Exotic dynamic behavior of the forced FitzHugh-Nagumo equations

Space-clamped FitzHugh-Nagumo nerve model subjected to a stimulating electrical current of form Io + I cos γt is investigated via Poincaré map and numerical continuation. If I = 0, it is known that Hopf bifurcation occurs when Io is neither too small nor too large. Given such an Io. If γ is chosen close to the natural frequency of the Hopf bifurcated oscillation, a series of exotic phenomena varying with I are observed numerically. Let 2πλ/γ denote the generic period we watched. Then the scenario consists of two categories of period-adding bifurcation. The first category consists of a sequence of hysteretic, λ → λ + 2 period-adding starting with λ = 1 at I = 0+, and ending at some finite I, say I*, as λ → ∞. The second category contains multiple levels of period-adding bifurcation. The top level consists of a sequence of λ → λ + 1, period-adding starting with λ = 2 at I = I*. From this sequence, a hierarchy of m → m + n → n, period-adding in between are derived. Such a regular pattern is sometimes interrupted by a series of chaos. This category of bifurcation also terminates at some finite I. Harmonic resonance sets in afterwards. Lyapunov exponents, power spectra, and fractal dimensions are used to assist these observations.