{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T14:46:13Z","timestamp":1766155573270,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T00:00:00Z","timestamp":1689033600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Research Foundation for Advanced Talents of West Anhui University","award":["WGKQ2021050","2008085MA20"],"award-info":[{"award-number":["WGKQ2021050","2008085MA20"]}]},{"DOI":"10.13039\/501100003995","name":"Anhui Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["WGKQ2021050","2008085MA20"],"award-info":[{"award-number":["WGKQ2021050","2008085MA20"]}],"id":[{"id":"10.13039\/501100003995","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper proposes a fractional-order chaotic system using a tri-stable locally active memristor. The characteristics of the memristor, dynamic mechanism of oscillation, and behaviors of the proposed system were analyzed, and then a visually meaningful image encryption scheme was designed based on the chaotic system, DNA encoding, and integer wavelet transform (IWT). Firstly, the mathematical model of the memristor was designed, which was nonvolatile, locally active, and tri-stable. Secondly, the stability, dynamic mechanism of oscillation, bifurcation behaviors, and complexity of the fractional-order memristive chaotic system were investigated and the conditions of stability were obtained. Thirdly, the largest Lyapunov exponent, bifurcation diagram, and complexity of the novel system were calculated and the coexisting bifurcation, coexisting attractors, spectral entropy, and so on are shown. Finally, a visually meaningful image encryption scheme based on the proposed system was designed, and its security was assessed by statistical analysis and different attacks. Numerical simulation demonstrated the effectiveness of the theoretical analysis and high security of the proposed image encryption scheme.<\/jats:p>","DOI":"10.3390\/sym15071398","type":"journal-article","created":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T01:40:02Z","timestamp":1689039602000},"page":"1398","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Novel Fractional-Order Memristive Chaotic Circuit with Coexisting Double-Layout Four-Scroll Attractors and Its Application in Visually Meaningful Image Encryption"],"prefix":"10.3390","volume":"15","author":[{"given":"Yuebo","family":"Wu","sequence":"first","affiliation":[{"name":"School of Electrical and Opto-Electronic Engineering, West Anhui University, Lu\u2019an 237012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Duansong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical and Opto-Electronic Engineering, West Anhui University, Lu\u2019an 237012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tan","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Electrical and Opto-Electronic Engineering, West Anhui University, Lu\u2019an 237012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinzhong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Electrical and Opto-Electronic Engineering, West Anhui University, Lu\u2019an 237012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Electrical and Opto-Electronic Engineering, West Anhui University, Lu\u2019an 237012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1109\/TCT.1971.1083337","article-title":"Memristor-The missing circuit element","volume":"18","author":"Chua","year":"1971","journal-title":"IEEE Trans. Circuit Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1038\/nature06932","article-title":"The missing memristor found","volume":"453","author":"Strukov","year":"2008","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1038\/s41467-020-20692-1","article-title":"Dynamic memristor-based reservoir computing for high-efficiency temporal signal processing","volume":"12","author":"Zhong","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1091","DOI":"10.1016\/j.rinp.2018.12.092","article-title":"Addressing the sneak-path problem in crossbar RRAM devices using memristor-based one schottky diode-one resistor array","volume":"12","author":"Gul","year":"2019","journal-title":"Results Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2050045","DOI":"10.1142\/S0218127420500455","article-title":"Memristor synapse-based morris-lecar model: Bifurcation analyses and FPGA-based validations for periodic and chaotic bursting\/spiking firings","volume":"30","author":"Bao","year":"2020","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2821","DOI":"10.1007\/s11071-020-06072-w","article-title":"Initial offset boosting coexisting attractors in memristive multi-double-scroll hopfield neural network","volume":"102","author":"Zhang","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"110751","DOI":"10.1016\/j.chaos.2021.110751","article-title":"A piezoelectric sensing neuron and resonance synchronization between auditory neurons under stimulus","volume":"145","author":"Zhou","year":"2021","journal-title":"Chaos Solit. Fract."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wu, X., He, S., Tan, W., and Wang, H. (2022). From Memristor-Modeled Jerk System to the Nonlinear Systems with Memristor. Symmetry, 14.","DOI":"10.3390\/sym14040659"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lei, T., Zhou, Y., Fu, H., Huang, L., and Zang, H. (2022). Multistability Dynamics Analysis and Digital Circuit Implementation of Entanglement-Chaos Symmetrical Memristive System. Symmetry, 14.","DOI":"10.3390\/sym14122586"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Yang, B., Wang, Z., Tian, H., and Liu, J. (2022). Symplectic Dynamics and Simultaneous Resonance Analysis of Memristor Circuit Based on Its van der Pol Oscillator. Symmetry, 14.","DOI":"10.3390\/sym14061251"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dai, W., Xu, X., Song, X., and Li, G. (2022). Audio Encryption Algorithm Based on Chen Memristor Chaotic System. Symmetry, 14.","DOI":"10.3390\/sym14010017"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.aeue.2019.05.031","article-title":"Dynamical investigation and chaotic associated behaviors of memristor chua\u2019s circuit with a non-ideal voltage-controlled memristor and its application to voice encryption","volume":"107","author":"Rajagopal","year":"2019","journal-title":"AEU-Int. J. Electron. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"110504","DOI":"10.1088\/1674-1056\/abbbfe","article-title":"Memristor-based hyper-chaotic circuit for image encryption","volume":"29","author":"Chen","year":"2020","journal-title":"Chin. Phys. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1016\/j.ins.2022.06.011","article-title":"A visually secure image encryption scheme using adaptive thresholding sparsification compression sensing model and newly-designed memristive chaotic map","volume":"607","author":"Zhu","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_15","first-page":"102944","article-title":"A memristive RLC oscillator dynamics applied to image encryption","volume":"61","author":"Tsafack","year":"2021","journal-title":"J. Info. Sec. App."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3435","DOI":"10.1142\/S0218127405014337","article-title":"Local activity is the origin of complexity","volume":"15","author":"Chua","year":"2005","journal-title":"Int. J. Bifur. Chaos"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1630009","DOI":"10.1142\/S0218127416300093","article-title":"Chua corsage memristor oscillator via Hopf bifurcation","volume":"26","author":"Mannan","year":"2016","journal-title":"Int. J. Bifur. Chaos."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"104001","DOI":"10.1088\/0268-1242\/29\/10\/104001","article-title":"If it\u2019s pinched it\u2019s a memristor","volume":"29","author":"Chua","year":"2014","journal-title":"Semicond. Sci. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Williams, R.S., and Pickett, M.D. (2013). The Art and Science of Constructing a Memristor Model: Memristors and Memristive Systems, Springer.","DOI":"10.1007\/978-1-4614-9068-5_3"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4979","DOI":"10.1109\/TCSI.2021.3126657","article-title":"NbO2-Mott memristor: A circuit-theoretic investigation","volume":"68","author":"Messaris","year":"2021","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1730011","DOI":"10.1142\/S0218127417300117","article-title":"Chua corsage memristor: Phase portraits, basin of attraction, and coexisting pinched hysteresis loops","volume":"27","author":"Mannan","year":"2017","journal-title":"Int. J. Bifur. Chaos"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/MCAS.2018.2821724","article-title":"Oscillation with 4-lobe Chua corsage memristor","volume":"18","author":"Mannan","year":"2018","journal-title":"IEEE Circuits Syst. Mag."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2018\/8405978","article-title":"Exact analysis and physical realization of the 6-lobe Chua corsage memristor","volume":"2018","author":"Mannan","year":"2018","journal-title":"Complexity"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105203","DOI":"10.1016\/j.cnsns.2020.105203","article-title":"A bistable nonvolatile locally-active memristor and its complex dynamics","volume":"84","author":"Dong","year":"2020","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"053118","DOI":"10.1063\/1.5143071","article-title":"A simple locally active memristor and its application in HR neurons","volume":"30","author":"Tan","year":"2020","journal-title":"Chaos"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4523","DOI":"10.1007\/s11071-021-06476-2","article-title":"A fractional-order multistable locally active memristor and its chaotic system with transient transition, state jump","volume":"104","author":"Xie","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4433","DOI":"10.1007\/s11071-022-07371-0","article-title":"Coexisting multi-stability of Hopfield neural network based on coupled fractional-order locally active memristor and its application in image encryption","volume":"108","author":"Ding","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"120515","DOI":"10.1088\/1674-1056\/ac1fdf","article-title":"Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation","volume":"30","author":"Yang","year":"2021","journal-title":"Chin. Phy. B."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.ins.2015.06.049","article-title":"Image encryption: Generating visually meaningful encrypted images","volume":"324","author":"Bao","year":"2015","journal-title":"Info. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.sigpro.2016.11.016","article-title":"A visually secure image encryption scheme based on compressive sensing","volume":"134","author":"Chai","year":"2017","journal-title":"Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.sigpro.2018.10.001","article-title":"A visually secure image encryption scheme based on parallel compressive sensing","volume":"155","author":"Wang","year":"2019","journal-title":"Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"103263","DOI":"10.1016\/j.dsp.2021.103263","article-title":"Generating visually secure encrypted images by partial block pairing-substitution and semi-tensor product compressed sensing","volume":"120","author":"Ping","year":"2022","journal-title":"Digit. Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2250130","DOI":"10.1142\/S0218127422501309","article-title":"A Novel Multiscroll Memristive Hopfield Neural Network","volume":"32","author":"Li","year":"2022","journal-title":"Int. J. Bifur. Chaos."},{"key":"ref_34","first-page":"112241","article-title":"A tristable locally-active memristor and its complex dynamics","volume":"160","author":"Ying","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Carpinteri, A., and Mainardi, F. (1997). Fractals and Fractional Calculus in Continuum Mechanics, Springer. International Centre for Mechanical Sciences.","DOI":"10.1007\/978-3-7091-2664-6"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-2789(96)00216-3","article-title":"An efficient QR based method for the computation of Lyapunov exponents","volume":"101","author":"Bremen","year":"1997","journal-title":"Physica D"},{"key":"ref_37","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_38","doi-asserted-by":"crossref","first-page":"7365","DOI":"10.1007\/s11042-021-11803-1","article-title":"An image encryption scheme based on chaotic logarithmic map and key generation using deep CNN","volume":"81","author":"Erkan","year":"2022","journal-title":"Multimed. Tools Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"112625","DOI":"10.1016\/j.chaos.2022.112625","article-title":"Visually meaningful image encryption scheme based on new-designed chaotic map and random scrambling diffusion strategy","volume":"164","author":"Wang","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"107629","DOI":"10.1016\/j.sigpro.2020.107629","article-title":"A robust meaningful image encryption scheme based on block compressive sensing and SVD embedding","volume":"175","author":"Zhu","year":"2020","journal-title":"Signal Process."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1398\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:10:21Z","timestamp":1760127021000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1398"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,11]]},"references-count":41,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071398"],"URL":"https:\/\/doi.org\/10.3390\/sym15071398","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,7,11]]}}}