A mathematical model of respiratory system exhibiting chaos (Q2706058)

The authors of this interesting paper investigate bifurcations of the periodic solutions of a respiratory model with nonlinear and damping terms and parametric excitation. This model is represented schematically by two major components: the Plant (Lungs+Body Tissues) in which \(CO_2\) exchanges take place, and the Controller (located in the brain stem), which responds to inputs from central and peripheral chemoreceptors (via the vagus nerves) by producing cyclic lung inflation (via the phrenic nerves). As a gross simplification, the interaction between the lung volume \(v\) and the controller can be modeled as \(\ddot{v}+F(v,\dot{v})=e(v,\dot{v},t)\), where \(\dot{v}\) is the derivative with respect to \(t\). The controller responds with a corrective action \(e(v,\dot{v},t)\). Setting different functions \(F\) one reads the Leńard's equation, Van der Pol's equation or Duffing's equation.NEWLINENEWLINENEWLINEThe analysis is carried out with respect to the forcing frequency \(f\). The phase portraits in the two-parameter space are obtained by means of a numerical calculation. The Poincaré maps and Lyapunov exponents for a representative selection of attractors are used. The transition to chaos is found to be via a Feigenbaum cascade of period-doubling bifurcations.