Bifurcation Diagram of Recurrence Equation X(t+1)=AX(t)(1−X(t)−X(t−1)−X(t−2))

Abstract

The properties of recurrence equation X(t+1)=AX(t)(1−X(t)−X(t−1)−X(t−2)) are investigated appearing in mutually connected nervous network having refractory period 3. The stable fixed point is 0 for A<1 and (1−1⁄A)⁄3 for 1≤A<2.5. For 2.5≤A<2.8 oscillation (or ultra long period periodic point) appears which shows critical behaviors. Usual analysis predicts instability at A=3, which contradicts with numerical results. For 2.8<A<2.85 chaos and window occur alternatively and large window from A=2.85 to 2.9077. Above 2.9077 usual period-doubling bifurcation and chaos take place. The Lorenz plots of chaotic state have different structures below and above A=2.9077.