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## Über dieses Buch

This book bridges the current gap between the theory of symmetry-based dynamics and its application to model and analyze complex systems. As an alternative approach, the authors use the symmetry of the system directly to formulate the appropriate models, and also to analyze the dynamics. Complex systems with symmetry arise in a wide variety of fields, including communication networks, molecular dynamics, manufacturing businesses, ecosystems, underwater vehicle dynamics, celestial and spacecraft dynamics and continuum mechanics. A general approach for their analysis has been to derive a detailed model of their 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 highly complicated models that are difficult to analyze even numerically. The alternative approach, while nonstandard, is not entirely new among the mathematics community. However, there is much less familiarity with the techniques of symmetry-breaking bifurcation, as they apply to the engineering, design and fabrication, of complex systems, in particular, nonlinear sensor devices with special emphasis on the conceptualization and development of new technologies of magnetic sensors such as fluxgate magnetometers and SQUID (Superconducting Quantum Interference Devices), E-- (electric-field) sensors, and communication and navigation systems that require multiple frequencies of operation, such as radar and antenna devices as well as gyroscopic systems.

## Inhaltsverzeichnis

### Chapter 1. A Unifying Theme

Abstract
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.
Visarath In, Antonio Palacios

### Chapter 2. Coupled-Core Fluxgate Magnetometer

Abstract
Magnetic sensors are inherently nonlinear systems, which have been been used over many years to detect weak magnetic signals for a wide variety of applications.
Visarath In, Antonio Palacios

### Chapter 3. Microelectric Field Sensor

Abstract
In this chapter, we explore the underlying dynamics of a coupled bistable system realized by coupling microelectronic circuits which belong to the same class of dynamics as the ferromagnetic system studied in the previous chapter.
Visarath In, Antonio Palacios

### Chapter 4. Superconductive Quantum Interference Devices (SQUID)

Abstract
This chapter focuses on conducting an extensive computational investigation and mathematical analysis into the average voltage response of arrays of Superconducting Quantum Interference Devices (SQUIDs). These arrays will serve as the basis for the development of a sensitive, low noise, significantly lower Size, Weight and Power (SWaP) antenna integrated with Low-Noise Amplifier (LNA) using the SQUID technology. The goal for this antenna is to be capable of meeting all requirements for Guided Missile Destroyers (DDG) 1000 class ships for Information Operations/Signals Intelligence (IO/SIGINT) applications in Very High Frequency/Ultra High Frequency (V/UHF) bands.
Visarath In, Antonio Palacios

### Chapter 5. Frequency Conversion

Abstract
Theoretical and experimental works on arrays of coupled oscillators have shown that symmetry alone can lead to periodic patterns with “multiple frequencies” of oscillation [15, 63, 144, 147, 387]. In Ref. [15], for instance, it is shown that a ring of N oscillators possessing $$\mathbf{D}_N$$-symmetry (symmetry of an N-gon) can induce, under certain conditions, an external oscillator to oscillate at N times the collective frequency of the ring. The actual conditions require that the ring oscillates in a traveling wave pattern and that the cross-coupling be directed from the ring to the external oscillator, i.e., a master-slave system. An alternative approach, which is pursued in this chapter, is to use the existing theoretical works in [144, 146, 147, 149], which show (within the context of a coupled cell system) that $$\mathbf{Z}_N$$ symmetry-breaking Hopf bifurcations can also lead to multifrequency patterns. In particular, the multi-frequency patterns discussed in this chapter are motivated by an application: a “channelizer”. This is a broad spectrum analyzer that can instantaneously lock onto Radio Frequency (RF) signals of multiple frequencies, see Fig. 5.1.
Visarath In, Antonio Palacios

### Chapter 6. ANIBOT: Biologically-Inspired Animal Robot

Abstract
We present the first (to the best of our knowledge) circuit realization of an animal (quadruped) robot controlled by a Central Pattern Generator (CPG) network of neurons, whose model and design are biologically-inspired by the work of Golubitsky et al. (Physica D 115:56–72 (1998)), [148], (J Math Biol 42: 291–326 (2001)), [61], (J Math Biol 42:327–346 (2001)), [62], (The Symmetry Perspective. Birkháuser, Switzerland (2000)), [146].
Visarath In, Antonio Palacios

### Chapter 7. Gyroscope Systems

Abstract
In this chapter, we investigate further the model-independent ideas of coupling-induced oscillations for signal detection.
Visarath In, Antonio Palacios

### Chapter 8. Energy Harvesting

Abstract
Energy harvesting devices are key to a wide range of technologies: wireless sensor networks, micro wind turbine, thermoelectric generators, health monitoring systems, human-generated, see Fig. 8.1, power devices for biomedical applications and, of course, consumer products such as automatic wristwatches and TV remote controls, to the cloud.
Visarath In, Antonio Palacios

### Chapter 9. Spin Torque Nano Oscillators

Abstract
Synchronization of Spin Torque Nano Oscillators has been a subject of extensive research as various groups try to harness the collective power of STNOs to produce a strong enough microwave signal at the nanoscale. Achieving synchronization has proven to be, however, rather difficult for even small arrays while in larger ones the task of synchronization has eluded theorist and experimentalists altogether.
Visarath In, Antonio Palacios

### Chapter 10. Precision Timing

Abstract
Precise time dissemination and synchronization have been some of the most important technological tasks for several centuries. No later than Harrison’s time, it was realized that precise time-keeping devices having the same stable frequency and precisely synchronized can have important applications in navigation.
Visarath In, Antonio Palacios

### Backmatter

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