{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:24:19Z","timestamp":1760243059077,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T00:00:00Z","timestamp":1434672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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, i.e., which sensors are coupled to each other. Under certain symmetry groups, the oscillations in the network emerge via an infinite-period bifurcation, so that at birth, they exhibit a very large period of oscillation. This characteristic renders the oscillatory wave highly sensitive to symmetry-breaking effects, thus leading to a new detection mechanism. Model equations and bifurcation analysis are discussed in great detail. Results from experimental works on networks of fluxgate magnetometers are also included.<\/jats:p>","DOI":"10.3390\/sym7021122","type":"journal-article","created":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T10:23:33Z","timestamp":1434709413000},"page":"1122-1150","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Symmetry-Breaking as a Paradigm to Design Highly-Sensitive Sensor Systems"],"prefix":"10.3390","volume":"7","author":[{"given":"Antonio","family":"Palacios","sequence":"first","affiliation":[{"name":"Nonlinear Dynamical Systems Group, Department of Mathematics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA"}]},{"given":"Visarath","family":"In","sequence":"additional","affiliation":[{"name":"Space and Naval Warfare Systems Center Pacific, 53560 Hull Street, San Diego, CA 92151-5001, USA"}]},{"given":"Patrick","family":"Longhini","sequence":"additional","affiliation":[{"name":"Space and Naval Warfare Systems Center Pacific, 53560 Hull Street, San Diego, CA 92151-5001, USA"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1090\/S0002-9947-1980-0561832-4","article-title":"Equivariant dynamical systems","volume":"259","author":"Field","year":"1980","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1017\/S0305004100064732","article-title":"Structurally stable heteroclinic cycles","volume":"103","author":"Guckenheimer","year":"1988","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"036103","DOI":"10.1103\/PhysRevE.70.036103","article-title":"Emergent oscillations in unidirectionally coupled overdamped bistable systems","volume":"70","author":"Bulsara","year":"2004","journal-title":"Phys. Rev. E."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.physleta.2007.01.089","article-title":"Coupled-core fluxgate magnetometer: Novel configuration scheme and the effects of a noise-contaminated external signal","volume":"367","author":"Palacios","year":"2007","journal-title":"Phys. Lett. A."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"066121","DOI":"10.1103\/PhysRevE.73.066121","article-title":"Complex Behavior in Driven Unidirectionally Coupled Overdamped Duffing Elements","volume":"73","author":"In","year":"2006","journal-title":"Phys. Rev. E."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"045102","DOI":"10.1103\/PhysRevE.68.045102","article-title":"Coupling induced oscillations in overdamped bistable systems","volume":"68","author":"In","year":"2003","journal-title":"Phys. Rev. E."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1017\/S0308210500024173","article-title":"Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry","volume":"113A","author":"Melbourne","year":"1989","journal-title":"Proc. Roy. Soc. Edinb."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1017\/S0143385700008270","article-title":"Asymptotic stability of heteroclinic cycles in systems with symmetry","volume":"15","author":"Krupa","year":"1995","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_9","unstructured":"Field, M. (1996). Lectures on Bifurcations, Dynamics and Symmetry, Addison-Wesley Longman Ltd."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF02677976","article-title":"Robust heteroclinic cycles","volume":"7","author":"Krupa","year":"1997","journal-title":"J. Nonlin. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1137\/0521081","article-title":"Bifurcations of relative equilibrias","volume":"21","author":"Krupa","year":"1990","journal-title":"SIAM J. Math. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/S0167-2789(00)00097-X","article-title":"Heteroclinic cycles in rings of coupled cells","volume":"143","author":"Buono","year":"2000","journal-title":"Phys. D."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1142\/S0218127495000909","article-title":"Cycling chaos","volume":"5","author":"Dellnitz","year":"1995","journal-title":"Int. J. Bifurc. Chaos."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/0924-4247(92)80159-Z","article-title":"Review of fluxgate sensors","volume":"33","author":"Ripka","year":"1996","journal-title":"Sens. Actuators A."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/S0924-4247(03)00094-3","article-title":"Advances in fluxgate sensors","volume":"106","author":"Ripka","year":"2003","journal-title":"Sens. Actuators A."},{"key":"ref_16","unstructured":"Primdahl, F. (1970). Bibliography of Fluxgate Magnetometers, Publications of the Earth Physics Branch, Department of Energy, Mines and Resources."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1016\/S0304-8853(00)00273-0","article-title":"New directions in fluxgate sensors","volume":"215\u2013216","author":"Ripka","year":"2000","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_18","unstructured":"Karlsson, M., Robinson, J., Gammaitoni, L., and Bulsara, A. (2001, January 11\u201313). The optimal achievable accuracy of the Advanced Dynamic Fluxgate Magnetometer (ADFM). Stockholm, Sweden."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3862","DOI":"10.1063\/1.125481","article-title":"Fundamental limits to magnetic-field sensitivity of flux-gate magnetic-field sensors","volume":"75","author":"Koch","year":"1999","journal-title":"Appl. Phys. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1109\/TMAG.1972.1067268","article-title":"Recent advances in fluxgate magnetometry","volume":"8","author":"Gordon","year":"1972","journal-title":"IEEE Trans. Magn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1109\/5.56910","article-title":"A Review of Magnetic Sensors","volume":"78","author":"Lenz","year":"1990","journal-title":"IEEE Proc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1126\/science.203.4382.745","article-title":"Initial Pioneer Venus magnetic field results: Dayside observations","volume":"203","author":"Russell","year":"1979","journal-title":"Science"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1109\/TMAG.1977.1059687","article-title":"A magnetometer for the Pioneer Venus mission","volume":"13","author":"Snare","year":"1997","journal-title":"IEEE Trans. Magn."},{"key":"ref_24","first-page":"102","article-title":"High sensitivity magnetic field sensors using amorphous alloy","volume":"53","author":"Vertesy","year":"1992","journal-title":"Electron. Horiz."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/0304-8853(95)01177-3","article-title":"Noise and stability of magnetic sensors","volume":"157\u2013158","author":"Ripka","year":"1996","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_26","first-page":"253","article-title":"A pulse position type fluxgate magnetometer","volume":"80","author":"Strycker","year":"1961","journal-title":"AIEE Trans"},{"key":"ref_27","unstructured":"Brailsford, F. (1951). Magnetic Materials, John Wiley & Sons, Inc."},{"key":"ref_28","unstructured":"Bertotti, G. (1998). Hystersis in Magnetism, Academic Press."},{"key":"ref_29","unstructured":"Stanley, H. (1971). Introduction to Phase Transitions and Critical Phenomena, Oxford University Press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1063\/1.1703954","article-title":"Time-Dependent statistics of the Ising model","volume":"4","author":"Glauber","year":"1963","journal-title":"J. Math. Phys."},{"key":"ref_31","unstructured":"Doedel, E., and Wang, X. (1994). Auto94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, California Institute of Technology."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.physleta.2005.11.077","article-title":"Coupling-Induced cooperative behavior in dynamic ferromagnetic cores in the presence of noise","volume":"353","author":"Bulsara","year":"2006","journal-title":"Phys. Lett. A."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Lukes, T. (1961). Space Correlation and Energy Levels in an Einstein Solid dagger. Proc. Phys. Soc., 78.","DOI":"10.1088\/0370-1328\/78\/5\/316"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ermentrout, B. (2002). Simulating, Analyzing, and Animating Dynamical Systems: A Guide to Xppaut for Researchers and Students, Society for Industrial and Applied Mathematics (SIAM).","DOI":"10.1137\/1.9780898718195"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1430029","DOI":"10.1142\/S0218127414300298","article-title":"Geometry of basins of attraction and heteroclinic connections in coupled bistable systems","volume":"24","author":"Lyons","year":"2014","journal-title":"Int. J. Bif. Chaos."},{"key":"ref_36","unstructured":"Lyons, D., Mahaffy, J., Wang, S., Palacios, A., and In, V. Available online: http:\/\/www.rohan.sdsu.edu\/antoniop\/research\/."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Davies, N., Vu, H., Palacios, A., In, V., and Longhini, P. (2013). Collective behavior of a coupled gyroscope system with coupling along the drive and sense modes. Int. J. Bifurc. Chaos., 23.","DOI":"10.1142\/S0218127413500065"},{"key":"ref_38","unstructured":"Davies, N. (2011). Ring of VIbratory Gyroscopes with Coupling Along the Drive and Sense Axes. [Master\u2019s Thesis, San Diego State University]."},{"key":"ref_39","unstructured":"Vu, H. (2011). Ring of Vibratory Gyroscopes with Coupling along the Drive Axis. [Ph.D. Thesis., San Diego State University]."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"013103","DOI":"10.1063\/1.3532802","article-title":"A drive-free vibratory gyroscope","volume":"21","author":"Vu","year":"2011","journal-title":"Chaos"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"031108","DOI":"10.1103\/PhysRevE.81.031108","article-title":"Two-time scale analysis of a ring of coupled vibratory gyroscopes","volume":"81","author":"Vu","year":"2010","journal-title":"Phys. Rev. E."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.physd.2014.09.004","article-title":"Dynamics and bifurcations in a Dn symmetric Hamiltonian network with applications to coupled gyroscopes","volume":"290","author":"Buono","year":"2015","journal-title":"Phys. D."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1550026","DOI":"10.1142\/S0218127415500261","article-title":"Dynamics, Bifurcations and Normal Forms in Arrays of Magnetostrictive Energy Harvesters with All-to-All Coupling","volume":"25","author":"Chan","year":"2015","journal-title":"Int. J. Bifurc. Chaos."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/7\/2\/1122\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:48:12Z","timestamp":1760215692000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/7\/2\/1122"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,19]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,6]]}},"alternative-id":["sym7021122"],"URL":"https:\/\/doi.org\/10.3390\/sym7021122","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2015,6,19]]}}}