{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:46:04Z","timestamp":1760233564991,"version":"build-2065373602"},"reference-count":50,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,18]],"date-time":"2021-01-18T00:00:00Z","timestamp":1610928000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a complete non-linear symmetric oscillator model using the Hamiltonian approach has been developed and used to describe the cardiovascular conduction process\u2019s dynamics, as the signal generated from the cardiovascular muscle is non-deterministic and random. Electrocardiogram (ECG) signal is a significant factor in the cardiovascular system as most of the medical diagnoses can be well understood by observing the ECG signal\u2019s amplitude. A non-linear cardiovascular muscle model has been proposed in this study, where a modified vanderPol symmetric oscillator-based equation is used. Gone are the days whena non-linear system had been designed using the describing function technique. It is better to design a non-linear model using the Hamiltonian dynamical equation for its high accuracy and flexibility. Varying a non-linear spring constant using this type of approach is more comfortable than the traditional describing function technique. Not only that but different initial conditions can also be taken for experimental purposes. It never affects the overall modeling. The Hamiltonian approach provides the energy of an asymmetric oscillatory system of that cardiovascular conduction system. A non-linear symmetric oscillator was initially depicted by the non-linear mass-spring (two degrees of freedom) model. The motion of an uncertain non-linear cardiovascular system has been solved considering second-order approximation, which also demonstrates the possibility of introducing spatial dimensions. Finally, the model\u2019s natural frequency expression has also been simulated and is composed of the previously published result.<\/jats:p>","DOI":"10.3390\/sym13010151","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T02:36:05Z","timestamp":1611196565000},"page":"151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Dynamics of Cardiovascular Muscle Using a Non-Linear Symmetric Oscillator"],"prefix":"10.3390","volume":"13","author":[{"given":"Soumyendu","family":"Bhattacharjee","sequence":"first","affiliation":[{"name":"Department of Electronics Science, West Bengal State University, Barasat 700126, West Bengal, India"}]},{"given":"Aishwarya","family":"Banerjee","sequence":"additional","affiliation":[{"name":"Electronics and Communication Engineering, Tech-Lateetud Private Limited, Kolkata 700091, India"}]},{"given":"Amit","family":"Rakshit","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Kanad Institute of Engineering &amp; Management, Mankar 713144, India"}]},{"given":"Swapan","family":"Bhattacharyya","sequence":"additional","affiliation":[{"name":"Department of Electronics &amp; Communication Engineering, Siliguri Institute of Technology, Salbari, P.O., Sukna, Dist., Darjeeling 734009, West Bengal, India"}]},{"given":"Swati","family":"Chowdhuri","sequence":"additional","affiliation":[{"name":"Institute of Engineering and Management, Salt Lake Electronics Complex, Sector V, Kolkata 700091, West Bengal, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6583-8416","authenticated-orcid":false,"given":"Biswajit","family":"Sarkar","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul 03722, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0981-5383","authenticated-orcid":false,"given":"Biswarup","family":"Neogi","sequence":"additional","affiliation":[{"name":"Department of Electronics &amp; Communication Engineering, JIS College of Engineering, Kalyani 741235, India"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Islam, M.R., and Zulkarnain, M.I. (2015, January 17\u201319). Effect of pump characteristic (HQ) curves on the efficiency of Total Artificial Heart. Proceedings of the 2015 International Conference on Advances in Electrical Engineering (ICAEE), Dhaka, Bangladesh.","DOI":"10.1109\/ICAEE.2015.7506795"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2908","DOI":"10.1161\/01.CIR.89.6.2908","article-title":"First use of an untethered, vented electric left ventricular assist device for long-term support","volume":"89","author":"Frazier","year":"1994","journal-title":"Circulation"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ghosal, S., Darbar, R., Neogi, B., Das, A., and Tibarewala, D.N. (2012, January 5\u20137). Application of swarm intelligence computation techniques in PID controller tuning: A review. Proceedings of the International Conference on Information Systems Design and Intelligent Applications 2012 (INDIA 2012), Visakhapatnam, India.","DOI":"10.1007\/978-3-642-27443-5_23"},{"key":"ref_4","unstructured":"Choi, S.W., and Min, B.G. (2014). Ventricular Assist Device Cannula and Ventricular Assist Device Including the Same. (8,771,165), U.S. Patent."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1016\/j.jclepro.2019.01.078","article-title":"Mathematical and analytical approach for the management of defective items in a multi-stage production system","volume":"218","author":"Sarkar","year":"2019","journal-title":"J. Clean. Prod."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"119055","DOI":"10.1016\/j.jclepro.2019.119055","article-title":"A sustainable development framework for a cleaner multi-item multi-stage textile production system with a process improvement initiative","volume":"246","author":"Tayyab","year":"2020","journal-title":"J. Clean. Prod."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Dey, B.K., Pareek, S., Tayyab, M., and Sarkar, B. (2020). Autonomation policy to control work-in-process inventory in a smart production system. Int. J. Prod. Res., in press.","DOI":"10.1080\/00207543.2020.1722325"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"323","DOI":"10.4028\/www.scientific.net\/AMM.703.323","article-title":"Axial suspension fuzzy pid control for axial artificial heart pump","volume":"703","author":"Zhao","year":"2015","journal-title":"Appl. Mech. Mater."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1960","DOI":"10.1016\/j.matpr.2017.11.299","article-title":"Design of advanced control strategies for cardiovascular system","volume":"5","author":"Castro","year":"2018","journal-title":"Mater. Today Proc."},{"key":"ref_10","first-page":"13543","article-title":"An imperfect production process for time-varying demand with inflation and time value of money\u2014An EMQ model","volume":"38","author":"Sarkar","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4881","DOI":"10.1016\/j.amc.2011.10.053","article-title":"An inventory model with reliability in an imperfect production process","volume":"218","author":"Sarkar","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8295","DOI":"10.1016\/j.amc.2012.01.053","article-title":"An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production","volume":"218","author":"Sarkar","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_13","first-page":"216","article-title":"Analytical study on the non-linear vibration of Euler-Bernoulli beams","volume":"14","author":"Pakar","year":"2012","journal-title":"J. Vibroeng."},{"key":"ref_14","first-page":"423","article-title":"On the Approximate Analytical solution for parametrically excited nonlinear oscillators","volume":"14","author":"Pakar","year":"2012","journal-title":"J. Vibroeng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/0167-2789(93)90249-Z","article-title":"Resonant normal forms, interpolating Hamiltonians and stability of area preserving maps","volume":"64","author":"Bazzani","year":"1993","journal-title":"Physica D"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Vesely, V., and Korosi, L. (2018). Robust PID Controller Design for Nonlinear Systems, De Gruyter.","DOI":"10.1515\/jee-2018-0009"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1080\/14786441108564652","article-title":"The heartbeat considered as a relaxation oscillation and an electrical model of the heart","volume":"6","year":"1928","journal-title":"Philos. Mag."},{"key":"ref_18","first-page":"193","article-title":"Review of Stability Analysis of Non-Linear Control Systems","volume":"4","author":"Boro","year":"2016","journal-title":"IJERG"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1109\/51.340753","article-title":"Development and test of a linear motor-driven total artificial heart","volume":"14","author":"Yamada","year":"1995","journal-title":"IEEE Eng. Med. Biol. Mag."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Love, H.C., Timms, D.L., Nestler, F., Frazier, O.H., and Cohn, W.E. (2014, January 26\u201330). A mock circulatory loop for designing and evaluating total artificial hearts. Proceedings of the 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Chicago, IL, USA.","DOI":"10.1109\/EMBC.2014.6944913"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kurita, N., Ishikawa, T., Saito, N., Masuzawa, T., and Timms, D. (2017, January 21\u201324). A double-sided stator type axial self-bearing motor development for total artificial heart. Proceedings of the 2017 IEEE International Electric Machines and Drives Conference (IEMDC), Miami, FL, USA.","DOI":"10.1109\/IEMDC.2017.8002190"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.12693\/APhysPolA.118.1183","article-title":"Results of experiments with fiber pressure sensor applied in the polish artificial heart prosthesis","volume":"118","author":"Konieczny","year":"2010","journal-title":"Acta Phys. Pol. Ser. A Gen. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2430","DOI":"10.1109\/TMAG.2007.894002","article-title":"Magnetic tracking system: Monitoring heart valve prostheses","volume":"43","author":"Baldoni","year":"2007","journal-title":"IEEE Trans. Magn."},{"key":"ref_24","unstructured":"Rosli, R.S.B., and Olanrewaju, R.F. (2016, January 26\u201327). Mobile Heart Rate Detection System (MoHeRDS) for Early Warning of Potentially-Fatal Heart Diseases. Proceedings of the 2016 International Conference on Computer and Communication Engineering (ICCCE), Kuala Lumpur, Malaysia."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Marom, G., Chiu, W.C., Slepian, M.J., and Bluestein, D. (2014, January 26\u201330). Numerical model of total artificial heart hemodynamics and the effect of its size on stress accumulation. Proceedings of the 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Chicago, IL, USA.","DOI":"10.1109\/EMBC.2014.6944909"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Shi, W., and Chew, M.S. (2009, January 9\u201311). Mathematical and physical models of a total artificial heart. Proceedings of the IEEE International Conference on Control and Automation, Christchurch, New Zealand.","DOI":"10.1109\/ICCA.2009.5410410"},{"key":"ref_27","unstructured":"Pohlmann, A., Le\u00dfmann, M., and Hameyer, K. (2010, January 10\u201313). Comparative study on optimization methods for a motor-drive of artificial hearts. Proceedings of the 2010 International Conference on Electrical Machines and Systems (ICEMS), Incheon, Korea."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Das, S., Nandi, D., Neogi, B., and Sarkar, B. (2020). Nonlinear system stability and behavioral analysis for effective implementation of artificial lower limb. Symmetry, 12.","DOI":"10.3390\/sym12101727"},{"key":"ref_29","first-page":"654","article-title":"Nonlinear free vibration analysis of tapered beams by Hamiltonian approach","volume":"13","author":"Pakar","year":"2011","journal-title":"J. Vibroeng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s11803-013-0182-0","article-title":"Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses","volume":"12","author":"Bayat","year":"2013","journal-title":"Earthq. Eng. Eng. Vib."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1643","DOI":"10.1016\/S0960-0779(01)00172-2","article-title":"Computational biology of propagation in excitable media models of cardiac tissue","volume":"13","author":"Holden","year":"2002","journal-title":"Chaos Solitons Fractals"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/BF02728793","article-title":"Normal forms for hamiltonian maps and non linear effects in a particle accelerator","volume":"102","author":"Bazzani","year":"1988","journal-title":"IL Nuovo Cim. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1016\/0010-4809(77)90011-8","article-title":"Mathematical model of automaticity in the sinus node and AV junction based on weakly coupled relaxation symmetric oscillators","volume":"10","author":"Katholi","year":"1977","journal-title":"Comput. Biomed. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1142\/S0218127498001418","article-title":"A model of two nonlinear coupled symmetric oscillator s for the study of heartbeat dynamics","volume":"8","author":"Signorini","year":"1998","journal-title":"Int. J. Bifurc. Chaos Appl. Sci. Eng."},{"key":"ref_35","first-page":"H1410","article-title":"Evidence of low and high-frequency oscillations in human AV interval variability: Evaluation with spectral analysis","volume":"267","author":"Nollo","year":"1994","journal-title":"Am. J. Physiol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"H397","DOI":"10.1152\/ajpheart.2000.279.1.H397","article-title":"Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node","volume":"279","author":"Zhang","year":"2000","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"102326","DOI":"10.1016\/j.jretconser.2020.102326","article-title":"The ramification of dynamic investment on the promotion and preservation technology for inventory management through a modified flower pollination algorithm","volume":"58","author":"Saha","year":"2021","journal-title":"J. Retail. Con. Ser."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF00275911","article-title":"The heart as a system of coupled nonlinear symmetric oscillators","volume":"18","author":"Honerkamp","year":"1983","journal-title":"J. Math. Biol."},{"key":"ref_39","first-page":"74","article-title":"Type I intermittency in nonstationary systems\u2014Models and human heart rate variability","volume":"336","author":"Baranowski","year":"2014","journal-title":"Phys. A"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Keener, J., and Sneyd, J. (1998). Mathematical Physiology, Interdisciplinary Applied Mathematics 8, Springer.","DOI":"10.1007\/b98841"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/0167-2789(85)90004-1","article-title":"Nonlinear dynamics of the heartbeat. The AV junction: Passive conduit or active symmetric oscillator?","volume":"17","author":"West","year":"1985","journal-title":"Physica D"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Jemai, J., and Sarkar, B. (2019). Optimum design of a transportation scheme for healthcare supply chain management: The effect of energy consumption. Energies, 12.","DOI":"10.3390\/en12142789"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.physa.2004.01.020","article-title":"Modeling cardiac pacemakers with relaxation symmetric oscillators","volume":"336","year":"2004","journal-title":"Phys. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1113\/jphysiol.1952.sp004764","article-title":"A quantitative description of membrane current and its applications to conduction and excitation in nerve","volume":"117","author":"Hodgkin","year":"1952","journal-title":"J. Physiol. Lond."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2799","DOI":"10.1103\/PhysRevE.60.2799","article-title":"Synchronization of diffusively coupled symmetric oscillator s near the homoclinic bifurcation","volume":"60","author":"Postnov","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_46","unstructured":"Meirovitch, L. (1997). Principles, and Techniques of Vibrations, Prentice-Hall."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1016\/S0006-3495(86)83559-7","article-title":"Propagation through electrically coupled cells: How a small SA node drives a large atrium","volume":"50","author":"Joyner","year":"1986","journal-title":"Biophys. J."},{"key":"ref_48","first-page":"97","article-title":"Application of Stability Theory of NonLinear Systems and Lyapnov Transformation in Control of Artificial Pneumatic Muscle","volume":"3","author":"Jadlovska","year":"2011","journal-title":"Ann. Fac. Eng."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1023\/A:1016514000260","article-title":"Describing Function Analysis of Systems with Impacts and Backlash","volume":"29","author":"Barbosa","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1016\/0005-1098(71)90027-6","article-title":"A computational method for determining quadratic Lyapunov functions for non Linear system","volume":"7","author":"Davison","year":"1971","journal-title":"Automatica"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/151\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:12:32Z","timestamp":1760159552000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,18]]},"references-count":50,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010151"],"URL":"https:\/\/doi.org\/10.3390\/sym13010151","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,1,18]]}}}