{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T17:41:16Z","timestamp":1764783676147,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2014,10,27]],"date-time":"2014-10-27T00:00:00Z","timestamp":1414368000000},"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>In this paper, we introduce an iterative method with lower positions of true numerical solutions located in the real orbit in order to investigate the property of the logistic map. The basic structure of the logistic map is presented, which consists of the root gene position, the common gene position and the individual gene position. The ergodicity and randomness of the logistic map are dependent on the individual gene position. We find that the lower positions of the true numerical solutions in the real orbits have the property of a half-life.<\/jats:p>","DOI":"10.3390\/e16115618","type":"journal-article","created":{"date-parts":[[2014,10,27]],"date-time":"2014-10-27T11:38:09Z","timestamp":1414409889000},"page":"5618-5632","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Property of Chaotic Orbits with Lower Positions of Numerical Solutions in the Logistic Map"],"prefix":"10.3390","volume":"16","author":[{"given":"Jiahui","family":"Liu","sequence":"first","affiliation":[{"name":"College of Computer Science and Technology, Harbin University of Science and Technology, P.O. Box 110, Harbin 150080, China"},{"name":"College of Computer Science and Technology, Harbin Institute of Technology, P.O. Box 320,Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongli","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Computer Science and Technology, Harbin Institute of Technology, P.O. Box 320,Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dahua","family":"Song","sequence":"additional","affiliation":[{"name":"Center of Educational Technology and Information, Mudanjiang Medical University, Mudanjiang 157011, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,10,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hogg, T., and Huberman, B.A. (1985). Attractors on finite sets: The dissipative dynamics of computing structures. Phys. Rev. A., 2338\u20132346.","DOI":"10.1103\/PhysRevA.32.2338"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/0885-064X(87)90024-0","article-title":"Do numerical orbits of chaotic dynamical processes represent true orbits?","volume":"3","author":"Hammel","year":"1987","journal-title":"J. Complexity."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Chia, T.T., and Tan, B.L. (1991). Maps with precision-dependent periods. Phys. Rev. A., R2231\u2013R2234.","DOI":"10.1103\/PhysRevA.44.R2231"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Longa, L., Curado, E.M.F., and Oliveira, F.A. (1996). Roundoff-induced coalescence of chaotic trajectories. Phys. Rev. E., R2201\u2013R2204.","DOI":"10.1103\/PhysRevE.54.R2201"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Dellago, C., and Hoover, W.G. (2000). Finite-precision stationary states at and away from equilibrium. Phys. Rev. E., 6275\u20136281.","DOI":"10.1103\/PhysRevE.62.6275"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"046206","DOI":"10.1103\/PhysRevE.68.046206","article-title":"Unstable periodic orbits and discretization cycles","volume":"68","author":"Binde","year":"2003","journal-title":"Phys. Rev. E."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1140\/epjst\/e2008-00850-4","article-title":"Increasing average period lengths by switching of robust chaos maps in finite precision","volume":"165","author":"Nagaraj","year":"2008","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lai, Y.C., Nagai, Y., and Grebogi, C. (1997). Characterization of the natural measure by unstable periodic orbits in chaotic attractors. Phys. Rev. Lett., 649\u2013652.","DOI":"10.1103\/PhysRevLett.79.649"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Binder, P.M., and Jensen, R.V. (1986). Simulating chaotic behavior with finite-state machines. Phys. Rev. A., 4460\u20134463.","DOI":"10.1103\/PhysRevA.34.4460"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Grebogi, C., Ott, E., and Yorke, J.A. (1988). Roundoff-induced periodicity and the correlation dimension of chaotic attractors. Phys. Rev. A., 3688\u20133692.","DOI":"10.1103\/PhysRevA.38.3688"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Grebogi, C., Hammel, S.M., Yorke, J.A., and Sauer, T. (1990). Shadowing of physical trajectories in chaotic dynamics: Containment and refinement. Phys. Rev. Lett., 1527\u20131530.","DOI":"10.1103\/PhysRevLett.65.1527"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1006\/jcph.2001.6645","article-title":"Chaos, number theory, and computers","volume":"166","author":"Adler","year":"2001","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3119","DOI":"10.1142\/S0218127405014052","article-title":"On the dynamical degradation of digital piecewise linear chaotic maps","volume":"15","author":"Li","year":"2005","journal-title":"Int. J. Bifurcat. Chaos."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"036214","DOI":"10.1103\/PhysRevE.76.036214","article-title":"Double precision errors in the logistic map: Statistical study and dynamical interpretation","volume":"76","author":"Oteo","year":"2007","journal-title":"Phys. Rev. E."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"043118","DOI":"10.1063\/1.3267510","article-title":"Error distribution in randomly perturbed orbits","volume":"19","author":"Marie","year":"2009","journal-title":"Chaos"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"013122","DOI":"10.1063\/1.2866487","article-title":"A relation on round-off error, attractor size and its dynamics in driven or coupled logistic map system","volume":"18","author":"Shi","year":"2008","journal-title":"Chaos"},{"key":"ref_17","unstructured":"Liu, J., Zhang, H., Song, D., Buza, M.K., Yang, B., and Guo, C. (, January October). A new property of logistic map with scalable precision, Dalian, China."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/11\/5618\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:17:26Z","timestamp":1760217446000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/11\/5618"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,27]]},"references-count":17,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2014,11]]}},"alternative-id":["e16115618"],"URL":"https:\/\/doi.org\/10.3390\/e16115618","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,10,27]]}}}