Unexpected Behaviors in Higher- Order Positive Feedback Loops

16 D-4455-22.4 Fourth-Order LoopThe fourth-order system is identical to the previous systems except for that it addsone more level to the chain (Figure 10).What surprising behavior can we obtain within the freedom of four level variables?As it happens, three distinct modes appear. The expected divergent exponential (a(0) =b(0) = c(0) = d(0) = 1) and convergent decay (a(0) = c(0)=1, b(0) = d(0) = -1) wehave seen already, but— as unbelievable as it may seem— we can now also have sustainedoscillations! With only positive feedback between the individual variables— and with theinitial conditions properly balanced— the delays between distant levels is great enough tocreate sustained oscillations. In Figure 11 a(0) = 1, b(0) = -1, and c(0) = d(0) = 0.(You may get expanding oscillations when you simulate this. If so, it is due only to DTerror, and it can be rectified by using a different integration method or a smaller DT. Iused DT=0.01; Runge-Kutta at DT=0.1 works just as well.)As is the case in the other systems, this peculiar behavior mode is not stable. A1/1000 perturbation— changing c(0)=0.001— creates the destabilization in Figure 12.ab+dbda++ddcdd+cFigure 10: Fourth-order positive feedback loop