Hysteresis Phenomena of Biological System.(Dept.E)

Dynamical investigation and FPGA implementation of a new Heartbeat model based on the Barrio-Varea-Aragon-Maini oscillator

“…Let's verify the condition 2. By replacing λ in the characteristic equation (6) by 'iω' we separate the real and imaginary parts and equation (11) is obtained; then by solving (11)-(b), the Hopf bifurcation frequency ω = ω Hopf that provides the frequency of stable oscillations is then given by equation (12) meanwhile equation (13) defines the critical value βc of β that corresponds to the Hopf bifurcation point for the system.…”

“…Various physical, chemical, biological or ecological phenomena, etc can be modelled thanks to mathematics, and then be analysed in order to predict their future behaviour and anticipate, for example, the action to be taken depending on whether the occurrence of the said behaviour will prove beneficial or harmful. In the theory of dynamic systems, these behaviours include among others: The route to chaos phenomenon highlighted in a neuron model [1][2][3] the chaotic dynamics characterised by sensitivity to initial conditions and highlighted in a system of interacting nephrons [4]; the coexistence of attractors [5,6], the antimonotonicity [7,8] and the hysteresis [9][10][11], just to name a few. To predict these different phenomena in a given system, several analysis tools for dynamic systems are needed, including : Time series, which allow the trajectory of the system to be observed over time; phase portraits, which characterise the presence of an attractor; bifurcation diagrams, which indicate the values taken asymptotically by a system as a function of its control parameter; the Lyapunov exponent, which provides information on the degree of sensitivity of the system to its initial conditions; and basins of attraction, which provide information on the set of initial conditions for which the trajectories of the system converge towards one of its attractors.…”