{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,20]],"date-time":"2026-06-20T05:19:16Z","timestamp":1781932756318,"version":"3.54.5"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2015,12,18]],"date-time":"2015-12-18T00:00:00Z","timestamp":1450396800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61073187"],"award-info":[{"award-number":["61073187"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61161006"],"award-info":[{"award-number":["61161006"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fundamental Research Funds for the Central Universities of Central South University","award":["2014zzts010"],"award-info":[{"award-number":["2014zzts010"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The fractional-order hyperchaotic Lorenz system is solved as a discrete map by applying the Adomian decomposition method (ADM). Lyapunov Characteristic Exponents (LCEs) of this system are calculated according to this deduced discrete map. Complexity of this system versus parameters are analyzed by LCEs, bifurcation diagrams, phase portraits, complexity algorithms. Results show that this system has rich dynamical behaviors. Chaos and hyperchaos can be generated by decreasing fractional order q in this system. It also shows that the system is more complex when q takes smaller values. SE and C                                                      0                                  complexity algorithms provide a parameter choice criteria for practice applications of fractional-order chaotic systems. The fractional-order system is implemented by digital signal processor (DSP), and a pseudo-random bit generator is designed based on the implemented system, which passes the NIST test successfully.<\/jats:p>","DOI":"10.3390\/e17127882","type":"journal-article","created":{"date-parts":[[2015,12,18]],"date-time":"2015-12-18T10:15:04Z","timestamp":1450433704000},"page":"8299-8311","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":207,"title":["Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System"],"prefix":"10.3390","volume":"17","author":[{"given":"Shaobo","family":"He","sequence":"first","affiliation":[{"name":"School of Physics and Electronics, Central South University Changsha, Changsha 410083, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kehui","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Physics and Electronics, Central South University Changsha, Changsha 410083, China"},{"name":"School of Physics Science and Technology, Xinjiang University, Urumqi 830046, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huihai","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Physics and Electronics, Central South University Changsha, Changsha 410083, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,12,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s11071-013-1073-7","article-title":"Stability analysis for nonlinear fractional-order systems based on comparison principle","volume":"75","author":"Wang","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1591","DOI":"10.1140\/epjst\/e2014-02181-3","article-title":"Dynamics of fractional-order sinusoidally forced simplified Lorenz system and its synchronization","volume":"223","author":"Wang","year":"2014","journal-title":"Eur. Phys. J. Special Top."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1016\/j.camwa.2009.07.003","article-title":"Chaos in fractional ordered Liu system","volume":"59","author":"Daftardar","year":"2010","journal-title":"Comp. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1007\/s11071-013-0894-8","article-title":"Circuit simulation for synchronization of a fractional-order and integer-order chaotic system","volume":"73","author":"Chen","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"729","DOI":"10.3390\/e16020729","article-title":"Robust control of a class of uncertain fractional-order chaotic systems with input nonlinearity via an adaptive sliding mode technique","volume":"16","author":"Tian","year":"2014","journal-title":"Entropy"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.1007\/s11071-014-1244-1","article-title":"Robust synchronization of two different uncertain fractional-order chaotic systems via adaptive sliding mode control","volume":"76","author":"Zhang","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1109\/9.159595","article-title":"Fractal system as represented by singularity function","volume":"37","author":"Charef","year":"1992","journal-title":"IEEE Trans. Auto. Contr."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0895-7177(90)90125-7","article-title":"Review of the decomposition method and some recent results for nonlinear equations","volume":"13","author":"Adomian","year":"1990","journal-title":"Math. Comp. Model."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1109\/TAC.1984.1103551","article-title":"Linear approximation of transfer function with a pole of fractional power","volume":"29","author":"Sun","year":"1984","journal-title":"IEEE Trans. Auto. Contr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1049\/iet-spr:20070053","article-title":"Unreliability of frequency-domain approximation in recognizing chaos in fractional-order systems","volume":"1","author":"Tavazoei","year":"2007","journal-title":"IET Sign. Proc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"030502","DOI":"10.7498\/aps.63.030502","article-title":"Solving of fractional-order chaotic system based on Adomian decomposition algorithm and its complexity analyses","volume":"63","author":"He","year":"2014","journal-title":"Acta Phys. Sin."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"013127","DOI":"10.1063\/1.3314277","article-title":"On the bound of the Lyapunov exponents for the fractional differential systems","volume":"20","author":"Li","year":"2010","journal-title":"Chaos"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/0375-9601(91)90958-B","article-title":"Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data","volume":"153","author":"Ellner","year":"1991","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.chaos.2013.03.001","article-title":"Evaluating Lyapunov exponent spectra with neural networks","volume":"51","author":"Maus","year":"2013","journal-title":"Chaos Solit. Fract."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1350050","DOI":"10.1142\/S0218127413500508","article-title":"An application of Adomian decomposition for analysis of fractional-order chaotic systems","volume":"23","author":"Caponetto","year":"2013","journal-title":"Int. J. Bifur. Chaos"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.physa.2005.05.025","article-title":"Intensive statistical complexity measure of pseudorandom number generators","volume":"356","author":"Larrondo","year":"2005","journal-title":"Phys. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3458","DOI":"10.3390\/e15093458","article-title":"Analysis of EEG via multivariate empirical mode decomposition for depth of anesthesia based on sample entropy","volume":"15","author":"Wei","year":"2013","journal-title":"Entropy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.medengphy.2008.04.005","article-title":"Measuring complexity using FuzzyEn, ApEn, and SampEn","volume":"31","author":"Chen","year":"2009","journal-title":"Med. Eng. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"011915","DOI":"10.1103\/PhysRevE.79.011915","article-title":"Rapidly detecting disorder in rhythmic biological signals: A spectral entropy measure to identify cardiac arrhythmias","volume":"79","author":"Phillip","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1007\/BF02507729","article-title":"Mathematical foundation of a new complexity measure","volume":"26","author":"Shen","year":"2005","journal-title":"Appl. Math. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"050506","DOI":"10.1088\/1674-1056\/22\/5\/050506","article-title":"Complexity analyses of multi-wing chaotic systems","volume":"22","author":"He","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"010501","DOI":"10.7498\/aps.62.010501","article-title":"Complexity analysis of chaotic pseudo-random sequence based on spectral entropy algorithm","volume":"62","author":"Sun","year":"2013","journal-title":"Acta Phys. Sin."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.jneumeth.2006.06.023","article-title":"Quantitative analysis of brain optical images with 2D C0 complexity measure","volume":"159","author":"Cao","year":"2007","journal-title":"J. Neurosci. Meth."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1845","DOI":"10.1142\/S0218127408021415","article-title":"Bifurcation and chaos in the fractional-order Chen system via a time-domain approach","volume":"18","author":"Cafagna","year":"2008","journal-title":"Int. J. Bifur. Chaos"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/S0096-3003(01)00167-9","article-title":"Analytical approximate solutions for nonlinear fractional differential equations","volume":"131","author":"Shawagfeh","year":"2002","journal-title":"Appl. Math. Comp."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.physleta.2006.09.042","article-title":"The generation and circuit implementation of a new hyper-chaos based upon Lorenz system","volume":"361","author":"Gao","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_29","unstructured":"NIST Computer Security Resource Center, Available online: http:\/\/csrc.nist.gov\/groups\/ST\/toolkit\/rng\/documentation_software.html."},{"key":"ref_30","unstructured":"Runkin, A., Soto, J., and Nechvatal, J. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, Available online: http:\/\/nvlpubs.nist.gov\/nistpubs\/Legacy\/SP\/nistspecialpublication800-22.pdf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1879","DOI":"10.1007\/s11071-015-2284-x","article-title":"FPGA realization of a chaotic communication system applied to image processing","volume":"82","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1049\/el:20010784","article-title":"Versatile DSP-based chaotic communication system","volume":"37","author":"Hidalgo","year":"2001","journal-title":"Electr. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/12\/7882\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:54:22Z","timestamp":1760216062000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/12\/7882"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,18]]},"references-count":32,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2015,12]]}},"alternative-id":["e17127882"],"URL":"https:\/\/doi.org\/10.3390\/e17127882","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,12,18]]}}}