{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:46:14Z","timestamp":1760237174870,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,4]],"date-time":"2020-03-04T00:00:00Z","timestamp":1583280000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Motion in biology is studied through a descriptive geometrical method. We consider a deterministic discrete dynamical system used to simulate and classify a variety of types of movements which can be seen as templates and building blocks of more complex trajectories. The dynamical system is determined by the iteration of a bimodal interval map dependent on two parameters, up to scaling, generalizing a previous work. The characterization of the trajectories uses the classifying tools from symbolic dynamics\u2014kneading sequences, topological entropy and growth number. We consider also the isentropic trajectories, trajectories with constant topological entropy, which are related with the possible existence of a constant drift. We introduce the concepts of pure and mixed bimodal trajectories which give much more flexibility to the model, maintaining it simple. We discuss several procedures that may allow the use of the model to characterize empirical data.<\/jats:p>","DOI":"10.3390\/math8030339","type":"journal-article","created":{"date-parts":[[2020,3,4]],"date-time":"2020-03-04T10:46:08Z","timestamp":1583318768000},"page":"339","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Kinematics in Biology: Symbolic Dynamics Approach"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6772-7920","authenticated-orcid":false,"given":"Carlos","family":"Correia Ramos","sequence":"first","affiliation":[{"name":"Department of Mathematics, CIMA, University of \u00c9vora, 7000 \u00c9vora, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2201","DOI":"10.1098\/rstb.2010.0078","article-title":"Stochastic modelling of animal movement","volume":"365","author":"Smouse","year":"2010","journal-title":"Philos. Trans. R. Soc. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"19052","DOI":"10.1073\/pnas.0800375105","article-title":"A movement ecology paradigm for unifying organismal movement research","volume":"105","author":"Nathan","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1111\/j.1749-8198.2010.00337.x","article-title":"Agent-based modeling of animal movement: A review","volume":"4","author":"Tang","year":"2010","journal-title":"Geogr. Compass"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2874","DOI":"10.1890\/04-1852","article-title":"Robust state\u2013space modeling of animal movement data","volume":"86","author":"Jonsen","year":"2005","journal-title":"Ecology"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.tree.2007.10.009","article-title":"State\u2013space models of individual animal movement","volume":"23","author":"Patterson","year":"2008","journal-title":"Trends Ecol. Evol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1890\/11-0326.1","article-title":"A general discrete-time modeling framework for animal movement using multistate random walks","volume":"82","author":"McClintock","year":"2012","journal-title":"Ecol. Monogr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2336","DOI":"10.1890\/11-2241.1","article-title":"Flexible and practical modeling of animal telemetry data: Hidden Markov models and extensions","volume":"93","author":"Langrock","year":"2012","journal-title":"Ecology"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"8704","DOI":"10.1073\/pnas.1015208108","article-title":"Variation in individual walking behavior creates the impression of a L\u00e9vy flight","volume":"108","author":"Petrovskii","year":"2011","journal-title":"PNAS"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"160566","DOI":"10.1098\/rsos.160566","article-title":"A random walk description of individual animal movement accounting for periods of rest","volume":"3","author":"Tilles","year":"2016","journal-title":"R. Soc. Open Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5464","DOI":"10.3934\/mbe.2019272","article-title":"Animal movement: Symbolic dynamics and topological classification","volume":"16","author":"Ramos","year":"2019","journal-title":"MBE"},{"key":"ref_11","unstructured":"Almeida, P., Lampreia, J.P., and Ramos, J.S. (1996). Topological invariants for bimodal maps. ECIT 1992, World Sci. Publishing."},{"key":"ref_12","first-page":"1","article-title":"Symbolic dynamics of bimodal maps","volume":"54","author":"Lampreia","year":"1997","journal-title":"Port. Math"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1155\/DDNS.2005.69","article-title":"Irreducible complexity of iterated symmetrical bimodal maps","volume":"1","author":"Lampreia","year":"2005","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Alexander, J.C. (1988). On Iterated Maps of the Interval. Dynamical System, Springer. Procededings Univ. Maryland 1986\u20131987, Lect. Notes in Math. n.1342.","DOI":"10.1007\/BFb0082819"},{"key":"ref_15","first-page":"257","article-title":"Renormalizations for cubic maps","volume":"13","author":"Lampreia","year":"1999","journal-title":"Ann. Math. Silesianae"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Sousa Ramos, J. (2006). Introduction to Nonlinear Dynamics of Electronic Systems: Tutorial, Nonlinear Dynamics, Springer.","DOI":"10.1007\/s11071-006-1930-8"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s002200050018","article-title":"On entropy and monotonicity of real cubic maps","volume":"209","author":"Milnor","year":"2000","journal-title":"Comm. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.1142\/S0218127403007552","article-title":"Isentropic real cubic maps","volume":"13","author":"Martins","year":"2003","journal-title":"Int. J. Bifurc. Chaos"}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/8\/3\/339\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:03:56Z","timestamp":1760173436000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/8\/3\/339"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,4]]},"references-count":18,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["math8030339"],"URL":"https:\/\/doi.org\/10.3390\/math8030339","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2020,3,4]]}}}