{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:22:14Z","timestamp":1787239334447,"version":"build-2736575974"},"reference-count":46,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,12,27]],"date-time":"2017-12-27T00:00:00Z","timestamp":1514332800000},"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>This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is its multiple attractors caused by different initial values. With the change of parameters, the system experiences mono-stability, bi-stability, mono-periodicity, bi-periodicity, one strange attractor, and two coexisting strange attractors. The complex dynamic behaviors of the system are revealed by analyzing the corresponding equilibria and using the numerical simulation method. In addition, an electronic circuit is given for implementing the chaotic attractors of the system. Using the new chaotic system, an S-Box is developed for cryptographic operations. Moreover, we test the performance of this produced S-Box and compare it to the existing S-Box studies.<\/jats:p>","DOI":"10.3390\/e20010012","type":"journal-article","created":{"date-parts":[[2017,12,27]],"date-time":"2017-12-27T11:16:00Z","timestamp":1514373360000},"page":"12","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":95,"title":["A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design"],"prefix":"10.3390","volume":"20","author":[{"given":"Qiang","family":"Lai","sequence":"first","affiliation":[{"name":"School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Akif","family":"Akgul","sequence":"additional","affiliation":[{"name":"Department of Electrical and Electronics Engineering, Faculty of Technology, Sakarya University, Serdivan 54187, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chunbiao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guanghui","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Electrical and Electronic Engineering, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"\u00dcnal","family":"\u00c7avu\u015fo\u011flu","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, Faculty of Computer and Information Sciences, Sakarya University, Serdivan 54187, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic non-periodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/0375-9601(76)90101-8","article-title":"An equation for continuous chaos","volume":"57","year":"1976","journal-title":"Phys. Lett. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1142\/S0218127499001024","article-title":"Yet another chaotic attractor","volume":"9","author":"Chen","year":"1999","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_4","first-page":"659","article-title":"A new chaotic attractor coined","volume":"3","author":"Chen","year":"2000","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s11071-013-1069-3","article-title":"Analysis of a new three-dimensional system with multiple chaotic attractors","volume":"75","author":"Guan","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1650098","DOI":"10.1142\/S021812741650098X","article-title":"A simple chaotic flow with a plane of equilibria","volume":"26","author":"Jafari","year":"2016","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1350152","DOI":"10.1142\/S0218127413501526","article-title":"Generation of multi-wing chaotic attractors from a Lorenz-like system","volume":"23","author":"Lai","year":"2013","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/s11071-012-0669-7","article-title":"Constructing a chaotic system with any number of equilibria","volume":"71","author":"Wang","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"026110","DOI":"10.1063\/1.2743611","article-title":"Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons","volume":"17","author":"Riecke","year":"2007","journal-title":"Chaos"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1098\/rstb.2011.0254","article-title":"Multistability in perception: Binding sensory modalities, an overview","volume":"367","author":"Schwartz","year":"2012","journal-title":"Philos. Trans. R. Soc. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"043833","DOI":"10.1103\/PhysRevA.92.043833","article-title":"Nonlinear mechanisms for multistability in microring lasers","volume":"92","author":"Yuan","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"10389","DOI":"10.1016\/j.ijleo.2016.08.046","article-title":"Variable-boostable chaotic flows","volume":"127","author":"Li","year":"2016","journal-title":"Optik"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1450131","DOI":"10.1142\/S0218127414501314","article-title":"Multistability in the Lorenz system: A broken butterfly","volume":"24","author":"Li","year":"2014","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1750160","DOI":"10.1142\/S0218127417501607","article-title":"Infinite multistability in a self-reproducing chaotic system","volume":"27","author":"Li","year":"2017","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1016\/j.physleta.2016.01.045","article-title":"Hypogenetic chaotic jerk flows","volume":"380","author":"Li","year":"2016","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1007\/s11071-016-3118-1","article-title":"Constructing chaotic systems with conditional symmetry","volume":"87","author":"Li","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"103126","DOI":"10.1063\/1.4934653","article-title":"Periodicity, chaos, and multiple attractors in a memristor-based Shinriki\u2019s circuit","volume":"25","author":"Kengne","year":"2015","journal-title":"Chaos"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s11071-015-2364-y","article-title":"Dynamical analysis of a simple autonomous jerk system with multiple attractors","volume":"83","author":"Kengne","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1007\/s11071-016-2988-6","article-title":"Coexisting infinitely many attractors in active band-pass filter-based memristive circuit","volume":"86","author":"Bao","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.chaos.2015.12.007","article-title":"Multiple attractors in a non-ideal active voltage-controlled memristor based Chua\u2019s circuit","volume":"83","author":"Xu","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3000","DOI":"10.1016\/j.ijleo.2015.12.089","article-title":"Research on a new 3D autonomous chaotic system with coexisting attractors","volume":"127","author":"Lai","year":"2016","journal-title":"Optik"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1007\/s12555-015-0056-5","article-title":"Coexisting attractors gnerated from a new 4D smooth chaotic system","volume":"14","author":"Lai","year":"2016","journal-title":"Int. J. Control Autom. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1650177","DOI":"10.1142\/S0218127416501777","article-title":"Generating multiple chaotic atractors from Sprott B system","volume":"26","author":"Lai","year":"2016","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s11071-015-2144-8","article-title":"Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system","volume":"82","author":"Wei","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.physleta.2007.02.021","article-title":"Theory and practice of chaotic cryptography","volume":"366","author":"Amigo","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2129","DOI":"10.1142\/S0218127406015970","article-title":"Some basic cryptographic requirements for chaos-based cryptosystems","volume":"16","author":"Alvarez","year":"2006","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8851","DOI":"10.1109\/ACCESS.2017.2693301","article-title":"A new chaotic system with stable equilibrium: From theoretical model to circuit implementation","volume":"5","author":"Wang","year":"2017","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.cnsns.2015.03.003","article-title":"FPGA realization of multi-scroll chaotic oscillators","volume":"27","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2143","DOI":"10.1007\/s11071-016-2820-3","article-title":"Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs","volume":"85","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5113","DOI":"10.1016\/j.amc.2012.11.021","article-title":"A survey on the integrated design of chaotic oscillators","volume":"219","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5491","DOI":"10.1016\/j.ijleo.2016.02.066","article-title":"A new four-scroll chaotic attractor and its engineering applications","volume":"127","author":"Akgul","year":"2016","journal-title":"Optik"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1002\/sec.1414","article-title":"A novel chaos based encryption algorithm over TCP data packet for secure communication","volume":"9","author":"Cavusoglu","year":"2016","journal-title":"Secur. Commun. Netw."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"7062","DOI":"10.1016\/j.ijleo.2016.05.010","article-title":"A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications","volume":"127","author":"Akgul","year":"2016","journal-title":"Optik"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s11071-015-2501-7","article-title":"Chaos-based engineering applications with a 3D chaotic system without equilibrium points","volume":"84","author":"Akgul","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4328","DOI":"10.1016\/j.cnsns.2012.01.029","article-title":"Integrated circuit generating 3- and 5-scroll attractors","volume":"17","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1007\/s11071-017-3505-2","article-title":"Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points","volume":"89","author":"Fraga","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1016\/j.chaos.2005.10.022","article-title":"An extended method for obtaining S-boxes based on three-dimensional chaotic Baker maps","volume":"31","author":"Chen","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2303","DOI":"10.1007\/s11071-012-0621-x","article-title":"A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems","volume":"70","author":"Khan","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3089","DOI":"10.1016\/j.cnsns.2008.12.005","article-title":"A block cipher with dynamic S-boxes based on tent map","volume":"14","author":"Wang","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3733","DOI":"10.1016\/j.physleta.2010.07.019","article-title":"A method for designing strong S-Boxes based on chaotic Lorenz system","volume":"374","author":"Ozkaynak","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1109\/81.904880","article-title":"Chaos and cryptography: Block encryption ciphers based on chaotic maps","volume":"48","author":"Jakimoski","year":"2001","journal-title":"IEEE Trans. Circuits Syst. I Fundam. Theory Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1791","DOI":"10.1007\/s11071-012-0573-1","article-title":"A novel approach for designing substitution-boxes based on nonlinear chaotic algorithm","volume":"70","author":"Hussain","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1901","DOI":"10.1016\/S0960-0779(04)00454-0","article-title":"A method for designing dynamical S-boxes based on discretized chaotic map","volume":"23","author":"Tang","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/3-540-39799-X_41","article-title":"On the design of S-Boxes","volume":"Volume 218","author":"Williams","year":"1986","journal-title":"Advances in Cryptology-CRYPTO\u201985 Proceedings"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2450","DOI":"10.1016\/j.camwa.2012.05.017","article-title":"Construction of S8 Liu J S-boxes and their applications","volume":"64","author":"Hussain","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/BF00630563","article-title":"Differential cryptanalysis of DES-like cryptosystems","volume":"4","author":"Biham","year":"1991","journal-title":"J. Cryptol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:55:41Z","timestamp":1760208941000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,27]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010012"],"URL":"https:\/\/doi.org\/10.3390\/e20010012","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12,27]]}}}