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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

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

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

Simple Nonlinear ModelsSuggest Variable Star

UniversalityJohn F. Lindner, Wooster College

Presented by John G. LearnedUniversity of Hawai’i at Mānoa

Page 2: 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. LindnerThe College of Wooster

Vivek KoharNorth Carolina State University

Behnam KiaNorth Carolina State University

Michael HippkeInstitute for Data AnalysisGermany

John G. LearnedUniversity of Hawai’iat Mānoa

William L. DittoNorth Carolina State University

Page 3: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Multi-Frequency Stars

Page 4: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

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

Page 5: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Petersen Diagram Rescaled

Page 6: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Spectral Distribution

Page 7: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Strobe signal at primary period and plot its values versus time

modulo secondary periodto form the Poincaré section

If the function

represents the section,is it smooth?

Page 8: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Expand function & its derivativesin Fourier series

Page 9: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

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

Page 10: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Since an averaged spectrum decreases with mode number or frequency

reinterpret

to be the number of super threshold spectral peaks

Page 11: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

So rich, rough spectrahave power law spectral distributions

Page 12: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Stellar Analysis

Page 13: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 14: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

KIC 5520878 Normalized Flux Sample

Page 15: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Lomb-Scargle Periodogram

Page 16: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Spectral Distribution

Page 17: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Gutenberg-Richter law forvolcanic Canary Islands earthquake distribution

Page 18: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Some Number Theory

Page 19: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Golden Ratio

Slow convergence suggests maximally irrational

Page 20: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Liouville Number

Example of nearly rational irrational

Page 21: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Model 1:Finite Spring

Network

Page 22: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 23: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Natural Frequencies

Page 24: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Model 2:Hierarchical

Spring Network

Page 25: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 26: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Natural Frequencies

Page 27: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Model 3:Asymmetric Quartic

Oscillator

Page 28: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Asymmetric Quartic Potential Energy

Sinusoidal Forcing

Drive Frequency a Golden Ratio Above Natural Frequency

Page 29: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 30: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 31: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Model 4:Pressure vs. Gravity

Oscillator

Page 32: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Adiabatic Simplification

Pressure & Volume

Page 33: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Potential Energy

Force

Sinusoidal Forcing

Page 34: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 35: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Model

Data

Page 36: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Model 5:Autonomous Flow

Page 37: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

generalized Lorenz convection flow caused bythermal & gravity gradients plus vibration

Page 38: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Adjust parameters so that

Page 39: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 40: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa
Page 41: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Model 6:Twist Map

Page 42: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Twist map is circles for vanishing push

Page 43: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Push perturbation has vanishing mean

Page 44: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Least resonant golden shift remains

Page 45: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Insights from Helioseismology

Page 46: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Helioseismology and asteroseismology have observed manyseismic spectral peaks in the sun and other nonvariable stars,

which correspond to thousands of normal modes

Yet, despite preliminary analysis, we have not discoveredpower law scaling in the solar oscillation spectrum

Page 47: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Stochasticity and turbulence dominate the pressure waves inthe sun that produce its standing wave normal modes

In contrast, a varying opacity feedback mechanism inside avariable star creates its regular pulsations

In golden stars, interactions with this pulsating mode maydissipate all other modes except those a golden ratio away

Page 48: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Discussion

Page 49: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

The simple nonlinear models suggest the importance of considering simple explanations

0: The golden ratio itself has unique and remarkableproperties; 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 hastwo normal modes whose frequency ratio is golden

2: An infinite network hierarchy can be mass terminated in twoways to naturally generate two modes whose frequency ratio is

golden, while a realistic truncation of the model generates aratio near golden, as observed in the golden stars

Summary

Page 50: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

3: A simple asymmetric nonlinear oscillator produces a richspectrum with a power-law spectral distribution

4: A more realistic oscillator model of pressure counteringgravity exhibits a recognizable but stylized golden

star attractor

5: An unforced Lorenz-like convection flow also produces asingular 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 goldenstate, because golden shifts are least resonant with any

oscillatory perturbation

Summary

Page 51: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

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

Page 52: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

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

Page 53: Simple Nonlinear Models Suggest Variable Star Universality John F. Lindner, Wooster College Presented by John G. Learned University of Hawai’i at Mānoa

Thanks for Listening


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