{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T03:44:38Z","timestamp":1773200678877,"version":"3.50.1"},"reference-count":82,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T00:00:00Z","timestamp":1630022400000},"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>In this paper, the robust stabilization and synchronization of a novel chaotic system are presented. First, a novel chaotic system is presented in which this system is realized by implementing a sigmoidal function to generate the chaotic behavior of this analyzed system. A bifurcation analysis is provided in which by varying three parameters of this chaotic system, the respective bifurcations plots are generated and evinced to analyze and verify when this system is in the stability region or in a chaotic regimen. Then, a robust controller is designed to drive the system variables from the chaotic regimen to stability so that these variables reach the equilibrium point in finite time. The robust controller is obtained by selecting an appropriate robust control Lyapunov function to obtain the resulting control law. For synchronization purposes, the novel chaotic system designed in this study is used as a drive and response system, considering that the error variable is implemented in a robust control Lyapunov function to drive this error variable to zero in finite time. In the control law design for stabilization and synchronization purposes, an extra state is provided to ensure that the saturated input sector condition must be mathematically tractable. A numerical experiment and simulation results are evinced, along with the respective discussion and conclusion.<\/jats:p>","DOI":"10.3390\/e23091110","type":"journal-article","created":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T09:53:23Z","timestamp":1630058003000},"page":"1110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7869-6373","authenticated-orcid":false,"given":"Ahmad Taher","family":"Azar","sequence":"first","affiliation":[{"name":"College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Faculty of Computers and Artificial Intelligence, Benha University, Benha 13511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8800-7578","authenticated-orcid":false,"given":"Fernando E.","family":"Serrano","sequence":"additional","affiliation":[{"name":"Instituto de Investigaci\u00f3n en Energ\u00eda, Universidad Nacional Autonoma de Honduras (UNAH), Tegucigalpa 11101, Honduras"},{"name":"Research Collaborator, Prince Sultan University, Riyadh 11586, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8173-1179","authenticated-orcid":false,"given":"Quanmin","family":"Zhu","sequence":"additional","affiliation":[{"name":"FET\u2013Engineering, Design and Mathematics, University of the West of England, Bristol BS16 1QY, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maamar","family":"Bettayeb","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, University of Sharjah, Sharjah 27272, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Fusco","sequence":"additional","affiliation":[{"name":"Department of Electrical and Information Engineering, Universita degli Studi di Cassino e del Lazio Meridionale, 03043 Cassino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Na","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical & Electrical Engineering, Kunming University of Science and Technology, No. 727 Jingming South Road, Chenggong, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weicun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nashwa Ahmad","family":"Kamal","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Cairo 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Fractals"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"110791","DOI":"10.1016\/j.chaos.2021.110791","article-title":"A fractional-order chaotic system with hidden attractor and self-excited attractor and its DSP implementation","volume":"145","author":"Liu","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/S0893-6080(03)00122-9","article-title":"Chaos control and synchronization, with input saturation, via recurrent neural networks","volume":"16","author":"Sanchez","year":"2003","journal-title":"Neural Netw."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.1016\/j.chaos.2006.04.048","article-title":"Chaos control of Lorenz systems using adaptive controller with input saturation","volume":"34","author":"Yau","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107209","DOI":"10.1016\/j.ymssp.2020.107209","article-title":"Robust attitude control of a 3-DOF helicopter considering actuator saturation","volume":"149","author":"Zhu","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.nahs.2019.01.005","article-title":"Saturation control of switched nonlinear systems","volume":"32","author":"Zhang","year":"2019","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"109457","DOI":"10.1016\/j.automatica.2020.109457","article-title":"Static output feedback control of a chain of integrators with input constraints using multiple saturations and delays","volume":"125","author":"Niu","year":"2021","journal-title":"Automatica"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"109268","DOI":"10.1016\/j.automatica.2020.109268","article-title":"Flexible performance-based robust control for a class of nonlinear systems with input saturation","volume":"122","author":"Yong","year":"2020","journal-title":"Automatica"},{"key":"ref_11","first-page":"839","article-title":"Robust Kinematic Control of Unmanned Aerial Vehicles with Non-holonomic Constraints","volume":"1261","author":"Azar","year":"2020","journal-title":"Proc. Int. Conf. Adv. Intell. Syst. Inform."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.chaos.2017.10.008","article-title":"Control and synchronization of chaos with sliding mode control based on cubic reaching rule","volume":"105","author":"Kocamaz","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1016\/j.chaos.2005.12.012","article-title":"Chaos suppression via observer based active control scheme: Application to Duffing\u2019s oscillator","volume":"32","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1016\/j.chaos.2005.10.058","article-title":"Robust control of chaos in Chua\u2019s circuit based on internal model principle","volume":"31","author":"Lee","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/j.chaos.2005.04.034","article-title":"Robust control of chaos in Lorenz systems subject to mismatch uncertainties","volume":"27","author":"Lin","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.ast.2015.09.026","article-title":"Design of the H-Infinity robust control for the piezoelectric actuator based on chaos optimization algorithm","volume":"47","author":"E","year":"2015","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"8265","DOI":"10.1016\/j.apm.2016.04.019","article-title":"Adaptive robust nonlinear feedback control of chaos in PMSM system with modeling uncertainty","volume":"40","author":"Hu","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.physleta.2006.10.074","article-title":"Robust adaptive dynamic surface control of chaos in permanent magnet synchronous motor","volume":"363","author":"Wei","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.matcom.2020.12.022","article-title":"New synchronization results for a class of nonlinear discrete-time chaotic systems based on synergetic observer and their implementation","volume":"185","author":"Hamiche","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hamoudi, A., Djeghali, N., and Bettayeb, M. (2021, January 22\u201325). Speech encryption based on synchronisation of chaotic Takagi-Sugeno systems using PI observer. Proceedings of the 2021 18th International Multi-Conference on Systems, Signals Devices (SSD), Monastir, Tunisia.","DOI":"10.1109\/SSD52085.2021.9429324"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s11071-019-05266-1","article-title":"Impulsive observer with predetermined finite convergence time for synchronization of fractional-order chaotic systems based on Takagi\u2013Sugeno fuzzy model","volume":"98","author":"Djennoune","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bououden, S., Chadli, M., Ziani, S., and Zelinka, I. (2021). Secure Color Image Transmission Based on the Impulsive Synchronization of Fractional-Order Chaotic Maps Over a Single Channel. Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Springer.","DOI":"10.1007\/978-981-15-6403-1"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"179","DOI":"10.2478\/amcs-2019-0014","article-title":"Synchronization of fractional\u2013order discrete\u2013time chaotic systems by an exact delayed state reconstructor: Application to secure communication","volume":"29","author":"Djennoune","year":"2019","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1016\/j.ijleo.2016.12.016","article-title":"Four-wing attractors in a novel chaotic system with hyperbolic sine nonlinearity","volume":"131","author":"Wang","year":"2017","journal-title":"Opt. Int. J. Light Electron Opt."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3171","DOI":"10.1049\/iet-cta.2017.0396","article-title":"High gain observer design for fractional-order non-linear systems with delayed measurements: Application to synchronisation of fractional-order chaotic systems","volume":"11","author":"Bettayeb","year":"2017","journal-title":"IET Control Theory Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3427","DOI":"10.1016\/j.ijleo.2012.10.055","article-title":"Inverse chaos synchronization between bidirectionally coupled variable multiple time delay systems","volume":"124","author":"Shahverdiev","year":"2013","journal-title":"Opt. Int. J. Light Electron Opt."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1292","DOI":"10.1016\/j.cnsns.2011.07.027","article-title":"Chaos synchronization for master slave piecewise linear systems: Application to Chua\u2019s circuit","volume":"17","author":"Mkaouar","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"105708","DOI":"10.1016\/j.cnsns.2021.105708","article-title":"Chaos synchronization in generalized Lorenz systems and an application to image encryption","volume":"96","author":"Moon","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Vaidyanathan, S., and Azar, A.T. (2016). Dynamic analysis, adaptive feedback control and synchronization of an eight-term 3-D novel chaotic system with three quadratic nonlinearities. Advances in Chaos Theory and Intelligent Control, Springer.","DOI":"10.1007\/978-3-319-30340-6_7"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Vaidyanathan, S., and Azar, A.T. (2016). A novel 4-D four-wing chaotic system with four quadratic nonlinearities and its synchronization via adaptive control method. Advances in Chaos Theory and Intelligent Control, Springer.","DOI":"10.1007\/978-3-319-30340-6_9"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Vaidyanathan, S., and Azar, A.T. (2016). Adaptive control and synchronization of Halvorsen circulant chaotic systems. Advances in Chaos Theory and Intelligent Control, Springer.","DOI":"10.1007\/978-3-319-30340-6_10"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Vaidyanathan, S., and Azar, A.T. (2016). Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. Advances in Chaos Theory and Intelligent Control, Springer.","DOI":"10.1007\/978-3-319-30340-6_11"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Vaidyanathan, S., and Azar, A.T. (2016). Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. Advances in Chaos Theory and Intelligent Control, Springer.","DOI":"10.1007\/978-3-319-30340-6_12"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0375-9601(95)00925-6","article-title":"An experimental study on stabilization of unstable periodic motion in magneto-elastic chaos","volume":"211","author":"Hikihara","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.chaos.2018.12.019","article-title":"On fractional\u2013order discrete\u2013time systems: Chaos, stabilization and synchronization","volume":"119","author":"Khennaoui","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1016\/j.egypro.2017.05.188","article-title":"Stabilization and Control of Chaos Based on Nonlinear Dynamic Inversion","volume":"117","author":"Mukherjee","year":"2017","journal-title":"Energy Procedia"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/0167-2789(94)90062-0","article-title":"Stabilization of period doubling bifurcations and implications for control of chaos","volume":"70","author":"Abed","year":"1994","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1016\/S0960-0779(97)00148-3","article-title":"Feedback-control to realize and stabilize chaos synchronization","volume":"9","author":"Duan","year":"1998","journal-title":"Chaos Solitons Fractals"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1007\/s13042-016-0566-3","article-title":"A new type of hybrid synchronization between arbitrary hyperchaotic maps","volume":"8","author":"Ouannas","year":"2017","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1504\/IJCAT.2017.082868","article-title":"New hybrid synchronisation schemes based on coexistence of various types of synchronisation between master-slave hyperchaotic systems","volume":"55","author":"Ouannas","year":"2017","journal-title":"Int. J. Comput. Appl. Technol."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Azar, A.T., and Vaidyanathan, S. (2016). Qualitative Study and Adaptive Control of a Novel 4-D Hyperchaotic System with Three Quadratic Nonlinearities. Advances in Chaos Theory and Intelligent Control, Springer International Publishing.","DOI":"10.1007\/978-3-319-30340-6_8"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1002\/mma.4099","article-title":"A robust method for new fractional hybrid chaos synchronization","volume":"40","author":"Ouannas","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"132927","DOI":"10.1016\/j.physd.2021.132927","article-title":"A novel antimonotic hyperjerk system: Analysis, synchronization and circuit design","volume":"424","year":"2021","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.physc.2018.11.005","article-title":"Chaos synchronization between Josephson junctions coupled in series and driven by a central junction","volume":"557","author":"Shahverdiev","year":"2019","journal-title":"Phys. C Supercond. Its Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"110459","DOI":"10.1016\/j.chaos.2020.110459","article-title":"Complete synchronization of chaos in systems with nonlinear inertial coupling","volume":"142","author":"Korneev","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_46","first-page":"115","article-title":"Chaos synchronization of stochastic reaction-diffusion time-delay neural networks via non-fragile output-feedback control","volume":"354","author":"Tai","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"126032","DOI":"10.1016\/j.optcom.2020.126032","article-title":"Effect of control parameters on chaos synchronization by means of optical feedback","volume":"472","author":"Ahmad","year":"2020","journal-title":"Opt. Commun."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.jmaa.2009.03.041","article-title":"Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions","volume":"362","author":"Xu","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"6829","DOI":"10.1016\/j.apm.2013.02.023","article-title":"Synchronization and anti-synchronization of chaotic oscillators under input saturation","volume":"37","author":"Rehan","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1922","DOI":"10.1016\/j.phpro.2012.02.283","article-title":"Stabilizing a Class of Chaotic Systems by Using Adaptive Feedback Control","volume":"24","author":"Wang","year":"2012","journal-title":"Phys. Procedia"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"104130","DOI":"10.1016\/j.rinp.2021.104130","article-title":"Chaos control and Penta-compound combination anti-synchronization on a novel fractional chaotic system with analysis and application","volume":"24","author":"Mahmoud","year":"2021","journal-title":"Results Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"103786","DOI":"10.1016\/j.rinp.2020.103786","article-title":"Chaos control and analysis of fractional order neural network under electromagnetic radiation","volume":"21","author":"Allehiany","year":"2021","journal-title":"Results Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"107989","DOI":"10.1016\/j.ymssp.2021.107989","article-title":"Piezoelectric autoparametric vibration energy harvesting with chaos control feature","volume":"161","author":"Tan","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"9524","DOI":"10.1016\/j.apm.2013.05.008","article-title":"Synchronization of chaos in simultaneous time-frequency domain","volume":"37","author":"Liu","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"3493","DOI":"10.1016\/j.aej.2017.11.017","article-title":"Chaos synchronization of nonlinear dynamical systems via a novel analytical approach","volume":"57","author":"Aziz","year":"2018","journal-title":"Alex. Eng. J."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1016\/j.joems.2013.11.007","article-title":"Circuit realization, chaos synchronization and estimation of parameters of a hyperchaotic system with unknown parameters","volume":"22","author":"Elsonbaty","year":"2014","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3516","DOI":"10.1016\/j.apm.2015.10.010","article-title":"Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system","volume":"40","author":"Nour","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"105503","DOI":"10.1016\/j.cnsns.2020.105503","article-title":"(INVITED) Homoclinic puzzles and chaos in a nonlinear laser model","volume":"93","author":"Pusuluri","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"5672","DOI":"10.1016\/j.jde.2019.11.024","article-title":"Multiple chaos arising from single-parametric perturbation of a degenerate homoclinic orbit","volume":"268","author":"Zhu","year":"2020","journal-title":"J. Differ. Equ."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1587","DOI":"10.1016\/j.neucom.2014.08.034","article-title":"Diverting homoclinic chaos in a class of piecewise smooth oscillators to stable periodic orbits using small parametrical perturbations","volume":"149","author":"Li","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"5533","DOI":"10.1016\/j.apm.2011.05.021","article-title":"Homoclinic bifurcation and chaos control in MEMS resonators","volume":"35","author":"Siewe","year":"2011","journal-title":"Appl. Math. Model."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1016\/j.jde.2018.07.078","article-title":"New global bifurcation diagrams for piecewise smooth systems: Transversality of homoclinic points does not imply chaos","volume":"266","author":"Franca","year":"2019","journal-title":"J. Differ. Equ."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.physd.2015.10.009","article-title":"Minimal topological chaos coexisting with a finite set of homoclinic and periodic orbits","volume":"315","author":"Huaraca","year":"2016","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1016\/j.na.2008.11.097","article-title":"Chaos induced by heteroclinic cycles connecting repellers and saddles in locally compact metric spaces","volume":"71","author":"Li","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1016\/j.chaos.2006.07.014","article-title":"Chaos induced by heteroclinic cycles connecting repellers in complete metric spaces","volume":"36","author":"Li","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_66","first-page":"125877","article-title":"Generating Shilnikov chaos in 3D piecewise linear systems","volume":"395","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Natiq, H., Kamel Ariffin, M.R., Asbullah, M.A., Mahad, Z., and Najah, M. (2021). Enhancing Chaos Complexity of a Plasma Model through Power Input with Desirable Random Features. Entropy, 23.","DOI":"10.3390\/e23010048"},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"S\u00e1ndor, B., Schneider, B., L\u00e1z\u00e1r, Z.I., and Ercsey-Ravasz, M. (2021). A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks. Entropy, 23.","DOI":"10.3390\/e23010103"},{"key":"ref_69","doi-asserted-by":"crossref","unstructured":"Petrzela, J. (2021). Evidence of Strange Attractors in Class C Amplifier with Single Bipolar Transistor: Polynomial and Piecewise-Linear Case. Entropy, 23.","DOI":"10.3390\/e23020175"},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Ribeiro, M., Henriques, T., Castro, L., Souto, A., Antunes, L., Costa-Santos, C., and Teixeira, A. (2021). The Entropy Universe. Entropy, 23.","DOI":"10.3390\/e23020222"},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Guo, Y., Liu, T., Zhao, T., Zhang, H., and Guo, X. (2021). Chaotic Time-Delay Signature Suppression and Entropy Growth Enhancement Using Frequency-Band Extractor. Entropy, 23.","DOI":"10.3390\/e23050516"},{"key":"ref_72","doi-asserted-by":"crossref","unstructured":"Marszalek, W., Sadecki, J., and Walczak, M. (2021). Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams. Entropy, 23.","DOI":"10.3390\/e23070876"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"105505","DOI":"10.1016\/j.cnsns.2020.105505","article-title":"Synchronization patterns in rings of time-delayed Kuramoto oscillators","volume":"93","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"105556","DOI":"10.1016\/j.cnsns.2020.105556","article-title":"Efficient synchronization estimation for complex time series using refined cross-sample entropy measure","volume":"94","author":"Shang","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"105494","DOI":"10.1016\/j.cnsns.2020.105494","article-title":"Hidden attractors and multistability in a modified Chua\u2019s circuit","volume":"92","author":"Wang","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"105479","DOI":"10.1016\/j.cnsns.2020.105479","article-title":"Simultaneous vibration mitigation and energy harvesting from a pendulum-type absorber","volume":"92","author":"Kecik","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"105860","DOI":"10.1016\/j.cnsns.2021.105860","article-title":"Lagrangian descriptors and regular motion","volume":"102","author":"Montes","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"105662","DOI":"10.1016\/j.cnsns.2020.105662","article-title":"Magnetic confinement of a neutral atom in a double-wire waveguide: A nonlinear dynamics approach","volume":"101","author":"Salas","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_79","first-page":"e00617","article-title":"Chaotic time series analysis of meteorological parameters in some selected stations in Nigeria","volume":"10","author":"Adewole","year":"2020","journal-title":"Sci. Afr."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.physletb.2018.04.014","article-title":"Chaos in charged AdS black hole extended phase space","volume":"781","author":"Chabab","year":"2018","journal-title":"Phys. Lett. B"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.cherd.2018.04.007","article-title":"Mixing enhancement using chaos theory in fluid dynamics: Experimental and numerical study","volume":"141","author":"Shirmohammadi","year":"2019","journal-title":"Chem. Eng. Res. Des."},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Azar, A.T., and Serrano, F.E. (2020). Stabilization of Port Hamiltonian Chaotic Systems with Hidden Attractors by Adaptive Terminal Sliding Mode Control. 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