Proportional feedback control of chaos in a simple electronic oscillator

An interesting feature of the return map for the KSS circuit is that the thin 1D curve bends back near itself, creating closely separated lower and upper branches (see Fig. 9 in Ref. 22), which are not resolved by the scale used in Fig. 3. Thus there are two intersections of the map and the $xn+1=xn$ line, which means there are two unstable period-one fixed points. We used recursive proportional feedback to stabilize the lower-branch fixed point. We were not able to gain control for the upper-branch fixed point, which suggests that it is not a saddle point. See Ref. 1 for details on classifying fixed points and Ref. 16 for a discussion of why a saddle point is needed for proportional feedback control.

If the sequence is generated with the odd values of $x$ corresponding to $p=p0$ and the even values corresponding to $p=p0+Δp$⁠, then the up map consists of $x2$ plotted against $x1$⁠, $x4$ plotted against $x3$⁠, and so on; the back map consists of $x3$ plotted against $x2$⁠, $x5$ plotted against $x4$⁠, and so on.