{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T05:14:26Z","timestamp":1787030066185,"version":"build-2736575974"},"reference-count":81,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,12]],"date-time":"2020-05-12T00:00:00Z","timestamp":1589241600000},"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>In many stochastic dynamical systems, ordinary chaotic behavior is preceded by a full-dimensional phase that exhibits 1\/f-type power spectra and\/or scale-free statistics of (anti)instantons such as neuroavalanches, earthquakes, etc. In contrast with the phenomenological concept of self-organized criticality, the recently found approximation-free supersymmetric theory of stochastics (STS) identifies this phase as the noise-induced chaos (N-phase), i.e., the phase where the topological supersymmetry pertaining to all stochastic dynamical systems is broken spontaneously by the condensation of the noise-induced (anti)instantons. Here, we support this picture in the context of neurodynamics. We study a 1D chain of neuron-like elements and find that the dynamics in the N-phase is indeed featured by positive stochastic Lyapunov exponents and dominated by (anti)instantonic processes of (creation) annihilation of kinks and antikinks, which can be viewed as predecessors of boundaries of neuroavalanches. We also construct the phase diagram of emulated stochastic neurodynamics on Spikey neuromorphic hardware and demonstrate that the width of the N-phase vanishes in the deterministic limit in accordance with STS. As a first result of the application of STS to neurodynamics comes the conclusion that a conscious brain can reside only in the N-phase.<\/jats:p>","DOI":"10.3390\/sym12050805","type":"journal-article","created":{"date-parts":[[2020,5,12]],"date-time":"2020-05-12T10:53:55Z","timestamp":1589280835000},"page":"805","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Criticality or Supersymmetry Breaking?"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6121-5422","authenticated-orcid":false,"given":"Igor V.","family":"Ovchinnikov","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, UCLA, Los Angeles, CA 90095, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenyuan","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, UCLA, Los Angeles, CA 90095, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuquan","family":"Sun","sequence":"additional","affiliation":[{"name":"LMIB &amp; School of Mathematics and Systems Science, BeiHang University, Beijing 100191, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew E.","family":"Hudson","sequence":"additional","affiliation":[{"name":"Department of Anesthesiology and Perioperative Medicine, David Geffen School of Medicine, UCLA, Los Angeles, CA 90095, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karlheinz","family":"Meier","sequence":"additional","affiliation":[{"name":"Kirchhoff-Institute for Physics, Heidelberg University, 69117 Heidelberg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4639-5936","authenticated-orcid":false,"given":"Robert N.","family":"Schwartz","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, UCLA, Los Angeles, CA 90095, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kang L.","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, UCLA, Los Angeles, CA 90095, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Aschwanden, M.J. (2011). Self-Organized Criticality in Astrophysics: The Statistics of Nonlinear Processes in the Universe, Springer.","DOI":"10.1007\/978-3-642-15001-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"7674","DOI":"10.1073\/pnas.1019484108","article-title":"Switching processes in financial markets","volume":"108","author":"Preis","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1038\/176795a0","article-title":"Magnitude and Energy of Earthquakes","volume":"176","author":"Gutenberg","year":"1955","journal-title":"Nature"},{"key":"ref_4","unstructured":"Schopf, T. (1972). Punctuated equilibria: An alternative to phyletic gradualism. Models in Paleobiology, Freeman, Cooper & Co."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7212","DOI":"10.1073\/pnas.1324045111","article-title":"Social interactions dominate speed control in poising natural flocks near criticality","volume":"111","author":"Bialek","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4786","DOI":"10.1073\/pnas.1118633109","article-title":"Statistical mechanics for natural flocks of birds","volume":"109","author":"Bialek","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1038\/nphys1803","article-title":"Emergent complex neural dynamics","volume":"6","author":"Chialvo","year":"2010","journal-title":"Nat. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5216","DOI":"10.1523\/JNEUROSCI.0540-04.2004","article-title":"Neuronal Avalanches Are Diverse and Precise Activity Patterns That Are Stable for Many Hours in Cortical Slice Cultures","volume":"24","author":"Beggs","year":"2004","journal-title":"J. Neurosci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"11167","DOI":"10.1523\/JNEUROSCI.23-35-11167.2003","article-title":"Neuronal Avalanches in Neocortical Circuits","volume":"23","author":"Beggs","year":"2003","journal-title":"J. Neurosci."},{"key":"ref_10","unstructured":"Musha, T., and Yamamoto, M. (November, January 30). 1\/f fluctuations in biological systems. Engineering in Medicine and Biology Society, 1997. Proceedings of the 19th Annual International Conference of the IEEE, Chicago, IL, USA."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1038\/nphys758","article-title":"Dynamical synapses causing self-organized criticality in neural networks","volume":"3","author":"Levina","year":"2007","journal-title":"Nat. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1103\/PhysRevLett.59.381","article-title":"Self-organized criticality: An explanation of the 1\/ f noise","volume":"59","author":"Bak","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"7425","DOI":"10.1103\/PhysRevB.40.7425","article-title":"1\/ f noise, distribution of lifetimes, and a pile of sand","volume":"40","author":"Jensen","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6345","DOI":"10.1103\/PhysRevE.57.6345","article-title":"How self-organized criticality works: A unified mean-field picture","volume":"57","author":"Vespignani","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s11214-015-0155-x","article-title":"25 Years of Self-organized Criticality: Concepts and Controversies","volume":"198","author":"Watkins","year":"2016","journal-title":"Space Sci. Rev."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Pruessner, G. (2012). Self-Organised Criticality, Cambridge University Press.","DOI":"10.1017\/CBO9780511977671"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1016\/S0039-3681(03)00046-3","article-title":"Self-organised criticality\u2014What it is and what it isn\u2019t","volume":"34","author":"Frigg","year":"2003","journal-title":"Stud. Hist. Philos. Sci. Part A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4793","DOI":"10.1103\/PhysRevLett.78.4793","article-title":"Order Parameter and Scaling Fields in Self-Organized Criticality","volume":"78","author":"Vespignani","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4528","DOI":"10.1103\/PhysRevLett.75.4528","article-title":"Avalanches, Barkhausen Noise, and Plain Old Criticality","volume":"75","author":"Dahmen","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Christensen, K., and Moloney, N. (2005). Complexity and Criticality, Imperial College Press.","DOI":"10.1142\/p365"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"22","DOI":"10.3389\/fnsys.2015.00022","article-title":"Criticality as a signature of healthy neural systems","volume":"9","author":"Massobrio","year":"2015","journal-title":"Front. Syst. Neurosci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"176","DOI":"10.3389\/fnsys.2014.00176","article-title":"Markers of criticality in phase synchronization","volume":"8","author":"Botcharova","year":"2014","journal-title":"Front. Syst. Neurosci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"166","DOI":"10.3389\/fnsys.2014.00166","article-title":"Self-organized criticality as a fundamental property of neural systems","volume":"8","author":"Hesse","year":"2014","journal-title":"Front. Syst. Neurosci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"154","DOI":"10.3389\/fnsys.2014.00154","article-title":"Critical role for resource constraints in neural models","volume":"8","author":"Roberts","year":"2014","journal-title":"Front. Syst. Neurosci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"118102","DOI":"10.1103\/PhysRevLett.97.118102","article-title":"Does the 1\/f Frequency Scaling of Brain Signals Reflect Self-Organized Critical States?","volume":"97","author":"Destexhe","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"302","DOI":"10.3389\/fphys.2012.00302","article-title":"Avalanche Analysis from Multielectrode Ensemble Recordings in Cat, Monkey, and Human Cerebral Cortex during Wakefulness and Sleep","volume":"3","author":"Dehghani","year":"2012","journal-title":"Front. Physiol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"108","DOI":"10.3389\/fnsys.2014.00108","article-title":"Spike avalanches in vivo suggest a driven, slightly subcritical brain state","volume":"8","author":"Priesemann","year":"2014","journal-title":"Front. Syst. Neurosci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.chaos.2013.05.009","article-title":"Criticality and avalanches in neural networks","volume":"55","author":"Zare","year":"2013","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"151","DOI":"10.3389\/fnsys.2014.00151","article-title":"Marginally subcritical dynamics explain enhanced stimulus discriminability under attention","volume":"8","author":"Tomen","year":"2014","journal-title":"Front. Syst. Neurosci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.1103\/PhysRevLett.65.1395","article-title":"Self-organized criticality: Goldstone modes and their interactions","volume":"65","author":"Obukhov","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1088\/0253-6102\/32\/2\/229","article-title":"Goldstone Modes in Driven Diffusion Systems","volume":"32","author":"LunBiao","year":"1999","journal-title":"Commun. Theor. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"051129","DOI":"10.1103\/PhysRevE.83.051129","article-title":"Self-organized criticality as Witten-type topological field theory with spontaneously broken Becchi-Rouet-Stora-Tyutin symmetry","volume":"83","author":"Ovchinnikov","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Ovchinnikov, I.V. (2016). Introduction to Supersymmetric Theory of Stochastics. Entropy, 18.","DOI":"10.3390\/e18040108"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/0370-1573(91)90117-5","article-title":"Topological field theory","volume":"209","author":"Birmingham","year":"1991","journal-title":"Phys. Rep."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/BF01218589","article-title":"Morse theory interpretation of topological quantum field theories","volume":"123","author":"Labastida","year":"1989","journal-title":"Commun. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/BF01223371","article-title":"Topological quantum field theory","volume":"117","author":"Witten","year":"1988","journal-title":"Commun. Math. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/BF01466725","article-title":"Topological sigma models","volume":"118","author":"Witten","year":"1988","journal-title":"Commun. Math. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0393-0440(93)90049-K","article-title":"The Mathai-Quillen formalism and topological field theory","volume":"11","author":"Blau","year":"1993","journal-title":"J. Geom. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0370-2693(88)91328-7","article-title":"A topological interpretation of stochastic quantization","volume":"212","author":"Baulieu","year":"1988","journal-title":"Phys. Lett. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/BF01217907","article-title":"The topological sigma model","volume":"125","author":"Baulieu","year":"1989","journal-title":"Commun. Math. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.jmaa.2013.01.027","article-title":"Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters","volume":"402","author":"Slavik","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1650086","DOI":"10.1142\/S021798491650086X","article-title":"Topological supersymmetry breaking: The definition and stochastic generalization of chaos and the limit of applicability of statistics","volume":"30","author":"Ovchinnikov","year":"2016","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_43","unstructured":"Ovchinnikov, I.V., and Di Ventra, M. (2017). Chaos or Order ?. arXiv."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1103\/RevModPhys.70.1455","article-title":"Topological analysis of chaotic dynamical systems","volume":"70","author":"Gilmore","year":"1998","journal-title":"Rev. Mod. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/0550-3213(81)90006-7","article-title":"Dynamical breaking of supersymmetry","volume":"188","author":"Witten","year":"1981","journal-title":"Nucl. Phys. B"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"11","DOI":"10.3389\/fnins.2013.00011","article-title":"Six Networks on a Universal Neuromorphic Computing Substrate","volume":"7","author":"Pfeil","year":"2013","journal-title":"Front. Neurosci."},{"key":"ref_48","first-page":"021023","article-title":"Effect of Heterogeneity on Decorrelation Mechanisms in Spiking Neural Networks: A Neuromorphic-Hardware Study","volume":"6","author":"Pfeil","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1103\/PhysRevE.60.222","article-title":"Overdamped sine-Gordon kink in a thermal bath","volume":"60","author":"Quintero","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_50","first-page":"199","article-title":"BREMSSTRAHLUNG FROM SOLITONS","volume":"27","author":"Eilenberger","year":"1977","journal-title":"Z. Phys. B Condens. Matter"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"5737","DOI":"10.1103\/PhysRevB.25.5737","article-title":"Soliton excitations in Josephson tunnel junctions","volume":"25","author":"Lomdahl","year":"1982","journal-title":"Phys. Rev. B"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(98)00029-5","article-title":"Nonlinear dynamics of the Frenkel\u2013Kontorova model","volume":"306","author":"Braun","year":"1998","journal-title":"Phys. Rep."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1103\/PhysRevB.54.1234","article-title":"Charge solitons in one-dimensional arrays of serially coupled Josephson junctions","volume":"54","author":"Hermon","year":"1996","journal-title":"Phys. Rev. B"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1070\/RM1977v032n04ABEH001639","article-title":"Characteristic Lyapunov exponents and smooth ergodic theory","volume":"32","author":"Pesin","year":"1977","journal-title":"Russ. Math. Surv."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Ziegler, F., and Schu\u00ebller, G.I. (1988). Lyapunov Exponents of Nonlinear Stochastic Systems. Nonlinear Stochastic Dynamic Engineering Systems: IUTAM Symposium Innsbruck Igls, Austria, 21\u201326 June 1987, Springer.","DOI":"10.1007\/978-3-642-83334-2"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/BF01015324","article-title":"Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors","volume":"51","author":"Grassberger","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"9283","DOI":"10.1073\/pnas.1408296111","article-title":"Recovery of consciousness is mediated by a network of discrete metastable activity states","volume":"111","author":"Hudson","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1152\/jn.1999.81.4.1795","article-title":"Mechanism of Anesthesia Revealed by Shunting Actions of Isoflurane on Thalamocortical Neurons","volume":"81","author":"Ries","year":"1999","journal-title":"J. Neurophysiol."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Touboul, J., and Destexhe, A. (2010). Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics?. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0008982"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Meisel, C., Storch, A., Hallmeyer-Elgner, S., Bullmore, E., and Gross, T. (2012). Failure of adaptive self-organized criticality during epileptic seizure attacks. PLoS Comput. Biol., 8.","DOI":"10.1371\/journal.pcbi.1002312"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.neuron.2012.03.016","article-title":"Not Noisy, Just Wrong: The Role of Suboptimal Inference in Behavioral Variability","volume":"74","author":"Beck","year":"2012","journal-title":"Neuron"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.pneurobio.2009.01.006","article-title":"Stochastic dynamics as a principle of brain function","volume":"88","author":"Deco","year":"2009","journal-title":"Prog. Neurobiol."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.brainres.2011.08.002","article-title":"Cortical attractor network dynamics with diluted connectivity","volume":"1434","author":"Rolls","year":"2012","journal-title":"Brain Res."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0006-3495(72)86068-5","article-title":"Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons","volume":"12","author":"Wilson","year":"1972","journal-title":"Biophys. J."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1002\/wsbm.1348","article-title":"Multiscale modeling of brain dynamics: From single neurons and networks to mathematical tools","volume":"8","author":"Siettos","year":"2016","journal-title":"Wiley Interdiscip. Rev. Syst. Biol. Med."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1090\/S0273-0979-02-00964-3","article-title":"Topological quantum computation","volume":"40","author":"Freedman","year":"2003","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0550-3213(82)90538-7","article-title":"Supersymmetric field theories and stochastic differential equations","volume":"206","author":"Parisi","year":"1982","journal-title":"Nucl. Phys. B"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/0550-3213(86)90592-4","article-title":"Renormalization and stochastic quantization","volume":"275","year":"1986","journal-title":"Nucl. Phys. B"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"095005","DOI":"10.1088\/1751-8113\/45\/9\/095005","article-title":"Stochastic processes, slaves and supersymmetry","volume":"45","author":"Drummond","year":"2012","journal-title":"J. Phys. A"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"661","DOI":"10.4310\/jdg\/1214437492","article-title":"Supersymmetry and Morse theory","volume":"17","author":"Witten","year":"1982","journal-title":"J. Differ. Geom."},{"key":"ref_71","first-page":"519","article-title":"Stochastic integral","volume":"20","year":"1944","journal-title":"Proc. Imp. Acad."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1137\/0304028","article-title":"A new representation for stochastic integrals and equations","volume":"4","author":"Stratonovich","year":"1966","journal-title":"SIAM J. Contr."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF01007642","article-title":"It\u00f3 versus Stratonovich","volume":"24","year":"1981","journal-title":"J. Stat. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"3944","DOI":"10.1063\/1.1489072","article-title":"Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of anti-linear symmetries","volume":"43","author":"Mostafazadeh","year":"2002","journal-title":"J. Math. Phys."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"085023","DOI":"10.1103\/PhysRevD.93.085023","article-title":"Kinematic dynamo, supersymmetry breaking, and chaos","volume":"93","author":"Ovchinnikov","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"8666","DOI":"10.1016\/j.jcp.2010.07.039","article-title":"An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere","volume":"229","author":"Li","year":"2010","journal-title":"J. Comput. Phys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1086\/150988","article-title":"Kinematic-Dynamo Theory","volume":"166","author":"Lerche","year":"1971","journal-title":"Astrophys. J."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1098\/rsta.1972.0074","article-title":"Kinematic Dynamo Models","volume":"272","author":"Roberts","year":"1972","journal-title":"R. Soc. Lond. Philos. Trans. Ser. A"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"037103","DOI":"10.1063\/1.4795546","article-title":"Revisiting the ABC flow dynamo","volume":"25","author":"Bouya","year":"2013","journal-title":"Phys. Fluids"},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Benkadda, S., and Zaslavsky, G.M. (1998). Lagrangian chaos and the fast kinematic dynamo problem. Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas, Proceedings of a Workshop Held in Carry-Le Rouet, France, 16\u201321 June 1997, Springer.","DOI":"10.1007\/BFb0106949"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1090\/S0002-9947-1965-0175106-9","article-title":"Topological entropy","volume":"114","author":"Adler","year":"1965","journal-title":"Trans. Am. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/805\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:28:10Z","timestamp":1760174890000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/805"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,12]]},"references-count":81,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050805"],"URL":"https:\/\/doi.org\/10.3390\/sym12050805","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,12]]}}}