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A large class of dynamic sensors have nonlinear input-output characteristics, often corresponding to a bistable potential energy function that controls the evolution of the sensor dynamics. These sensors include magnetic field sensors, e.g., the simple fluxgate magnetometer and the superconducting quantum interference device (SQUID), ferroelectric sensors and mechanical sensors, e.g., acoustic transducers, made with piezoelectric materials. Recently, the possibilities offered by new technologies and materials in realizing miniaturized devices with improved performance have led to renewed interest in a new generation of inexpensive, compact and low-power fluxgate magnetometers and electric-field sensors. In this article, we review the analysis of an alternative approach: a symmetry-based design for highly-sensitive sensor systems. The design incorporates a network architecture that produces collective oscillations induced by the coupling topology,

Symmetry in nonlinear dynamical systems can force certain regions of their phase-space to be invariant under the governing equations. That is, any solution trajectory that starts in one of those regions remains there forever. Furthermore, it is well-known that the presence of invariant regions in continuous dynamical systems can lead to cyclic trajectories that connect, through these invariant subspaces, steady states, periodic oscillations and even chaotic sets [

In this review article, we provide a self-contained description of the basic ideas and results of the modeling and analysis of this new paradigm. The description is aimed mainly at networks of fluxgate magnetometers coupled via magnetic flux. The network architecture is a ring with a preferred orientation, _{N}_{N}

The review article is organized as follows. In Section 2, we present a brief overview of the basic definition of heteroclinic cycles, followed by a few representative examples from ordinary and partial differential equations. In Section 3, we present a review of the bifurcation analysis for the model equations that govern the behavior of the networks of magnetic- and electric-field sensors. Similarities and differences between the bifurcations and response of these two systems are presented in great detail, followed by a discussion of the sensitivity response. In Section 4, the results of experimental works are presented. For brevity, only the magnetic field sensor array is described. The delay introduced by electronic equipment has positive effects, since it tends to increase the basin of attraction of the global oscillations. Details can be found in the references. Finally, some concluding remarks are included in Section 5.

Loosely speaking, a heteroclinic cycle is a robust collection of solution trajectories that connects sequences of equilibria, periodic solutions or chaotic invariant sets via saddle-sink connections. For a more precise description of heteroclinic cycles and their stability, see Melbourne

Melbourne ^{N} →^{N}

Note that _{j}_{j}_{j}_{j} ∩_{j}_{+1}, such that dim Fix(_{j}_{j}_{j}_{j}_{+1}) must be a sink in Fix(_{j}

Such configurations of subgroups have the possibility of leading to heteroclinic cycles if saddle-sink connections between equilibria in Fix(Σ_{j}_{j}_{+1}) exist in Fix(_{j}_{j}_{j}_{+1}) is a sink in the fixed-point subspace Fix(_{j}

Near the points of the Hopf bifurcation, this method for constructing heteroclinic connections can be generalized to include time periodic solutions, as well as equilibria. Melbourne, Chossat and Golubitsky [^{1}, the symmetry group of Poincare–Birkhoff normal form at points of Hopf bifurcation, and using the 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 symmetry is broken. However, when that cycle is asymptotically stable, then the cycling-like behavior remains, even when the equations are not in the normal form. In later work, Buono, Golubitsky and Palacios [_{n}

The group

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 three. The actual system of ODEs can be written in the following form:

In the related work that describes cycling chaos, Dellnitz _{i}

Guckenheimer and Holmes [_{1}_{2}-plane (which is forced by the internal symmetry of the cells to be an invariant plane for the dynamics). As Dellnitz _{2}_{3}-plane and the _{3}_{1}-plane, leading to a heteroclinic connection between three equilibrium solutions.

Overdamped bistable dynamics, of the generic form:
^{2}(^{−}^{1} ln(cosh(^{2} + ^{4}. Absent an external forcing term, the state-point

In its most basic form, a fluxgate magnetometer consists of two detection coils wound around a ferromagnetic core (usually a single core) in opposite directions from one another [

One must, however, take these performance quantifiers with the caveat that, when operated unshielded in a practical application, the detection of target signals above the noise floor is limited by the ambient magnetic field (which, in the case of the terrestrial magnetic field, can have non-stationary, as well as random components). Hence, various techniques often involving a reference magnetometer for the purposes of subtraction of the noise floor from the output of the target-sensing device must be employed if one wishes to take advantage of the low noise floor. Among recent advances in fluxgate sensor technology, the so-called “fluxset” devices [

A simple way to model the ferromagnetic core dynamics in a fluxgate is through an Ising-type model. We will assume the core to be composed of a set of atomic “spins” arranged on a regular lattice representing the crystal structure of the core [_{c}

A further simplification is to consider spin 1/2 magnetic materials, so that only two distinct orientations at each lattice point _{i}_{i}_{i}_{i}_{j}^{ext}_{ij}_{j}_{i}_{B}T_{B}_{i}_{i}_{ij}h_{j}^{ext}_{i}_{ij}^{ext}_{B}^{ext}^{2}

The model

A coupled-core fluxgate magnetometer (CCFM) is then constructed by uni-directionally coupling _{i}_{0} is an external DC “target” magnetic flux, _{0} being the energy barrier height (absent the coupling) for each of the elements (assumed identical for theoretical purposes); the parameter _{i}_{i}_{+1}. Effectively, the unidirectional coupling leads to a network with _{N}_{N}

Indeed, a bifurcation analysis [

The oscillations commence when the coupling coefficient exceeds a threshold value:
_{c}_{c}_{c}_{c}_{HB} ≤ λ ≤ λ_{c}

The bifurcation diagram for the

At the birth of the oscillations, the amplitudes are fully grown due to the global nature of the bifurcation. As _{c}_{c}

The individual oscillations (in each elemental response) are separated in phase by 2_{c} − λ

The summed output oscillates at period _{+} = _{i}/N

To measure an external signal, we rely on a readout mechanism: the residence times detection (RTD), which is based on a threshold crossing strategy of measuring the symmetry-breaking effect of an external signal. More specifically, RTD consists of measuring the “residence times” of the oscillations of the sensor device about the two stable states of the potential energy function _{+} = _{−}_{+} _{−}_{+}_{−}

The system responsivity, defined via the derivative

For small target signals, one may do a small-_{c}_{3} now represents the critical coupling obtained using the signal from _{3} as a reference point. Further details can be found in [

Laboratory experiments seem to indicate that the sensitivity of a CCFM-based system of fluxgates increases by simply alternating the orientation of each individual fluxgate [

In the absence of noise and of a target signal, _{i} vs_{i} vs

It follows that:

In other words, the sensitivity of the AO configuration improves, linearly, by a factor of _{1}

We expect noise in a coupled-core fluxgate magnetometer network to arise from three sources: a magnetic noise floor (due to the core material and, in particular, magnetic domain motion), contamination of the target signal and noise from the electronics in the coupling circuitry. In recent work, we studied, numerically, the effects of an additive magnetic noise floor [_{c}^{2}. This type of noise is a good approximation (except for a small 1

In general, we would expect somewhat different noise in each equation, since, realistically, the reading of the external signal _{i}_{i}_{i}_{i}_{i}_{i}_{c}_{c}_{c}_{i}

We now present first the results from the numerical simulations of the coupled system (_{i}_{M}_{K}

Finally, we address the issue of increasing the number of elements in the coupled array. Changing

We now consider the effects of signal contamination by noise. We assume the temperature-related parameter _{c}_{F}_{c}_{F}_{c}_{i}

Next, we study the response of the CCFM through the signal-to-noise ratio (SNR).

It is worth mentioning that these small fluctuations are also present in the more ideal case, wherein we take identical noise functions in each element of the dynamics. We now investigate the AO configuration, which, as suggested by the results of the preceding section, as well as our laboratory experiments, holds the promise of further performance enhancements.

Finally, we study the effects of noise on the CCFM with the AO configuration. We use the output of the “favored” element that gives the best deterministic RTD response (see the preceding section) in this configuration. Calculations of the SNR output of this “favored” element (not shown for brevity) show, at first glance, similar results to those of the standard-orientation configuration. A point-wise comparison between the SNR output of the two cases indicates, however, that the SNR output of (the favored element of) the AO configuration can be larger than that of the standard (CCFM) system (

Modeling and analysis of high-dimensional dynamical systems is usually focused on finding conditions for the existence and stability of typical invariant sets, _{N}

There are a number of methods for finding the many equilibria for System (_{ie}_{ie}_{i}_{ie}_{je}_{ie}_{je}

With these equilibria established for System (_{1}_{e}, x_{2}_{e}, x_{3}_{e}_{2}_{e}, x_{3}_{e}, x_{1}_{e}_{3}_{e}, x_{1}_{e}, x_{2}_{e}_{1}_{e}, −x_{2}_{e}, −x_{3}_{e}_{2}_{e}, −x_{3}_{e}, −x_{1}_{e}_{3}_{e}, −x_{1}_{e}, −x_{2}_{e}_{1}_{e}, x_{2}_{e}, x_{3}_{e}

Even though there are 27 equilibria, there are only six qualitatively different ones. The eight stable equilibria divide into two categories with the two symmetric equilibria and the six asymmetric equilibria. There are twelve asymmetric equilibria with 2D stable manifolds, which also divide into two categories. The first six are the ones that, when _{1} = _{2} = _{3}, starting at

Similarly, these equilibria are symmetrically related and share stability properties, since they are in the same (second) group orbit generated by Γ. The small curves track from gray to black as _{1}_{e}, x_{2}_{e}, x_{3}_{e}_{ie}_{ie}_{ie}_{je}

In summary, the 24 asymmetric equilibrium points can be arranged into one of three distinct group orbits. Thus,

We consider a system of _{0}:
_{i}

A bifurcation analysis [

The oscillations commence when the coupling coefficient exceeds a threshold value:
_{0}. This expression for the critical coupling is found to agree quite well with the results of numerical simulations on the coupled system (_{0} (compared to the energy barrier height _{0}). As was the case of the coupled magnetic sensors, the oscillations are again non-sinusoidal, with a frequency that increases as the coupling strength increases away from _{c}_{c}_{c}

The individual oscillations (in each elemental response) are separated in phase by 2

Increasing

The residence time difference in the two states is easily evaluated, via the summed response, as:
_{c}_{i}_{0}, as well as the critical coupling _{c}

Typically, the integrator output contains a DC component that must be removed before the signal is passed to the other fluxgates. This is accomplished by employing a Sallen–Key second-order high pass filter immediately after the integrator. The signal then passes through an amplifier to achieve adequate gain to drive the adjacent fluxgate. After this, the signal passes through a voltage-to-current converter (V-I converter) in its final step to drive the primary coil of the adjacent fluxgate. The setup repeats for the remaining two fluxgates, and all values of the coupling circuit parameters are closely matched from one set to the other.

The oscillations observed from this setup are quite striking; see

Over the past twelve years, we have conducted various analytical, computational and experimental works to investigate the fundamental idea that coupling-induced oscillations that emerge through heteroclinic connections can be exploited to develop a new generation of highly-sensitive, low-powered, dynamic sensors. In this review article, we use the fluxgate magnetometer, which is essentially a coil sensor with a ferromagnetic core, and a network of electric field sensors made up of overdamped Duffing oscillators as case studies to illustrate the basic ideas and methods to enhance sensor performance. Both sensor systems are coupled in a ring configuration with no preferred orientation,

In more recent years, we have expanded the ideas and methods to model, analyze and develop highly sensitive sensor systems beyond magnetic- and electric-field sensors. In particular, we have extensively studied networks of vibratory gyroscopes for navigation systems. Current prototype MEMS (micro-electro-mechanical systems) gyroscopes are compact and inexpensive to produce, but their performance characteristics, in particular drift rate, fail to meet the requirements for an inertial grade guidance system. As an alternative approach, a coupled inertial navigation sensor, made up of

In yet another project at the interface between symmetric dynamical systems and engineering, we have also shown a proof of concept that an array of vibratory energy harvesters, coupled mechanically and inductively, can produce, under certain conditions that depend mainly on the coupling strength, collective patterns of oscillations [

The authors acknowledge support from the Space and Naval Warfare Systems Command (SPAWAR) internal funding (In-house Laboratory Independent Research (ILIR) and Naval Innovative Science and Engineering (NISE) Programs) and the Office of Naval Research (Code 30). The work of Antonio Palacios was supported in part by National Science Foundation Grants CMMI-0638814 and CMMI-0625427.

Antonio Palacios conducted the mathematical modeling and theoretical analysis of governing equations; Visarath In conceived and designed the experiments; Patrick Longhini performed computational bifurcation analysis; Antonio Palacios and Visarath In and Patrick Longhini wrote the paper.

The authors declare no conflict of interest.

_{n}

Pattern inside a lattice of subgroups that suggests the existence of heteroclinic cycles.

Heteroclinic cycle found between three equilibrium points of the Guckenheimer and Holmes system. (

Typical hardware design of a fluxgate magnetometer.

Atomic spins in two states: (left) a paramagnet state and (right) a ferromagnetic state.

(Left) Bifurcation diagram for a system of three identical bistable elements coupled unidirectionally and without delay. Solid (dotted) lines indicate stable (unstable) equilibrium points. Filled-in (empty) circles represent stable (unstable) periodic oscillations. (Insert) Close-up view of the region of bistability between large-amplitude oscillations and synchronous equilibria. (Right) Family of limit cycles oscillations in phase space. The bold curve corresponds to oscillations very close to the onset of the heteroclinic connection.

Sensitivity response of an array of fluxgate magnetometers as a function of the number of cores _{c}

Residence times detection (RTD) response of a coupled-core fluxgate magnetometer (CCFM) as a function of ring size

Simulated modal oscillation (or switching) frequency _{M}^{2} (normalized potential barrier height _{M}_{K}

Frequency (Hz) response of simulations of a CCFM system, subject to parametric noise, as a function of coupling strength _{c}_{c}

Signal-to-noise ratio output of (a single element) of a CCFM system in the presence of parametric noise. The parameters are: _{c}

Difference in SNR response between AO and SO configurations. Observe that near the onset of coupling-induced oscillations, the SNR response of the AO configuration is significantly better. The parameters are: _{c}

There are 27 equilibria shown in this diagram at various values of

Montage of the geometric structure of the basins of attraction of a ring of bistable units connected unidirectionally.

(Left) Bifurcation diagram for a system of three identical bistable elements coupled unidirectionally and without delay. Solid (dotted) lines indicate stable (unstable) equilibrium points. Filled-in (empty) circles represent stable (unstable) periodic oscillations.

Prototype design of a coupled core fluxgate magnetometer with three fluxgates.

Flow diagram for the coupled fluxgate experiment. Each fluxgate consists of two coils, the sensing coil and the driving coil. Starting with Fluxgate 1, the signal from the sensing coil, first, goes through the current-to-voltage converter. Then, it passes through the “leaky” integrator, followed by a Sallen second-order filter before going through the main gain stage. Thereafter, the signal goes through the voltage-to-current converter, and then, it connects to the drive coil of the adjacent fluxgate (Fluxgate 2). The other two fluxgates are connected in the same manner.

Top: The numerical data for c = 4,