1. Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

2. Collaboration John F. Lindner The College of Wooster Vivek Kohar North Carolina State University Behnam Kia North Carolina State University Michael Hippke Institute for Data Analysis Germany John G. Learned University of Hawai’i at Mānoa William L. Ditto North Carolina State University

3. Multi-Frequency Stars

4. P. Moskalik, “Multi-mode oscillations in classical cepheids and RR Lyrae-type stars”, Proceedings of the International Astronomical Union 9 (S301) 249 (2013). Petersen Diagram

5. Petersen Diagram Rescaled

6. Spectral Distribution

7. Strobe signal at primary period and plot its values versus time modulo secondary period to form the Poincaré section If the function represents the section, is it smooth?

8. Expand function & its derivatives in Fourier series

9. For smooth sections, expect Fourier coefficients to decay exponentially, so all the derivatives also decay For nonsmooth sections, expect Fourier coefficients to decay slower, as a power law, so that some derivatives diverge Invert to get

10. Since an averaged spectrum decreases with mode number or frequency reinterpret to be the number of super threshold spectral peaks

11. So rich, rough spectra have power law spectral distributions

12. Stellar Analysis

14. KIC 5520878 Normalized Flux Sample

15. Lomb-Scargle Periodogram

16. Spectral Distribution

17. Gutenberg-Richter law for volcanic Canary Islands earthquake distribution

18. Some Number Theory

19. Golden Ratio Slow convergence suggests maximally irrational

20. Liouville Number Example of nearly rational irrational

21. Model 1: Finite Spring Network

23. Natural Frequencies

24. Model 2: Hierarchical Spring Network

26. Natural Frequencies

27. Model 3: Asymmetric Quartic Oscillator

28. Asymmetric Quartic Potential Energy Sinusoidal Forcing Drive Frequency a Golden Ratio Above Natural Frequency

31. Model 4: Pressure vs. Gravity Oscillator

32. Adiabatic Simplification Pressure & Volume

33. Potential Energy Force Sinusoidal Forcing

35. Model Data

36. Model 5: Autonomous Flow

37. generalized Lorenz convection flow caused by thermal & gravity gradients plus vibration

38. Adjust parameters so that

41. Model 6: Twist Map

42. Twist map is circles for vanishing push

43. Push perturbation has vanishing mean

44. Least resonant golden shift remains

45. Insights from Helioseismology

46. Helioseismology and asteroseismology have observed many seismic spectral peaks in the sun and other nonvariable stars, which correspond to thousands of normal modes Yet, despite preliminary analysis, we have not discovered power law scaling in the solar oscillation spectrum

47. Stochasticity and turbulence dominate the pressure waves in the sun that produce its standing wave normal modes In contrast, a varying opacity feedback mechanism inside a variable star creates its regular pulsations In golden stars, interactions with this pulsating mode may dissipate all other modes except those a golden ratio away

48. Discussion

49. The simple nonlinear models suggest the importance of considering simple explanations 0: The golden ratio itself has unique and remarkable properties; as the irrational number least well approximated by rational numbers, it is the least “resonant” number 1: A finite network model of identical springs and masses has two normal modes whose frequency ratio is golden 2: An infinite network hierarchy can be mass terminated in two ways to naturally generate two modes whose frequency ratio is golden, while a realistic truncation of the model generates a ratio near golden, as observed in the golden stars Summary

50. 3: A simple asymmetric nonlinear oscillator produces a rich spectrum with a power-law spectral distribution 4: A more realistic oscillator model of pressure countering gravity exhibits a recognizable but stylized golden star attractor 5: An unforced Lorenz-like convection flow also produces a singular spectrum with a power-law spectral distribution, provided its parameters are tuned so that a golden ratio characterizes its orbit 6: An ensemble of twist maps naturally evolve to a golden state, because golden shifts are least resonant with any oscillatory perturbation Summary

51. The Feigenbaum constant delta ~4.67, which characterizes the period doubling route to chaos, has been observed in many diverse experiments Does the golden ratio ~1.62, or equivalently the inverse golden ratio ~0.62, play a similar role? Or does the mysterious factor of ~0.62, which characterizes many multifrequency stars, merely result from nonradial stellar oscillation modes? Simplicity vs. Complexity

52. Some natural dynamical patterns result from universal features common to even simple models Other patterns are peculiar to particular physical details Is the frequency distribution of variable stars universal or particular? Universality vs. Particularity

53. Thanks for Listening