Strange Nonchaotic Stars

The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler’s “golden” stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars.

Figure 1

Spectral content. (a) Raster pattern of disks with brightness proportional to the KIC 5520878 stellar flux at intervals of $\delta t\approx 1766\mathrm{s}\approx 0.5\mathrm{h}$. (b) Fourier transform magnitude of stellar flux sampled at intervals $\delta t$ has primary and secondary frequencies (red arrows) at $f_1\approx 1/(0.266\mathrm{d})$ and $f_2\approx 1/(0.169\mathrm{d})$, where $f_2/f_1\approx 1.58\approx \phi$ is the golden ratio.

Figure 2

Attractor reconstruction. Two views of a three-dimensional plot of normalized flux $F_N$ at $t=n\delta t$ successively delayed by $\tau =2\delta t$. This delay coordinate embedding “unfolds” the time series into a warped torus, suggesting two-frequency nonlinear dynamics. Equal-sized spheres locate data. Rainbow colors code time, from red to violet. Flux triplets straddling data gaps appear far from torus.

Figure 3

Spectral scaling. (a) Fourier transform magnitude of stellar flux strobed at the primary period $T_1$ has a wide distribution of peaks. Peaks above threshold $\sigma$ (horizontal line) are highlighted (gold). (b) Log-log plot of number of superthreshold peaks $N$ versus the threshold $\sigma$ has a linear regime (in leg below knee of curve) that indicates power law scaling and fractal structure. Spectral exponent slope $-2<s<-1$ indicates strange nonchaotic dynamics. Deviation for low thresholds arises from light curve’s finite temporal resolution.

Figure 4

Other stars. (a) Spectral scalings of the four Kepler golden stars all exhibit power law scaling, over 2 decades of thresholds, and nearly collapse onto each other when the axes are proportionally scaled (as denoted by primes). (b) By contrast, spectral scalings of two Kepler nongolden stars with primary and secondary frequency ratios near $3/2$ do not exhibit power law scaling but are in fact concave or multiscaled.