Strange nonchaotic stars

SNR

KIC 5520878 attractor reconstructed from light curve. Three-dimensional plot of normalized light flux at successively delayed times. 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.


Synopsis

While the brightness of stars like the sun is nearly constant, the brightness of other stars changes with time. Exploiting the unprecedented capabilities of the planet-hunting Kepler space telescope, which stared at 150 000 stars for four years, we report evidence that certain stars dim and brighten in complex patterns with fractal features. Such stars pulsate at primary and secondary frequencies whose ratios are near the famous golden mean, the most irrational number. A nonlinear system driven by an irrational ratio of frequencies is generically attracted toward a “strange” behavior that is geometrically fractal without displaying the “butterfly effect” of chaos. Strange nonchaotic attractors have been observed in laboratory experiments, but a bluish white star 16 000 light years from Earth in the constellation Lyra may manifest, in the scale-free distribution of its frequency components, the first strange nonchaotic attractor observed in the wild. The recognition of stellar strange nonchaotic dynamics may improve the classification of these stars and refine the physical modeling of their interiors.

Attractors

Some physical systems, like a pendulum or a child on a swing, tend toward simple motions, like rest or regular repetition, which are represented mathematically as point or circle attractors. However, other systems tend toward bounded but irregular motion, even in the absence of noise, which are represented as strange chaotic attractors, where “strange” means fractal geometry and “chaos” means extreme sensitivity to initial conditions. The fractal geometry provides infinitely many unstable periodic orbits, while the chaos forces initially nearby orbits to exponentially diverge.

A system is linear if doubling its input doubles its output. However, chaotic systems are nonlinear. For example, Newtonian gravity is nonlinear, as doubling the distances between masses quarters (not doubles) the corresponding forces. Consequently, chaos pervades the clockwork of the solar system, as demonstrated especially in the orientation of Mars’ spin axis, the location of Pluto in its orbit, and (to our peril) the long-term orbits of many comets and asteroids.

Strange nonchaotic attractors

In the 1980s, theorists Celso Grebogi, Edward Ott, Steven Pelikan, and James Yorke (GOPY) first realized, in simple mathematical and computer models, a kind of motion between order and chaos, where the attractors are strange but not chaotic. In fact, theorists quickly established that strange nonchaotic motion is generic for nonlinear systems quasiperiodically forced by two drivers whose frequencies are not rationally related. In the 1990s, experimentalists first observed strange nonchaotic attractors in the lab. The original such experiment, by coauthor William Ditto and colleagues, featured a vertical magnetoelastic ribbon whose stiffness varied with an applied magnetic field causing it to alternately stiffen and sag. The stiffness was varied by driving the magnetic field quasiperiodically with two frequencies whose ratio was the golden mean, the most irrational number, ϕ=(1+√5)/2≈1.62≈1/0.62.

The golden mean or golden ratio has captivated the human imagination for generations in diverse disciplines, from art to science. Its infinite continued fraction expansion ϕ=1+1/ϕ=1+1/(1+1/(1+⋯))) involves only the smallest nonzero digit 1. Consequently, finite truncations of the expansion converge slowly to its exact value, making the golden mean the irrational number least well approximated by rational numbers. Although rational numbers exist arbitrarily close to the golden mean, infinitely many irrationals that are poorly approximated by rational numbers also exist nearby (obtained by changing some of the 1s in the expansion to other digits). A famous theorem by Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser (KAM) implies that quasiperiodic dynamics with two frequencies in the golden ratio maximally resist perturbations.

Scientists have argued that a strange nonchaotic attractor may best describe the electrochemical activity of the brain by providing the flexibility of the strange never-ending fractal patterns without the penalty of the chaotic exponential divergence of nearby orbits.

Attractor reconstruction

In 1981, Floris Takens proved that even a higher-dimensional attractor can be reconstructed (up to geometrical distortions) by recording a single variable, like stellar brightness, over time, and plotting that variable versus successively later, delayed versions of itself. The delay embedding of periodic motion is a circular attractor, with the period being the time the state takes to circumnavigate the circle. The delay embedding of two-frequency quasiperiodic motion is a toroidal attractor, like a donut with hole, with one period being the time the state takes to circumnavigate the toroid in the toroidal direction, the long way around, and the second period being the time the state takes to circumnavigate in the poloidal direction, the short way through the hole.

When we reconstruct the stellar attractors from their light curves of brightness versus time, we find warped toroids, suggesting two-frequency nonlinear dynamics plus fine features that account for bumps in the curves.

Variable stars

The evolution of life on Earth depends on the constancy of the sun, which is in hydrostatic balance between pressure outward and gravity inward. But even the sun’s brightness varies slightly (by about 0.1% over its 11-year sunspot cycle), and helioeismologists record the sun ringing like a bell with a fundamental frequency and many overtones.

Other stars periodically swell and shrink. The most famous are the Cepheid variable stars, which pulsate with periods of days. Less luminous, but very important are the Lyrae, which pulsate with periods of hours. In the 1930s, Arthur Eddington related such pulsations to the degree of ionization of the outer layers of the stars: greater ionization means more opaque and vice versa, alternately capturing and releasing core radiation, swelling and shrinking the star.

In 1912, Henrietta Leavitt discovered a simple relationship between the average brightness of these stars and their periods. In the 1920s, observing variable star periods, inferring intrinsic brightness, and comparing with apparent brightness (via an inverse square law) enabled Edwin Hubble to discover the distances to other galaxies and infer the expansion of the universe.

Today astronomers continue to fine-tune this “standard candle” technique, a critical rung on the cosmic distance ladder. Because RR Lyrae variable stars are not all perfectly identical, their light output varies slightly, and this must be accounted for when using them as standard candles. Thus detailed understanding of these stars is crucial to astronomy.

Most RR Lyraes are old, at least 10 billion years, and are divided into three classes, according to their light curve characteristics: RRab, RRc, and RRd. They were among the first stars created in our (and any) galaxy. Their composition tells us about what elements were present shortly after the creation of the universe. Furthermore, because we can determine where they are, we can trace different elements to different locations (in galactic spiral arms or globular clusters). The prototype for this class of stars, RR Lyrae itself, was in the Kepler field of view, along with the blue-white RRc Lyrae star KIC 5520878, which is 16 000 light years from Earth in the constellation Lyra.

Kepler mission

Our work was made possible by NASA’s planet-hunting Kepler space telescope. Launched into an Earth trailing orbit in 2009, Kepler’s main 1.4-meter mirror is the largest outside Earth orbit. During its 4-year primary mission, Kepler stared at about 150 thousand stars for four years monitoring exoplanets transits. In addition to discovering thousands of planets orbiting other stars, Kepler has reinvigorated the study of variable stars by recording apparent stellar brightness versus time. This is especially important for the RR Lyraes, whose periods of a quarter to a full day combined with Earth’s day-night cycle frustrate terrestrial recording.

Although of unprecedented quantity and quality, Kepler’s data is marred in a number of ways. During its primary mission, to keep its solar cells in sunlight, Kepler rotated 90 degrees every quarter solar orbit causing each star to illuminate different pixels of its charge-coupled device. Different pixels have different responses and appear to have contributed to shifts and skews in the data segments. In addition, events such as cosmic ray strikes, monthly science data downlinks, and safe modes caused both small and large gaps in the data.

Spectral scaling

A sine wave of a single amplitude and frequency can represent a regular ripple on a pond or a pure tone in the air. Any wave or sound can be synthesized by suitably combining sine waves of different amplitudes and frequencies. The spectrum of a wave lists the amplitudes of the frequencies of its component sine waves.

If the ratio of a system’s two largest frequencies is near the golden ratio, we call the system golden. Each of Kepler’s golden stars, the RRc Lyraes, oscillates mainly at a primary frequency, but with fine features that exhibit subtle fractal characteristics in the distribution of its secondary frequencies. The secondary sine waves have multiple sizes, with many smaller sizes and fewer larger sizes. Indeed, the number of sizes above a threshold decrease as the threshold increases in a scale-invariant way, like a regular fractal scales under magnification, picking out no special intermediate sizes. Furthermore, balancing order and chaos, this decrease is neither too fast nor too slow. Such balanced scale-free spectral scaling is a robust classical signature of strange nonchaotic behavior.

Data limitations, including noise and gaps and a limited number of cycles, make it difficult to directly observe the fractal geometry and the nonchaotic dynamics in the KIC 5520878 light curve or waveform, especially because the amplitude of the primary sine wave is much larger than the amplitude of the secondary sine wave. Fortunately, the light curve exhibits a subtle but distinctive strange nonchaotic spectral scaling.

We sample the light curve at the primary period and decompose the resulting time series into sine waves of different amplitudes and frequencies. The number of amplitudes depends on the detection threshold. In fact, the number of amplitudes greater than the threshold decreases with increasing threshold in a scale-invariant way: if the threshold quadruples the number eighths (that is, the number of amplitudes greater than four times the threshold is one-eighth the number greater than the threshold), for a wide range of thresholds.

Similarly, in a famous example of spatial scaling, the length of the rugged, fractal Norwegian coast depends on the length of the measuring stick. In fact, the coast length decreases with increasing stick length in a scale invariant way, just like our frequency amplitudes: if the stick length quadruples, the coast length eighths (that is, multiplying the stick length by four divides the coast length by eight), for a wide range of stick lengths.

Implications

Our star, the sun, shines nearly constantly, but the so-called variable stars do not. Why is this so? We are interested in the differences, which must include their interiors. We cannot look into them, but we watch them from outside and try to understand.

One-dimensional mathematical and computer models have had some success simulating their interiors. The models are good enough to create pulsations, and in some cases even double pulsations. Our nonlinear analyses of the Kepler golden stars may provide important constraints on such models, which in turn restricts their interiors, including their elemental compositions and the depths and temperatures of their different layers.

Spectral scaling may be useful in classifying such stars. Scale-free power law scaling appears to characterize both strange nonchaotic attractors and the RRc Lyrae golden stars. Strange nonchaotic dynamics, somewhere between order and chaos, may characterize both stars and the brains of the minds that contemplate them.


“Strange nonchaotic stars” paper on arXiv: http://arxiv.org/abs/1501.01747 .