{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T03:24:34Z","timestamp":1781234674432,"version":"3.54.1"},"reference-count":38,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T00:00:00Z","timestamp":1692748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The propagation of electromagnetic waves in a closed domain with a reflecting boundary amounts, in the eikonal approximation, to the propagation of rays in a billiard. If the inner medium is uniform, then the symplectic reflection map provides the polygonal rays\u2019 paths. The linear response theory is used to analyze the stability of any trajectory. The Lyapunov and reversibility error invariant indicators provide an estimate of the sensitivity to a small initial random deviation and to a small random deviation at any reflection, respectively. A family of chaotic billiards is considered to test the chaos detection effectiveness of the above indicators.<\/jats:p>","DOI":"10.3390\/e25091251","type":"journal-article","created":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T08:01:21Z","timestamp":1692777681000},"page":"1251","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards"],"prefix":"10.3390","volume":"25","author":[{"given":"Gabriele","family":"Gradoni","sequence":"first","affiliation":[{"name":"Department of Electrical and Electronic Engineering, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giorgio","family":"Turchetti","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Alma Mater Studiorum, University of Bologna, Viale Berti Pichat 6\/2, 40127 Bologna, Italy"},{"name":"INdAM Gruppo Nazionale per la Fisica Matematica Piazzale Aldo Moro, 00185 Roma, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Federico","family":"Panichi","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Alma Mater Studiorum, University of Bologna, Viale Berti Pichat 6\/2, 40127 Bologna, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3072","DOI":"10.1103\/PhysRevLett.65.3072","article-title":"Experimental demonstration of chaotic scattering of microwaves","volume":"65","author":"Doron","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"064101","DOI":"10.1103\/PhysRevLett.89.064101","article-title":"Experimental Test of a Trace Formula for a Chaotic Three-Dimensional Microwave Cavity","volume":"89","author":"Dembowski","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","unstructured":"Buzea, C.G., Agop, M., and Butler, L. (2019). Progress in Relativity, IntechOpen. Chapter 10."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1016\/j.wavemoti.2014.02.003","article-title":"Predicting the statistics of wave transport through chaotic cavities by the random coupling model: A review and recent progress","volume":"51","author":"Gradoni","year":"2014","journal-title":"Wave Motion"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/0003-4916(77)90334-7","article-title":"Electromagnetic waves near perfect conductors. I. Multiple scattering expansions. Distribution of modes","volume":"104","author":"Balian","year":"1977","journal-title":"Ann. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0003-4916(78)90083-0","article-title":"Electromagnetic waves near perfect conductors. II. Casimir effect","volume":"112","author":"Balian","year":"1978","journal-title":"Ann. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"383001","DOI":"10.1088\/1751-8121\/ac87e0","article-title":"Microwave studies of the spectral statistics in chaotic systems","volume":"55","author":"Kuhl","year":"2022","journal-title":"J. Phys. Math. Theor."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1103\/PhysRevLett.67.785","article-title":"Experimental observation of scarred eigenfunctions of chaotic microwave cavities","volume":"67","author":"Sridhar","year":"1991","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1296","DOI":"10.1103\/PhysRevLett.69.1296","article-title":"Distribution of eigenmodes in a superconducting stadium billiard with chaotic dynamics","volume":"69","author":"Harney","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2662","DOI":"10.1103\/PhysRevLett.74.2662","article-title":"Wave Chaos Experiments with and without Time Reversal Symmetry: GUE and GOE Statistics","volume":"74","author":"So","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"056209","DOI":"10.1103\/PhysRevE.70.056209","article-title":"Experimental investigation of nodal domains in the chaotic microwave rough billiard","volume":"70","author":"Savytskyy","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"150403","DOI":"10.1103\/PhysRevLett.106.150403","article-title":"Exceptional Points in a Microwave Billiard with Time-Reversal Invariance Violation","volume":"106","author":"Dietz","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1016\/j.wavemoti.2013.09.006","article-title":"Universal behavior of a wave chaos based electromagnetic reverberation chamber","volume":"51","author":"Gros","year":"2014","journal-title":"Wave Motion"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1038\/385045a0","article-title":"Ray and wave chaos in asymmetric resonant optical cavities","volume":"385","author":"Stone","year":"1997","journal-title":"Nature"},{"key":"ref_15","unstructured":"Stone, A.D. (2001). Quantum Chaos Y2K, World Scientific."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Chernov, N., and Markarian, R. (2006). Chaotic Billiards, American Mathematical Soc.. Number 127.","DOI":"10.1090\/surv\/127"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1038\/nphoton.2013.108","article-title":"Enhanced energy storage in chaotic optical resonators","volume":"7","author":"Liu","year":"2013","journal-title":"Nat. Photonics"},{"key":"ref_18","first-page":"015201","article-title":"Differences between emission patterns and internal modes of optical resonators","volume":"85","author":"Creagh","year":"2012","journal-title":"Phys. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1121\/1.426252","article-title":"One-channel time-reversal in chaotic cavities: Experimental results","volume":"105","author":"Draeger","year":"1999","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1109\/MEMC.2022.9780346","article-title":"Reverberation Chambers at the Edge of Chaos: Discussion Forum at EMC Europe 2020","volume":"11","author":"Serra","year":"2022","journal-title":"IEEE Electromagn. Compat. Mag."},{"key":"ref_21","first-page":"20170419","article-title":"On the integrability of Birkhoff billiards","volume":"376","author":"Kaloshin","year":"2018","journal-title":"Philos. Trans. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1142\/S0217979211058006","article-title":"An exact map for a chaotic billiard","volume":"25","author":"Mikoss","year":"2011","journal-title":"Int. J. Mod. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/BF02183643","article-title":"Numerical experiments on billiards","volume":"83","author":"Artuso","year":"1996","journal-title":"J. Stat. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s00220-019-03552-y","article-title":"Periodic ellipsoidal billiard trajectories and extremal polynomials","volume":"372","year":"2019","journal-title":"Commun. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/s00283-019-09951-2","article-title":"Can elleptic billiards still surprise us?","volume":"42","author":"Reznik","year":"2020","journal-title":"Math. Intell."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"123119","DOI":"10.1063\/5.0071426","article-title":"Spherical billiards with almost complete escape","volume":"31","author":"Dettmann","year":"2021","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4983","DOI":"10.1038\/s41598-022-08897-4","article-title":"Interaction between a robot and Bunimovich stadium billiards","volume":"12","author":"Vasconcelos","year":"2022","journal-title":"Sci. Rep."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"022214","DOI":"10.1103\/PhysRevE.98.022214","article-title":"3D billiards: Visualization of regular structures and trapping of chaotic trajectories","volume":"98","author":"Firmbach","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"022902","DOI":"10.1103\/PhysRevE.89.022902","article-title":"Visualization and comparison of classical structures and quantum states of four-dimensional maps","volume":"89","author":"Richter","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"064001","DOI":"10.1088\/1751-8121\/aa5192","article-title":"Errors, Correlations and Fidelity for noisy Hamilton flows. Theory and numerical examples","volume":"50","author":"Turchetti","year":"2015","journal-title":"J. Phys. Math. Theor."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1093\/mnras\/stx374","article-title":"The reversibility error method (REM): A new, dynamical fast indicator for planetary dynamics","volume":"468","author":"Panichi","year":"2017","journal-title":"MNRAS"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.cnsns.2015.10.016","article-title":"Fidelity and reversibility in the restricted three body problem","volume":"35","author":"Panichi","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"043138","DOI":"10.1063\/5.0043782","article-title":"Propagation of rays in 2D and 3D waveguides: A stability analysis with Lyapunov and Reversibility fast indicators","volume":"31","author":"Gradoni","year":"2021","journal-title":"Chaos"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Turchetti, G., and Panichi, F. (2019). Birkhoff normal forms and stability indicators for betatronic motion. Nonlinear Dyn. Collect. Eff. Part. Beam Phys., 47\u201369.","DOI":"10.1142\/9789813279612_0004"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0167-2789(03)00103-9","article-title":"Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits","volume":"182","author":"Cincotta","year":"2003","journal-title":"Phys. Nonlinear Phenom."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1088\/0143-0807\/2\/2\/006","article-title":"Regularity and chaos in classical mechanics, illustrated by three deformations of a circular \u2018billiard\u2019","volume":"2","author":"Berry","year":"1981","journal-title":"Eur. J. Phys."},{"key":"ref_37","unstructured":"Markus Himmelstrand (2023, August 01). A Survey of Dynamical Billiards. Available online: https:\/\/www.diva-portal.org\/smash\/get\/diva2:650284\/FULLTEXT01.pdf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1063\/1.165962","article-title":"A scattering approach to the quantization of billiards- The inside-outside duality","volume":"3","author":"Dietz","year":"1993","journal-title":"Chaos"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1251\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:40:53Z","timestamp":1760128853000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1251"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,23]]},"references-count":38,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["e25091251"],"URL":"https:\/\/doi.org\/10.3390\/e25091251","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,23]]}}}