{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T04:44:22Z","timestamp":1761108262398,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,1,19]],"date-time":"2016-01-19T00:00:00Z","timestamp":1453161600000},"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":["No.51305355"],"award-info":[{"award-number":["No.51305355"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the China Scholarship","award":["No.201406295017"],"award-info":[{"award-number":["No.201406295017"]}]},{"name":"the National Fundamental Research Program of China (973 Program)","award":["No.2012CB026002"],"award-info":[{"award-number":["No.2012CB026002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The dynamics of network traffic are complex and nonlinear, and chaotic behaviors and their prediction, which play an important role in local area networks (LANs), are studied in detail, using the largest Lyapunov exponent. With the introduction of phase space reconstruction based on the time sequence, the high-dimensional traffic is projected onto the low dimension reconstructed phase space, and a reduced dynamic system is obtained from the dynamic system viewpoint. Then, a numerical method for computing the largest Lyapunov exponent of the low-dimensional dynamic system is presented. Further, the longest predictable time, which is related to chaotic behaviors in the system, is studied using the largest Lyapunov exponent, and the Wolf method is used to predict the evolution of the traffic in a local area network by both Dot and Interval predictions, and a reliable result is obtained by the presented method. As the conclusion, the results show that the largest Lyapunov exponent can be used to describe the sensitivity of the trajectory in the reconstructed phase space to the initial values. Moreover, Dot Prediction can effectively predict the flow burst. The numerical simulation also shows that the presented method is feasible and efficient for predicting the complex dynamic behaviors in LAN traffic, especially for congestion and attack in networks, which are the main two complex phenomena behaving as chaos in networks.<\/jats:p>","DOI":"10.3390\/e18010032","type":"journal-article","created":{"date-parts":[[2016,1,19]],"date-time":"2016-01-19T12:55:32Z","timestamp":1453208132000},"page":"32","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent"],"prefix":"10.3390","volume":"18","author":[{"given":"Yan","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering, Northwestern Polytechnical University, Xi\u2019an 710072, China"}]},{"given":"Jiazhong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Energy and Power Engineering, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}]}],"member":"1968","published-online":{"date-parts":[[2016,1,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"012061","DOI":"10.1088\/1742-6596\/96\/1\/012061","article-title":"Nonlinear Dynamics of the Small-World Networks-Hopf Bifurcation, Sequence of Period-Doubling Bifurcations and Chaos","volume":"96","author":"Liu","year":"2008","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1016\/j.comnet.2013.02.012","article-title":"Real-Time Volume Control for Interactive Network Traffic","volume":"57","author":"Chu","year":"2013","journal-title":"Comput. Netw."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"050506","DOI":"10.1088\/1674-1056\/20\/5\/050506","article-title":"Correlation Dimension Based Nonlinear Analysis of Network Traffics with Different Application Protocols","volume":"20","author":"Wang","year":"2011","journal-title":"Chin. Phys. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1016\/j.nonrwa.2007.11.025","article-title":"Nonlinear Analysis of Wireless LAN Traffic","volume":"10","author":"Feng","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2364","DOI":"10.1016\/j.comnet.2013.04.010","article-title":"Congestion Removing Performance of Stackable ROADM in WDM Networks under Dynamic Traffic","volume":"57","author":"Nooruzzaman","year":"2013","journal-title":"Comput. Netw."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.comnet.2014.10.007","article-title":"Congestion Control in Wireless Sensor Networks through Dynamic Alternative Path Selection","volume":"75","author":"Sergiou","year":"2014","journal-title":"Comput. Netw."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1109\/TFUZZ.2003.814860","article-title":"Fuzzy Adaptive Predictive Flow Control of ATM Network Traffic","volume":"11","author":"Chen","year":"2003","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.compeleceng.2004.11.004","article-title":"A Comparison of AR Full Motion Video Traffic Models in B-ISDN","volume":"31","author":"Alheraish","year":"2005","journal-title":"Comput. Electr. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1061\/(ASCE)0733-947X(2009)135:9(658)","article-title":"Multivariate Traffic Forecasting Technique Using Cell Transmission Model and SARIMA Model","volume":"135","author":"Szeto","year":"2009","journal-title":"J. Transp. Eng. ASCE"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"85","DOI":"10.3141\/2215-09","article-title":"Seasonal Autoregressive Integrated Moving Average and Support Vector Machine Models Prediction of Short-Term Traffic Flow on Freeways","volume":"2215","author":"Zhang","year":"2011","journal-title":"Transp. Res. Rec."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1016\/j.engappai.2008.04.019","article-title":"A Study on the Network Traffic of Connexion by Boeing: Modeling with Artificial Neural Networks","volume":"21","author":"Swift","year":"2008","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_12","first-page":"361","article-title":"Detecting Strange Attractors in Turbulence","volume":"898","author":"Takens","year":"1981","journal-title":"Lect. Notes Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/0167-2789(96)00054-1","article-title":"State Space Reconstruction Parameters in the Analysis of Chaotic Time Series","volume":"95","author":"Kugiumtzis","year":"1996","journal-title":"Physica D"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/S0167-2789(98)00240-1","article-title":"Nonlinear Dynamics, Delay Times, and Embedding Windows","volume":"127","author":"Kim","year":"1999","journal-title":"Physica D"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/0167-2789(83)90113-6","article-title":"Five Turbulent Problems","volume":"7","author":"David","year":"1983","journal-title":"Physica D"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2291","DOI":"10.1088\/0305-4470\/29\/10\/009","article-title":"Strong Chaos without the Butterfly Effect in Dynamical Systems with Feedback","volume":"29","author":"Boffetta","year":"1996","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2146","DOI":"10.1103\/PhysRevA.39.2146","article-title":"Lyapunov Exponents from Time Series of Acoustic Chaos","volume":"39","author":"Joach","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov Exponent from a Time Series","volume":"16","author":"Wolf","year":"1985","journal-title":"Physica D"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/0167-2789(93)90009-P","article-title":"A Practical Method for Calculating Largest Lyapunov Exponents from Small Data Sets","volume":"65","author":"Rosenstein","year":"1993","journal-title":"Physica D"},{"key":"ref_20","first-page":"28","article-title":"A Prediction of Network Traffic Flow Based on Lyapunov Exponent","volume":"27","author":"Luo","year":"2004","journal-title":"J. Chongqing Univ."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/1\/32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:17:54Z","timestamp":1760210274000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/1\/32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,19]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1]]}},"alternative-id":["e18010032"],"URL":"https:\/\/doi.org\/10.3390\/e18010032","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,1,19]]}}}