Chima Sanchez STEEM Research Presentation
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Transcript of Chima Sanchez STEEM Research Presentation
![Page 1: Chima Sanchez STEEM Research Presentation](https://reader035.fdocuments.in/reader035/viewer/2022062522/58a7f2611a28abfa488b4613/html5/thumbnails/1.jpg)
Chaotic Dynamics of a One-Parameter Third-Order Polynomial FamilyFrancisco Chima SanchezMathematical SciencesBjörn BirnirDepartment of MathematicsUCSB California NanoSystems Institute
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Courtesy Google Images
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Chaos is “unpredictability”Think “Butterfly Effect”
Animation by Brian Weinstein
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Models Require Analysis For models using maps of the form
where a is any real number. Some subclasses will exhibit chaotic dynamics,
corresponding to unpredictability, like this galactic revolution model
𝑓𝑎ሺ𝑥ሻ= 𝑎−𝑎𝑥−𝑎𝑥2 −𝑎𝑥3
Source: N.D. Caranicolas
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Dynamical Systems𝑓ሺ𝑥ሻ= 2𝑥 𝑓2ሺ𝑥ሻ= 2ሺ2𝑥ሻ= 4𝑥 𝑓3ሺ𝑥ሻ= 8𝑥 𝑓𝑛ሺ𝑥ሻ= 2𝑛𝑥
Note that any point x greater than zero eventually “blows up” to infinity
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Computational ApproachUsing Matlab, I designed bifurcation diagrams for each subclass of time-evolution maps
Bifurcation diagram for logistic family
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Lyapunov Exponents
Measures how quickly orbits move apartNegative → asymptotically stableZero → Lyapunov stablePositive → chaotic
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Globally Stable SubclassesOrbits converge on fixed pointMost favorable physical case
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Attractor SubclassesOrbits remain near an attractorAlso physically favorable
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Period-Doubling SubclassesSideways “family tree”Leads to chaotic regime
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Feigenbaum CascadesBecause the map has a negative Schwarzian
derivative locally, location of bifurcations given by Feigenbaum constant
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Chaotic DynamicsFeigenbaum cascade “route” to chaosLyapunov exponents are positive in
chaotic regimes
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Physical Model RecommendationsGlobally stable
subclasses are optimal
Attractor subclasses work well too
Avoid period-doubling cascades
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So, what next?Higher-order polynomialsMay include exponentials and
trigonometric equationsMore than one parameter
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Special Thanks to…
Jens-Uwe KuhnMariateresa NapoliArica LubinBjörn BirnirOfelia Aguirre