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Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa Collaboration John F. Lindner The College of Wooster Michael Hippke Institute for Data Analysis Germany Vivek Kohar North Carolina State University John G. Learned University of Hawai’i at Mānoa Behnam Kia North Carolina State University William L. Ditto North Carolina State University Multi-Frequency Stars Petersen Diagram 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 Rescaled Spectral Distribution 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? Expand function & its derivatives in Fourier series 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 Since an averaged spectrum decreases with mode number or frequency reinterpret to be the number of super threshold spectral peaks So rich, rough spectra have power law spectral distributions Stellar Analysis KIC 5520878 Normalized Flux Sample Lomb-Scargle Periodogram Spectral Distribution Gutenberg-Richter law for volcanic Canary Islands earthquake distribution Some Number Theory Golden Ratio Slow convergence suggests maximally irrational Liouville Number Example of nearly rational irrational Model 1: Finite Spring Network Natural Frequencies Model 2: Hierarchical Spring Network Natural Frequencies Model 3: Asymmetric Quartic Oscillator Asymmetric Quartic Potential Energy Sinusoidal Forcing Drive Frequency a Golden Ratio Above Natural Frequency Model 4: Pressure vs. Gravity Oscillator Pressure & Volume Adiabatic Simplification Force Potential Energy Sinusoidal Forcing Model Data Model 5: Autonomous Flow generalized Lorenz convection flow caused by thermal & gravity gradients plus vibration Adjust parameters so that Model 6: Twist Map Twist map is circles for vanishing push Push perturbation has vanishing mean Least resonant golden shift remains Insights from Helioseismology 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 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 Discussion Summary 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 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 Simplicity vs. Complexity 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? Universality vs. Particularity 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? Thanks for Listening