Nonlinear Dynamics and Chaos in Fractional-Order Cardiac Action Potential Duration Mapping Model

Abstract

This study introduces a novel one-dimensional fractional-order model 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 $t_s = 307ms$) without preceding period-doubling bifurcations.

2) Novel rhythm alternations between 5 : 5 and 3 : 3 patterns.

3) Unique bistability phenomena, including 2 : 2 $\leftrightarrow$ chaos and 5 : 5 $\leftrightarrow$ 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.