{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:39:40Z","timestamp":1760243980784,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2009,4,29]],"date-time":"2009-04-29T00:00:00Z","timestamp":1240963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper provides a new approach for the analysis and eventually the classification of dynamical systems. The objective is pursued by extending the concept of the entropy of plane curves, first introduced within the theory of the thermodynamics of plane curves, to Rn space. Such a generalised entropy of a curve is used to evaluate curves that are obtained by connecting several points in the phase space. As the points change their coordinates according to the equations of a dynamical system, the entropy of the curve connecting them is used to infer the behaviour of the underlying dynamics. According to the proposed method all linear dynamical systems evolve at constant zero entropy, while higher asymptotic values characterise nonlinear systems. The approach proves to be particularly efficient when applied to chaotic systems, in which case it has common features with other classic approaches. Performances of the proposed method are tested over several benchmark problems.<\/jats:p>","DOI":"10.3390\/e11020249","type":"journal-article","created":{"date-parts":[[2009,4,29]],"date-time":"2009-04-29T10:32:23Z","timestamp":1241001143000},"page":"249-270","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Generalised Entropy of Curves for the Analysis and Classification of Dynamical Systems"],"prefix":"10.3390","volume":"11","author":[{"given":"Aldo","family":"Balestrino","sequence":"first","affiliation":[{"name":"Department of Electrical Systems and Automation, University of Pisa, Via Diotisalvi 2, 56100, Pisa, Italy"}]},{"given":"Andrea","family":"Caiti","sequence":"additional","affiliation":[{"name":"Department of Electrical Systems and Automation, University of Pisa, Via Diotisalvi 2, 56100, Pisa, Italy"}]},{"given":"Emanuele","family":"Crisostomi","sequence":"additional","affiliation":[{"name":"Department of Electrical Systems and Automation, University of Pisa, Via Diotisalvi 2, 56100, Pisa, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2009,4,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/S0165-1684(98)00029-2","article-title":"A method for nonlinear system classification in the time-frequency plane","volume":"65","author":"Galleani","year":"1998","journal-title":"Signal Processing"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1109\/9.763209","article-title":"A Lie-B\u00e4cklund approach to equivalence and flatness of nonlinear systems","volume":"44","author":"Fliess","year":"1999","journal-title":"IEEE Transactions on Automatic Control"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ott, E. (2002). Chaos in dynamical systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511803260"},{"key":"ref_4","unstructured":"Strogatz, S.H. (2000). Nonlinear dynamics and chaos, Westview Press."},{"key":"ref_5","first-page":"5","article-title":"Les courbes chaotiques","volume":"51","year":"1983","journal-title":"Courrier du Centre National de la Recerche Scientifique"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/BF00280431","article-title":"Can one measure the temperature of a curve?","volume":"94","author":"Dupain","year":"1986","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1023\/A:1021302922108","article-title":"Using the entropy of curves to segment a time or spatial series","volume":"34","author":"Denis","year":"2002","journal-title":"Mathematical Geology"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Balestrino, A., Caiti, A., and Crisostomi, E. (,  2007). Entropy of curves for nonlinear systems classification. IFAC symposium of nonlinear control systems (NOLCOS), Pretoria, South Africa.","DOI":"10.3182\/20070822-3-ZA-2920.00012"},{"key":"ref_9","first-page":"119","article-title":"A classification of nonlinear systems: an entropy based approach","volume":"11","author":"Balestrino","year":"2007","journal-title":"Chemical Engineering transactions"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Balestrino, A., Caiti, A., Crisostomi, E., and Grioli, G. (,  2008). A generalised entropy of curves: an approach to the analysis of dynamical systems. IEEE Conference on Decision and Control, Cancun, Mexico.","DOI":"10.1109\/CDC.2008.4738723"},{"key":"ref_11","unstructured":"Balestrino, A., Caiti, A., and Crisostomi, E. (,  2008). Nonlinear dynamics and generalised entropy of curves. Conference on Nonlinear Science and Complexity, Porto, Portugal."},{"key":"ref_12","unstructured":"Moore, R., and Van Der Potten, A. (,  1989). On the thermodynamics of curves and other curlicues. Conference on Geometry and Physics, Canberra."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell System Technical Journal"},{"key":"ref_14","unstructured":"Hopp, T.H., and Reeve, C.P. (1996). Technical Report NISTIR 5831, National Institute of Standards and Technology."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1287\/mnsc.19.1.96","article-title":"The minimum covering sphere problem","volume":"19","author":"Elzinga","year":"1972","journal-title":"Management science"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1023\/A:1008276418601","article-title":"Fast Lyapunov Indicators. Application to asteroidal motion","volume":"67","author":"Lega","year":"1997","journal-title":"Celestial Mechanics and Dynamical Astronomy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1023\/A:1008307332442","article-title":"Detection of ordered and chaotic motion using the dynamical spectra","volume":"73","author":"Voglis","year":"1999","journal-title":"Celestial Mechanics and Dynamical Astronomy"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"10029","DOI":"10.1088\/0305-4470\/34\/47\/309","article-title":"Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits","volume":"34","author":"Skokos","year":"2001","journal-title":"Journal of Physics A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1907","DOI":"10.1016\/j.physa.2007.11.024","article-title":"The production of Tsallis entropy in the limit of weak chaos and a new indicator of chaoticity","volume":"387","author":"Voglis","year":"2008","journal-title":"Physica A"},{"key":"ref_20","unstructured":"Baran, V., and Bonasera, A. Fingerprints of chaos. available online at http:\/\/arxiv.org\/pdf\/chao-dyn\/9804023."},{"key":"ref_21","first-page":"779","article-title":"A new characterization of chaotic dynamics: the d\u221e parameter","volume":"6","author":"Bonasera","year":"2003","journal-title":"Nonlinear Phenomena in Complex Systems"},{"key":"ref_22","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":"Russian Math. Surveys"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1137\/S0036142993247311","article-title":"On the computation of Lyapunov exponents for continuous dynamical systems","volume":"34","author":"Dieci","year":"1997","journal-title":"SIAM Journal of Numerical Analysis"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1063\/1.1741751","article-title":"Estimating the Lyapunov exponents of discrete systems","volume":"14","author":"Li","year":"2004","journal-title":"Chaos"},{"key":"ref_25","unstructured":"Skokos, C. The Lyapunov Characteristic Exponents and their computation. available online at http:\/\/arxiv.org\/pdf\/0901.4418."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1142\/S0218127407017732","article-title":"Time-varying linearization and the Perron effects","volume":"17","author":"Leonov","year":"2007","journal-title":"Int. Journal of Bifurcation and Chaos"},{"key":"ref_27","unstructured":"Salwi, D. (1997). Scientists of India, CBT Publications."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1103\/PhysRevLett.65.389","article-title":"Characterizing loss of memory in a dynamical system","volume":"65","author":"Gade","year":"1990","journal-title":"Physical Review Letters"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bucolo, M., Di Grazia, F., Sapuppo, F., and Virz\u00ec, M.C. (,  2008). A new approach for nonlinear time series characterization, \u201cDivA\u201d. Mediterranean Conference on Control and Automation, Ajaccio.","DOI":"10.1109\/MED.2008.4602067"},{"key":"ref_30","unstructured":"MacKay, D.J.C. (2007). Information theory, inference, and learning algorithms, Cambridge University Press. [6th Ed.]."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1023\/A:1009637014772","article-title":"Spectra of Dynamical Systems","volume":"6","author":"Contopoulos","year":"1999","journal-title":"Open Systems & Information Dynamics"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/11\/2\/249\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:10:20Z","timestamp":1760220620000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/11\/2\/249"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,4,29]]},"references-count":31,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2009,6]]}},"alternative-id":["e11020249"],"URL":"https:\/\/doi.org\/10.3390\/e11020249","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2009,4,29]]}}}