system. Each “cell” or unit has its own internal dynamics governed and the cells are coupled to one another, in this case, unidirectionally with varying coupling strength λ j i between nodes j and i
f1 f2
λN3
In what follows, we assume N identical cells, and consider the internal dynamics of each cell to be governed by a k-dimensional continuous-time system of differential equations of the form
d Xi
dt = f (Xi ,λ), (1.13)
where Xi = (xi1, . . . , xik) ∈ Rk denotes the state variables of cell i and λ = (λ1, . . . ,λp) is a vector of parameters. Observe that f is independent of i because the cells are assumed to be identical. In engineering applications, such as in nonlinear antenna technology, for instance, it is common for the cell dynamics to be described by electrical oscillators, typically a Van der Pol oscillator [115, 166].
Definition 8 A network of N cells is a collection of identical cells interconnected in some fashion. We model the network by the following system of coupled differential equations
d Xi
ci j h(Xi , X j ), (1.14)
where h is the coupling function between two cells, the summation is taken over those cells j that are coupled to cell i , and ci j is a matrix of coupling strengths.
Additionally, if we let X = (X1, . . . , X N ) denote the state variable of the network, then we can write (1.14) in the simpler form
d X
dt = F(X),
where the dependence in the parameters λ has been omitted for brevity.
Local and Global Symmetries. Following the work of Dionne et al. [97, 98], we distinguish local symmetries from global symmetries as follows.
20 1 A Unifying Theme
Definition 9 Let O(N ) be the group of orthogonal transformations in Rn . Then L ⊂ O(k) is the group of local or internal symmetries of individual cells if, for all l ∈ L, we have
f (l Xi ,λ) = l f (Xi ,λ).
While local symmetries are dictated by f , global symmetries are described by the coupling pattern. More precisely,
Definition 10 G ⊂ O(N ) is the group of global symmetries of the network if, for all σ ∈ G, we have
F(σ X) = σF(X).
Considering again the case of Hopf bifurcation with symmetry, it turns out that the set of all spatial symmetries also forms a group:
K = {γ ∈ Γ : γX (t) = X (t) ∀t}.
In addition, there is also a subgroup H that preserves the overall trajectories of a periodic solution but not necessarily the phases. This group is defined as
H = {γ ∈ Γ : γ{X (t)} = {X (t)} ∀t}.
It is easier to explain the differences between these two subgroups within the context of a coupled cell system, as is shown in the next example.
Example 5 Consider the four-cell network of Fig. 1.15. The state of the entire net- work is described by the spatio-temporal pattern X (t) = (x(t), x(t + T/2), x(t), x(t + T/2)), where T is the common period of oscillations. Observe that exchang- ing cells x1 and x3 or cells x2 and x4 leaves the pattern unchanged. In other words, the network exhibits purely spatial symmetry generated by reflections across both diagonals. These generators form the group K = Z2 of purely spatial symmetries. Now if we were to permute the cells cyclically, i.e., x1 → x2, x2 → x3, x3 → x4 and x4 → x1, the overall combined trajectory of X (t) would remain the same, without paying attention to the phase differences. It follows that H = Z4 is the group of symmetries that preserves the trajectory. In this sense, H consists of the spatial parts (without phases) of the spatio-temporal symmetries of X (t). Together, (H, K ) define the spatio-temporal symmetries of the collective pattern, which can be identifiedwith the isotropy subgroup ΣX (t) that was defined earlier on.
While these examples are representative cases of symmetry-breaking bifurca- tions we should point out that it also possible for the changes in parameters to lead to symmetry-preserving bifurcations. This can occur, for instance, within the context of networks of interconnected nodes xi when all the individual nodes in the emerging pattern behave exactly the same, i.e., x1 = x2 = · · · = xN . If the bifurcation is of steady-state type then the emerging solution is classified as a symmetric equilib- rium. But if the nodes oscillate then we have symmetry-preserving Hopf bifurcation
1.6 Coupled Cell Systems 21
Fig. 1.15 Representative example of a four-cell network with identical nodes or cells coupled unidirectionally
x1
x(t)
x2
λ
λ
yielding synchronized oscillations Xsync = (x1(t) = x2(t) = . . . xN (t)). The syn- chronized solution is associatedwith the irreducible representationV = [1, 1, . . . , 1] and its isotropy subgroup is the original group of symmetries, i.e.,ΣXsync = Γ , which explains why the bifurcation is called symmetry-preserving.
Further example of Hopf symmetry-breaking patterns of oscillations will be described later on in Chap.10 within the context of coupled crystal oscillators for precision timing devices.
1.7 Heteroclinic Connections
In addition to the advantages of low power consumption that can be achieved by coupling-induced oscillations, the presence of symmetry in coupled nonlinear devices can lead to non-generic behavior, e.g., heteroclitic cycles, which can be exploited to enhance further sensitivity performance. In this case, the cycle involves solution trajectories that connect sequences of equilibrium points. Near the onset of the cycle, infinite-period oscillations emerge. Then, due to the large-period of oscil- lation, the presence of a target signal can create a large asymmetry, thus rendering a heteroclinicwaveformhighly sensitive to detect target signals via symmetry-breaking effects. Heteroclinic cycles are said to be non-generic features of nonlinear systems because they typically do not exist or it is very difficult to produce them. The presence of symmetry can lead, however, to invariant subspaces of the phase of a dynamical system through which cyclic connections are facilitated. We make extensive use of these type of solutions, so this section is dedicated to describing heteroclinic cycles and their existence in greater detail.
Definition 11 In simple terms, a heteroclinic cycle is a collection of solution tra- jectories that connects sequences of equilibria, periodic solutions, and/or chaotic sets [63, 122, 238–240, 278]. As time evolves, a typical nearby trajectory stays for increasingly longer periods near each solution before it makes a rapid excursion to the next solution.
For a more precise description of heteroclinic cycles and their stability, see Mel- bourne et al. [278], Krupa and Melbourne [240], the monograph by Field [122], and
22 1 A Unifying Theme
the survey article by Krupa [239]. The existence of structurally stable heteroclinic cycles is considered a highly degenerate feature of both types of systems, continuous and discrete. In other words, typically they do not exist. In continuous systems, where the governing equations normally consist of systems of differential equations, it is well-known that the presence of symmetry can, however, lead to structurally stable, asymptotically stable, cycles [121, 156]. First, symmetry forces certain subspaces of the phase-space to be invariant under the governing equations. Then, cycles are formed through saddle-sink connections between equilibria and/or periodic solutions that lie on the invariant subspaces. Since saddle-sink connections are structurally sta- ble so are the cycles. Homoclinic cycles, on the other hand, are a specific case of heteroclinic cycles in which the sequence of connections joins invariant solutions (equilibria, periodic solutions or chaotic sets) which belong to the same group orbit. Why are heteroclinic connections important to nonlinear sensor devices?
From a dynamical systems point of view, it is desirable to construct symmet- ric networks of sensors that can produce oscillations via symmetry-breaking global bifurcations of heteroclinic connections because near a heteroclinic cycle the emer- gent oscillations tend to have a very large period of oscillation, which renders their waveform highly sensitive to symmetry-breaking effects caused by external signals. Indeed, as one gets closer to the bifurcation point of the oscillations, their period increases exponentially and so does the symmetry-breaking response. This effect offers the possibility of significantly increasing the sensitivity response of a network- based sensor device by careful-tuning its operation close to the onset of oscillations. These facts have lead us to consider a new readout mechanism referred to as Res- idence Times Detection or (RTD) mechanism, as is described in Chap.2, Sect. 2.5. Next we present a few more details about the existence and stability of heteroclinic connections.
Finding Heteroclinic Cycles. For systems whose symmetries are described by the continuous group O(2), i.e., the group of rotations and reflections on the plane, Armbruster et al. [16] show that heteroclinic cycles between steady-states can occur stably, and Melbourne et al. [278] provide a method for finding cycles that involve steady-states as well as periodic solutions. Let Γ ⊂ O(N ) be a Lie subgroup (where O(N ) denotes the orthogonal group of order N ) and let g : RN → RN be Γ -equivariant, that is,
g(γX) = γg(X),
d X
dt = g(X).
Note that N = kn in an n-cell system with k state variables in each cell. Equivari- ance of g implies that whenever X (t) is a solution, so is γX (t). Using fixed-point subspaces, Melbourne et al. [278] suggest a method for constructing heteroclinic cycles connecting equilibria. Suppose that Σ ⊂ Γ is a subgroup. Then the fixed- point subspace
1.7 Heteroclinic Connections 23
Fig. 1.16 Pattern inside lattice of subgroups that suggests the existence of heteroclinic cycles
Fix(Σ) = {X ∈ RN : σX = X ∀σ ∈ Σ}
is a flow invariant subspace. The idea is to find a sequence of maximal subgroups Σ j ⊂ Γ such that dim Fix(Σ j ) = 1 and submaximal subgroups Tj ⊂ Σ j ∩ Σ j+1
such that dim Fix(Tj ) = 2, as is shown schematically in Fig. 1.16. In addition, the equilibrium in Fix(Σ j ) must be a saddle in Fix(Tj ) whereas the equilibrium in Fix(Σ j+1) must be a sink in Fix(Tj ).
Such configurations of subgroups have the possibility of leading to heteroclinic cycles if saddle-sink connections between equilibria in Fix(Σ j ) and Fix(Σ j+1) exist in Fix(Tj ). It should be emphasized that more complicated heteroclinic cycles can exist. Generally, all that is needed to be known is that the equilibria in Fix(Σ j )
is a saddle and the equilibria in Fix(Σ j+1) is a sink in the fixed-point subspace Fix(Tj ) (see Krupa andMelbourne [240]) though the connections can not, in general, be proved. Since saddle-sink connections are robust in a plane, these heteroclinic cycles are stable to perturbations of g so long as Γ -equivariance is preserved by the perturbation. For a detailed discussion of asymptotic stability and nearly asymptotic stability of heteroclinic cycles, which are also very important topics, see Krupa and Melbourne [240].
Near points of Hopf bifurcation, this method for constructing heteroclinic connec- tions can be generalized to include time periodic solutions as well as equilibria. Mel- bourne, Chossat, and Golubitsky [278] do this by augmenting the symmetry group of the differential equations with S1 — the symmetry group of Poincare–Birkhoff normal form at points of Hopf bifurcation—and using phase-amplitude equations in the analysis. In these cases the heteroclinic cycle exists only in the normal form equations since some of the invariant fixed-point subspaces disappear when symme- try is broken. However, when that cycle is asymptotically stable, then the cycling like behavior remains even when the equations are not in normal form. This is proved by using asymptotic stability to construct a flow invariant neighborhood about the cycle and then invoking normal hyperbolicity to preserve the flow invariant neigh- borhood when normal symmetry is broken. Indeed, as is shown by Melbourne et al. [278], normal form symmetry can be used to produce stable cycling behavior even in systems without any spatial symmetry. More generally, it also follows that if an asymptotically stable cycle can be produced in a truncated normal form equation (say truncated at third or fifth order), then cycling like behavior persists in equations with higher order terms— even when those terms break symmetry— and the cycling like behavior is robust.
24 1 A Unifying Theme
X
Z
Y
(a)
0 100 200 300 400 500 600 700 800 900 1000 0
0.5
1
1.5
x
0 100 200 300 400 500 600 700 800 900 1000 0
0.5
1
1.5
y
0 100 200 300 400 500 600 700 800 900 1000 0
0.5
1
1.5
z
t
(b)
Fig. 1.17 Heteroclinic cycle found between three equilibrium points of the Guckenheimer and Holmes system. a Saddle-sink connections in phase-space, b Time series evolution of a typical nearby trajectory. Parameters are: μ = 1.0, a = 1.0, b = 0.55, c = 1.5
The Guckenheimer–Holmes Cycle. Figure1.17 illustrates a cycle involving three steady-states of a system of ODE’s proposed by Guckenheimer and Holmes [156]. Observe that as time evolves a nearby trajectory stays longer on each equilibrium.
The group Γ in this example has 24 elements and is generated by the following symmetries
(x, y, z) → (±x,±y,±z) (x, y, z) → (y, z, x)
Note that, in fact, this is a homoclinic cycle since the three equilibria are on the group orbit given by the cyclic generator of order 3. The actual system of ODE’s can be written in the following form
x1 = μx1 − (ax2 1 + bx2
2 + cx2 3 )x1
3 + cx2 1 )x2
1 + cx2 2 )x3.
In related work that describes cycling chaos, Dellnitz et al. [94] point out that the Guckenheimer–Holmes system can be interpreted as a coupled cell system (with three cells) in which the internal dynamics of each cell is governed by a pitchfork bifurcation of the form
xi = μxi − ax3 i ,
where i = 1, 2, 3 is the cell number. As μ varies from negative to positive through zero, a bifurcation from the trivial equilibrium xi = 0 to nontrivial equilibria xi = ±√
μ occurs. Guckenheimer and Holmes [156] show that when the strength of the remaining terms in the system of ODE’s (which can be interpreted as cou- pling terms) is large, an asymptotically stable heteroclinic cycle connecting these bifurcated equilibria exists. The connection between the equilibria in cell one to the
1.7 Heteroclinic Connections 25
equilibria in cell two occurs through a saddle-sink connection in the x1x2−plane (which is forced by the internal symmetry of the cells to be an invariant plane for the dynamics). As Dellnitz et al. (1995) further indicate, the global permutation sym- metry of the three-cell system guarantees connections in both the x2x3-plane and the x3x1-plane, leading to a heteroclinic connection between three equilibrium solutions.
A Cycle in a System with Circular Symmetry. Melbourne et al. [278] prove the existence of robust, asymptotically stable heteroclinic cycles involving time periodic solutions in steady-state/Hopf and Hopf/Hopf mode interactions in systems with O(2)-symmetry. In these symmetry breaking bifurcations each critical eigenvalue is doubled by symmetry—so the centermanifold for a steady-state/Hopfmode interac- tion is six-dimensional and for a Hopf/Hopf mode interaction it is eight-dimensional. It is well known that O(2) symmetry-breaking Hopf bifurcations at invariant equi- libria lead to two types of periodic solutions: “standing waves” (solutions invariant under a single reflection for all time) and ”rotating waves” (solutions whose time evolution is the same as spatial rotation).
Figure1.18 shows a cycle connecting a steady-state with a standingwave obtained from a steady-state/Hopf mode interaction by numerically integrating a general sys- tem of ODE’s with O(2)-symmetry, which has the form
Fig. 1.18 Heteroclinic cycle connecting a steady-state with a standing wave in a system with O(2) × S1 symmetry with S1 symmetry due to normal form. Parameters are: c1 = λ − 1.5ρ − 4N , c2 = 1.3, c3 = −9, p1 = 1.2λ − 3ρ − N , p2 = 4, p3 = 4, q1 = 0.8λ + 7, and all other coefficients set to zero
26 1 A Unifying Theme
dz
P(z) = P1
[ z1 z2
] ,
where δ = |z2|2 − |z1|2, C j = c j + iδc j+1, c j are real-valued O(2) × S1-invariant functions and P j = p j + q j i are complex-valued O(2) × S1-invariant functions depending on two parameters λ and μ. The time series in this figure are taken from three different coordinates: x0 is a coordinate in the steady-state mode and x1, x2 are coordinates in the Hopf mode. In these coordinates a sta