{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T14:27:11Z","timestamp":1785421631001,"version":"3.56.0"},"reference-count":30,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T00:00:00Z","timestamp":1781568000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171004"],"award-info":[{"award-number":["12171004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["12171004"],"award-info":[{"award-number":["12171004"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this research, based on Adomian decomposition method (ADM), we construct true fractional-order differential equations. Due to the boosting function brought by the sine function, the system can output infinite coexistence attractors on y\u2013z planes. In particular, this grid effect becomes increasingly obvious as the fractional order increases. Based on this boosting grid idea, in combination with the fractal dynamics, we construct some fractal patterns, e.g., Koch snows. These fractal diagrams all present grid fractal shapes. And then, we design a grid image encryption algorithm. This algorithm is proven to have higher security. The combination of chaos and fractals explores a new research direction. It provides new ideas for research in related fields.<\/jats:p>","DOI":"10.3390\/e28060698","type":"journal-article","created":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T05:56:31Z","timestamp":1781675791000},"page":"698","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Complexity Entropy Analysis of Grid Chaotic System: Image Encryption and DSP Implementation"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8755-8829","authenticated-orcid":false,"given":"Gang","family":"Hu","sequence":"first","affiliation":[{"name":"School of Mathematics, Anshan Normal University, Anshan 114005, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Baolin","family":"Kang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Anshan Normal University, Anshan 114005, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8765-3212","authenticated-orcid":false,"given":"Xiaolin","family":"Ye","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, University of Science and Technology Liaoning, Anshan 114051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Zhang, X., and Hu, H. (2026). Image Encryption Algorithm Based on a New Two-Dimensional Chaotic System and Rotating Dial Model. Entropy, 28.","DOI":"10.3390\/e28050530"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wang, D. (2026). Chaos and Coexisting Attractors of Kolmogorov-Type Permanent-Magnet Synchronous Generators. Entropy, 28.","DOI":"10.3390\/e28050512"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Pastur, L., and Shamis, M. (2026). Area Law for the Entanglement Entropy of Free Fermions in Nonrandom Ergodic Field. Entropy, 28.","DOI":"10.3390\/e28050509"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2893","DOI":"10.1007\/s11071-018-4730-z","article-title":"Memristor-based neural networks with weight simultaneous perturbation training","volume":"95","author":"Wang","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"111992","DOI":"10.1016\/j.chaos.2022.111992","article-title":"A 3D memristive chaotic system with conditional symmetry","volume":"158","author":"Wang","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","first-page":"1035","article-title":"A New Memristive Secure Circuit Against Power Analysis Attack","volume":"68","author":"Wang","year":"2021","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2250057","DOI":"10.1142\/S0218127422500572","article-title":"A Conservative Memristive System with Amplitude Control and Offset Boosting","volume":"32","author":"Zhang","year":"2022","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3200","DOI":"10.1007\/s11071-020-05465-1","article-title":"Infinitely many coexisting conservative flows in a 4D conservative system inspired by LC circuit","volume":"99","author":"Wang","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"30802","DOI":"10.1038\/s41598-024-81423-w","article-title":"Fractional order system identification using a joint multi-innovation fractional gradient descent algorithm","volume":"14","author":"Wang","year":"2024","journal-title":"Sci. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1420","DOI":"10.1007\/s11071-024-10279-6","article-title":"Robust sliding mode control of discrete fractional difference chaotic system","volume":"113","author":"Fu","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1007\/s10473-025-0216-5","article-title":"Approximate controllability of nonlinear evolution fractional control system with delay","volume":"45","author":"Mishra","year":"2025","journal-title":"Acta Math. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s10773-024-05812-7","article-title":"Effects of Noise and Fractional Derivative on the Exact Solutions of the Stochastic Conformable Fractional Fokas System","volume":"63","author":"Liu","year":"2024","journal-title":"Int. J. Theor. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"26133","DOI":"10.1007\/s11071-025-11465-w","article-title":"Data-driven system inference for Caputo discrete fractional chaotic systems","volume":"113","author":"Yang","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"124004","DOI":"10.1088\/1402-4896\/aca2f9","article-title":"Chaotic dynamics in tritrophic interaction with volatile compounds in plants with power law kernel","volume":"97","author":"Sami","year":"2022","journal-title":"Phys. Scr."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"075268","DOI":"10.1088\/1402-4896\/ad56d5","article-title":"Time-fractional lorenz type chaotic systems with asymmetric gaussian uncertainty: Series solutions via extended He-Mohand algorithm in fuzzy-caputo sense","volume":"99","author":"Qayyum","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1007\/s11036-023-02155-1","article-title":"Multistability Analysis of a Fractional-Order Multi-Wing Chaotic System and its Circuit Realization","volume":"29","author":"Liu","year":"2024","journal-title":"Mob. Netw. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"26815","DOI":"10.1007\/s11071-025-11477-6","article-title":"Fractional chaotic dynamics in the rucklidge system and its application to image encryption","volume":"113","author":"Lqbal","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1140\/epjp\/i2018-12309-2","article-title":"Characteristic analysis of the fractional-order hyperchaotic memristive circuit based on the Wien bridge oscillator","volume":"133","author":"Ye","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"075212","DOI":"10.1088\/1402-4896\/ab8eec","article-title":"Characteristic analysis of a simple fractional-order chaotic system with infinitely many coexisting attractors and its DSP implementation","volume":"95","author":"Ye","year":"2020","journal-title":"Phys. Scr."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"112017","DOI":"10.1016\/j.chaos.2022.112017","article-title":"Hidden coexisting hyperchaos of new memristive neuron model and its application in image encryption","volume":"158","author":"Lai","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.neucom.2020.12.109","article-title":"Bicomplex-valued twin-hyperbolic Hopfield neural networks","volume":"434","author":"Kobayashi","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_22","first-page":"4770","article-title":"Wien-Bridge Chaotic Oscillator Based on a Memristor","volume":"66","author":"Wang","year":"2019","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"111994","DOI":"10.1016\/j.chaos.2022.111994","article-title":"IEHC: An efficient image encryption technique using hybrid chaotic map","volume":"158","author":"Kumar","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"112103","DOI":"10.1016\/j.chaos.2022.112103","article-title":"A new method of constructing cyclic symmetric conservative chaotic systems and improved offset boosting control","volume":"158","author":"Zhang","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"112114","DOI":"10.1016\/j.chaos.2022.112114","article-title":"Multi-parameter analysis of transition from conservative to dissipative behaviors for a reversible dynamic system","volume":"159","author":"Li","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","first-page":"20320","article-title":"Chimera states and the global domain of safety in a multi\u2013peril dynamic system","volume":"113","author":"Ye","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1140\/epjp\/s13360-023-03904-7","article-title":"Complex chaotic attractor via fractal process with parabolic map and triangular map","volume":"138","author":"Yan","year":"2023","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1007\/s11071-022-07825-5","article-title":"Desired number of coexisting chaotic attractors using quaternionic fractal","volume":"111","author":"Du","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s12083-025-02186-3","article-title":"Separable reversible data hiding in an encrypted image using unaltered adjacent pixels to enhance information security","volume":"19","author":"Anushiadevi","year":"2026","journal-title":"Peer-To-Peer Netw. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2155","DOI":"10.1038\/s41598-025-31792-7","article-title":"A custom hash algorithm for hosting secure gray scale image repository in public cloud","volume":"16","author":"Murugesan","year":"2026","journal-title":"Sci. Rep."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/6\/698\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T04:36:46Z","timestamp":1782362206000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/6\/698"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,16]]},"references-count":30,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2026,6]]}},"alternative-id":["e28060698"],"URL":"https:\/\/doi.org\/10.3390\/e28060698","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,16]]}}}