Theory of hybrid dynamical systems and its applications to biological and medical systems by Kazuyuki Aihara, and Hideyuki Suzuki Philosophical Transactions A Volume 368(1930): November 13, 2010 ©2010 by The Royal Society

Schematic of a piecewise smooth hybrid system where each arrow shows a discrete transition accompanied by a reset map of the continuous state. Kazuyuki Aihara, and Hideyuki Suzuki Phil. Trans. R. Soc. A 2010;368: ©2010 by The Royal Society

(a) 1/3-periodic and (b) 2/3-periodic responses in the Nagumo–Sato model with k = 0.85, α = 1, and (a) a = 0.3 and (b) a = 0.7. Kazuyuki Aihara, and Hideyuki Suzuki Phil. Trans. R. Soc. A 2010;368: ©2010 by The Royal Society

Response characteristics of (a) the bifurcation diagram, (b) the average firing rate ρ, and (c) the Lyapunov exponent λ in the Nagumo–Sato model where k = 0.85 and α = 1. Kazuyuki Aihara, and Hideyuki Suzuki Phil. Trans. R. Soc. A 2010;368: ©2010 by The Royal Society

(a) 1/3-periodic and (b) 2/3-periodic responses of the membrane potential in a squid giant axon that is in the resting state and stimulated by periodic pulses with a pulse width of 0.5 ms and a pulse interval of 5.0 ms; pulse amplitude A = (a) 7.6 μA ( Kazuyuki Aihara, and Hideyuki Suzuki Phil. Trans. R. Soc. A 2010;368: ©2010 by The Royal Society

A circuit of a hysteresis 2P(Port)-VCCS (voltage-controlled current source) chaotic oscillator. Kazuyuki Aihara, and Hideyuki Suzuki Phil. Trans. R. Soc. A 2010;368: ©2010 by The Royal Society

Strange attractors obtained by numerical simulation with SPICE (a(i),b(i)) and by electronic experiments with implemented analogue ICs (a(ii),b(ii)). Kazuyuki Aihara, and Hideyuki Suzuki Phil. Trans. R. Soc. A 2010;368: ©2010 by The Royal Society