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| Nonlinear Dynamics and Chaos in Fractional-Order Cardiac Action Potential Duration Mapping Model |
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| KeyWord:Fractional-order calculus, cardiac dynamics, action potential duration, nonlinear dynamics, chaos |
| Author Name | Affiliation | | Rabah Bououden | Laboratory of Mathematics and their Interactions, Abdelhafid Boussouf University, Mila 43000, Algeria | | Tarek Houmor | Applied Mathematics and Modeling Laboratory, Constantine 1-Mentouri University, Constantine 25017, Algeria | | Fahd Jarad | Department of Mathematics, \c{C}ankaya University, Etimesgut 06790, Ankara, T\"urkiye
Department of Engineering Mathematics and Artificial Intelligence, Azerbaijan Technical University, H\"useyn Cavid Av., Baku 1073, Azerbaijan | | Mohammed S. Abdelouahab | Laboratory of Mathematics and their Interactions, Abdelhafid Boussouf University, Mila 43000, Algeria |
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| Abstract: |
| This study introduces a novel one-dimensional fractional-order mo-del for cardiac action potential duration (APD) dynamics, incorporating memory effects through discrete fractional calculus. By generalizing the classical APD map using the Caputo fractional difference operator, we uncover complex nonlinear behaviors not observed in traditional integer-order models. Through comprehensive numerical simulations, including bifurcation analysis and Lyapunov exponent calculations validated by the 0-1 test, we demonstrate that the fractional-order system exhibits:
1) Early onset of chaos (at $t_s = 307 ms$) without preceding period-doubling bifurcations.
2) Novel rhythm alternations between $5:5$ and $3:3$ patterns.
3) Unique bistability phenomena, including {$2:2 \longleftrightarrow chaos$} and {$5:5 \longleftrightarrow 3:3$} states.
4) Memory-dependent dynamics where current APD depends on all previous states.
Our results reveal that fractional calculus provides a more physiologically realistic framework for modeling cardiac dynamics by naturally incorporating memory effects. The identified dynamical regimes offer new insights into the transition mechanisms from normal rhythms to potentially arrhythmic states, with particular clinical relevance to understanding alternans as precursors to ventricular fibrillation. The fractional-order approach demonstrates superior capability for capturing the complex, history-dependent nature of cardiac excitation compared with conventional models. |
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