Experimental observation of a complex periodic window

The existence of a special periodic window in the two-dimensional parameter space of an experimental Chua’s circuit is reported. One of the main reasons that makes such a window special is that the observation of one implies that other similar periodic windows must exist for other parameter values. However, such a window has never been experimentally observed, since its size in parameter space decreases exponentially with the period of the periodic attractor. This property imposes clear limitations for its experimental detection.

Figure 1

(a) Scheme of Chua’s circuit. Their component values are $R_{10}\approx 1.4\text{k}\Omega$, $R_{20}\approx 37\Omega$, $C_1\approx 4.7\text{nF}$, $C_2\approx 56\text{nF}$, and $L\approx 9.2\text{mH}$. (b) Parameter space for the experimental Chua’s circuit showing the period-3 and period-4 windows. Solid black circles represent parameters for which the lowest-period POs are observed, solid light gray circles represent the higher period PO that appear by periodic-doubling bifurcations, and solid dark gray circles represent parameters for the closest chaotic attractor to the PWs. The straight lines indicate parameter values for the sets of time series $X$ and $Y$.

Figure 2

(Color online) Return maps (black points) of the Poincaré section of the chaotic attractors obtained using the parameters indicated by the borders $a$, $b$, $c$, and $d$ in box III of Fig. 1. We also show the return maps (blue circles) of the periodic attractors obtained for the closest parameters to these borders. The vertical lines, passing through the maximum and the minimum, define the partition points. In $\left(d^{\prime }\right)$ is shown a zoomed-in view of the minimum of the return map in (d).

Figure 3

(Color online) (a) The encoding of all the period-4 POs found in the CPW. (b) Sketch of the critical lines ($S_{V1}$ and $S_{V2}$) structure of the CPW, disregarding the existence of characteristic (iii) that causes the appearance of structures as illustrated in Fig. 4.

Figure 4

Illustration of the structures that might appear in a CPW due to characteristic (iii).