Chaos and complexity in a cardiac pacemaker model with incommensurate fractional-order dynamics: Empirical validation and cardiovascular implications

doi:10.1088/1674-1056/ae0b3c

Abstract The dynamics of heart rhythms plays a pivotal role in physiological health, where chaotic behavior is often associated with pathological cardiac states. In this work, we investigate a fractional-order model of cardiac pacemaker dynamics using incommensurate derivatives to capture the complex memory effects and non-local interactions inherent in biological systems. We demonstrate that slight variations in the fractional orders induce rich dynamics, including chaos and coexisting attractors, signifying transitions between normal and dysfunctional rhythms. Crucially, our model exhibits wider chaotic regions than classical integer-order counterparts when the incommensurate derivatives are perturbed. Furthermore, we reveal pronounced multistability, where distinct chaotic attractors coexist under identical parameters, reflecting the system's capacity for abrupt transitions between physiological and pathological states. These findings advance the mechanistic understanding of rhythm disorders and highlight the critical role of fractional calculus in modeling cardiac dynamics.

Keywords: cardiac pacemaker model incommensurate fractional-order rhythm dynamics chaos bistability

Received: 05 August 2025
Revised: 07 September 2025
Accepted manuscript online: 25 September 2025

PACS:	87.19.Hh	(Cardiac dynamics)
	05.45.-a	(Nonlinear dynamics and chaos)
	02.30.Hq	(Ordinary differential equations)
	05.45.Pq	(Numerical simulations of chaotic systems)

Corresponding Authors: Haneche Nabil
E-mail: nabil.haneche@doc.umc.edu.dz

Cite this article:

Haneche Nabil and Hamaizia Tayeb Chaos and complexity in a cardiac pacemaker model with incommensurate fractional-order dynamics: Empirical validation and cardiovascular implications 2026 Chin. Phys. B 35 078702

[1] Zhang J, Chatham J C and Young M E 2020 Annu. Rev. Physiol. 82 79
[2] da Silva Lima G, Amorim Savi M and Moreira BessaW2024 Sci. Rep. 14 23284
[3] Cheffer A, Savi M A, Pereira T L and de Paula A S 2021 Appl. Math. Model. 96 152
[4] Van Der Pol B and Van Der Mark J 1928 Lond. Edinb. Dubl. Phil. Mag. 6 763
[5] dos Santos A M, Lopes S R and Viana R R L 2004 Phys. A 338 335
[6] Gois S R and Savi M A 2009 Chaos Solitons Fractals 41 2553
[7] Glass L, Guevara M R, Shrier A and Perez R 1983 Phys. D 7 89
[8] Henriques T, Ribeiro M, Teixeira A, Castro L, Antunes L and Costa- Santos C 2020 Entropy 22 309
[9] Quiroz-Juárez M A, Vázquez-Medina R, Ryzhii E, Ryzhii M and Aragón J L 2017 Commun. Nonlinear Sci. Numer. Simul. 42 370
[10] Wu G Q, Arzeno N M, Shen L L, Tang D K, Zheng D A, Zhao N Q, Eckberg D L and Poon C S 2009 PloS One 4 e4323
[11] Fonkou R F and Savi M A 2023 Fractal Fract. 7 592
[12] Cheffer A and Savi M A 2020 Biosystems 196 104177
[13] Vellappandi M and Lee S 2025 Nonlinear Dyn. 113 12117
[14] Tarasov V E 2021 Commun. Nonlinear Sci. Numer. Simul. 102 105947
[15] Fernández-Anaya G, Godínez F A, Valdés R, Quezada-Téllez R A and Polo-Labarrios M A 2025 Fractal Fract. 9 264
[16] Barros L C, Lopes M M, Pedro F S, Esmi E, Santos J P C and Sánchez D E 2021 Comput. Appl. Math. 40 72
[17] Haneche N and Hamaizia T 2024 Phys. Scr. 99 115269
[18] Haneche N and Hamaizia T 2025 Phys. Scr. 100 075253
[19] Kumar A and Kumar S 2022 Chaos Solitons Fractals 156 111697
[20] Gratz D, Onal B, Dalic A and Hund T J 2018 Front. Phys. 6 63
[21] Templos-Hernandez D J, Quezada-Tellez L A, Gonzalez-Hernandez B M, Rojas-Vite G, Pineda-Sanchez J E, Fernández-Anaya G and Rodriguez-Torres E E 2021 Chaos Solitons Fractals 147 110942
[22] Bellout A, Bououden R, Houmor T and Berkal M 2025 Gulf J. Math. 19 369
[23] Belmiloudi A 2021 AIMS Math. 6 821
[24] Barbosa R S, Machado J T, Vinagre B M and Calderon A J 2007 J. Vib. Control 13 1291
[25] Saleem O, Ahmed D and Iqbal J 2025 Fractal Fract. 9 424
[26] Yürdem B, AksuMF and Saǧbaş M2024 Circuits Syst. Signal Process. 43 7423
[27] Grudziński K and Zebrowski J 2004 Phys. A 336 153
[28] Savi M A 2005 J. Braz. Soc. Mech. Sci. Eng. 27 157
[29] Al-Sharify N T, Rzaij D R, Nahi Z M and Al-Sharify Z T 2020 IOP Conf. Ser.: Mater. Sci. Eng. 870 012020
[30] Petráš I 2011 Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Berlin, Heidelberg: Springer)
[31] Heymans N and Podlubny I 2006 Rheol. Acta 45 765
[32] Murillo-Arcila M, Peris A and Vargas-Moreno A 2025 Fract. Calc. Appl. Anal. 28 1717
[33] Scherer R, Kalla S L, Tang Y and Huang J 2011 Comput. Math. Appl. 62 902
[34] Adams D E and Allemang R J 2000 Mech. Syst. Signal Process. 14 637
[35] Brandt A and Brincker R 2014 Measurement 58 511
[36] Diethelm K, Ford N J and Freed A D 2002 Nonlinear Dyn. 29 3
[37] Vinagre B M, Chen Y Q and Petráš I 2003 J. Frankl. Inst. 340 349
[38] Danca M F and Fečkan M 2024 Int. J. Bifurc. Chaos Appl. Sci. Eng. 34 2450156
[39] Danca M F and Kuznetsov N 2018 Int. J. Bifurc. Chaos Appl. Sci. Eng. 28 1850067
[40] Li H, Shen Y, Han Y, Dong J and Li J 2023 Chaos Solitons Fractals 168 113167
[41] Gottwald G A and Melbourne I 2009 SIAM J. Appl. Dyn. Syst. 8 129
[42] Sun K H, Liu X and Zhu C X 2010 Chin. Phys. B 19 110510
[43] Wang S, He S, Yousefpour A, Jahanshahi H, Repnik R and Perc M 2020 Chaos Solitons Fractals 131 109521
[44] He S, Sun K and Wang H 2015 Entropy 17 8299
[45] Hammouch Z and Mekkaoui T 2014 Nonauton. Dyn. Syst. 1 61
[46] Zhang Y, Yue X, Du L, Wang L and Fang T 2016 Shock Vib. 2016 6109062
[47] Bao H, Ding R, Chen B, Xu Q and Bao B 2023 Chaos Solitons Fractals 169 113228
[48] Ye X and Wang X 2020 Phys. Scr. 95 075212
[49] Han X, Mou J, Xiong L, Ma C, Liu T and Cao Y 2021 Integration 81 43

[1]	Spectral statistics and wave-chaos transition in three-dimensional acoustic cavities Xiaodong Zhang(张晓东). Chin. Phys. B, 2026, 35(6): 060510.
[2]	Symmetrical Turing instability in Chua corsage memristor siblings-based two-cell network Zhicheng Tian(田桎成), Peipei Jin(靳培培), Shutong Liu(刘姝彤), Meiyuan Gu(顾梅园), Long Chen(陈龙), and Guangyi Wang(王光义). Chin. Phys. B, 2026, 35(6): 060511.
[3]	Intelligent identification for discrete memristive neuron map: An adaptive chaos game optimization algorithm studied from the perspectives of different sample sizes and objective functions Yuexi Peng(彭越兮), Xinyi Luo(罗馨怡), Zhijun Li(李志军), Mengjiao Wang(王梦蛟), and Minglin Ma(马铭磷). Chin. Phys. B, 2026, 35(6): 060507.
[4]	Studying relationships from the perspective of chaos theory Xiyu Ren(任玺谕), Xianying Xu(徐宪莹), Xiaodong Liu(刘晓东), Minghui Zhang(张明会), Santo Banerjee, Suo Gao(高锁), and Jun Mou(牟俊). Chin. Phys. B, 2026, 35(6): 060504.
[5]	Quench dynamics of Anderson-localized interacting exciton-polaritons Zhi-Hao Sun(孙志豪), Yong-Jia Chong(冲勇嘉), Yu Guan(管煜), Zheng-Ye Wang(王正叶), Meng Chen(陈猛), Lei Chen(陈雷), and Xingran Xu(许星然). Chin. Phys. B, 2026, 35(5): 057106.
[6]	Frequency-tunable spiking dynamics of Chua corsage memristors Qijian Wu(吴奇键), Peipei Jin(靳培培), Xiameng Wu(吴夏萌), Meiyuan Gu(顾梅园), Wei Zhou(周玮), Yujiao Dong(董玉姣), Yan Liang(梁燕), and Long Chen(陈龙). Chin. Phys. B, 2026, 35(4): 048703.
[7]	Bistable Goos-Hänchen shift owing to leaky-mode excitation in a slab waveguide with Kerr nonlinear medium Yuan-Ping Cai(蔡园平), Li Jiang(姜丽), and Ren-Gang Wan(万仁刚). Chin. Phys. B, 2026, 35(4): 044203.
[8]	Impact of altruistic preference on the dynamic decision and coordination of joint emission reduction in low-carbon supply chain Jinchai Lin(林金钗), Chunyan Zheng(郑春艳), Ruguo Fan(范如国), Yuanyuan Wang(王圆缘), Yingqing Zhang(张应青), and Han Song(宋寒). Chin. Phys. B, 2026, 35(4): 040501.
[9]	Machine learning of chaotic characteristics in classical nonlinear dynamics using variational quantum circuit Sheng-Chen Bai(白生辰) and Shi-Ju Ran(冉仕举). Chin. Phys. B, 2026, 35(2): 020303.
[10]	Chaos of cavity optomechanical system with Coulomb coupling Yingjia Yang(杨应佳), Liwei Liu(刘利伟), Lianchun Yu(俞连春), Weizheng Kong(孔伟正), Haiyan Jiao(焦海燕), Xiaoyan Deng(邓小燕), and Xiaoyong Li(李小勇). Chin. Phys. B, 2025, 34(8): 080503.
[11]	Complexity dynamics analysis of a dual-channel green supply chain with government intervention and cap-and-trade regulation Yuhao Zhang(张玉豪), Lin Huang(黄林), and Man Yang(杨满). Chin. Phys. B, 2025, 34(7): 070501.
[12]	Resonant tunneling diode cellular neural network with memristor coupling and its application in police forensic digital image protection Fei Yu(余飞), Dan Su(苏丹), Shaoqi He(何邵祁), Yiya Wu(吴亦雅), Shankou Zhang(张善扣), and Huige Yin(尹挥戈). Chin. Phys. B, 2025, 34(5): 050502.
[13]	Model-free prediction of chaotic dynamics with parameter-aware reservoir computing Jianmin Guo(郭建敏), Yao Du(杜瑶), Haibo Luo(罗海波), Xuan Wang(王晅), Yizhen Yu(于一真), and Xingang Wang(王新刚). Chin. Phys. B, 2025, 34(4): 040505.
[14]	Study and circuit design of stochastic resonance system based on memristor chaos induction Qi Liang(梁琦), Wen-Xin Yu(于文新), and Qiu-Mei Xiao(肖求美). Chin. Phys. B, 2025, 34(4): 040502.
[15]	Hybrid image encryption scheme based on hyperchaotic map with spherical attractors Zhitang Han(韩智堂), Yinghong Cao(曹颖鸿), Santo Banerjee, and Jun Mou(牟俊). Chin. Phys. B, 2025, 34(3): 030503.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Chaos and complexity in a cardiac pacemaker model with incommensurate fractional-order dynamics: Empirical validation and cardiovascular implications

Cite this article:

Online attention