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Fractional-Order Dynamics of Cardio-Respiratory Oxygen Transport: A Stable Regulator of Human Endurance and the Boundary of Chaos |
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PP: 569-583 |
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doi:10.18576/pfda/120306
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Author(s) |
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Aashima Bangia,
Rashmi Bhardwaj,
Asokan Vasudevan,
Naeema Darwish Khamis Al Maashari,
Yafang Yan,
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Abstract |
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| This work develops a fractional-order dynamical model of coupled cardio-respiratory oxygen transport among the lungs, the heart and bloodstream, and the body tissues, and studies the conditions under which the human body sustains or loses stable oxygen regulation during sustained effort. The three-compartment system is formulated with Caputo fractional derivatives of order q, so that physiological memory arising from delayed oxygen uptake, circulatory transport lags, and gradual tissue utilisation is represented intrinsically rather than through added delay terms. The equilibrium structure is derived, and the associated Jacobian matrix is obtained in closed form as a real, non-symmetric operator. Local stability is established rigorously using the Matignon criterion appropriate to fractional-order systems: at the physiological operating point the eigenvalues give a minimum argument of 1.605 radians, yielding a critical order of 1.02, so the regulated state is a stable focus for every admissible order between zero and one. This analytical result is confirmed numerically by the largest Lyapunov exponent, computed with a variational method validated on the classical Lorenz system, which returns a value of −0.10 independent of the initial condition. The model therefore describes a robustly self-regulating oxygen- transport system, and this stable regime is interpreted as the dynamical basis of endurance. A physiologically motivated extension in which the cardiac response gain is separated from and exceeds the pulmonary transfer coefficient and the metabolic dissipation is raised to realistic levels undergoes a genuine transition to chaotic dynamics, offering a candidate description of cardio-respiratory dysregulation and fatigue. Notably, stronger memory (a lower fractional order) suppresses this chaos, indicating that the fractional formulation enhances the robustness of the regulated endurance state. A parameter-sensitivity analysis confirms that stability persists across the physiological ranges of the transport, breathing, and metabolic parameters. The findings connect fractional-order nonlinear dynamics to measurable endurance and recovery in sports science, physiotherapy, and cardio-respiratory health monitoring. Numerical simulations employ a fractional Adams-Bashforth-Moulton (ABM) scheme.
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