Chaotic Dynamics of a One-Parameter Third-Order Polynomial FamilyFrancisco Chima SanchezMathematical SciencesBjörn BirnirDepartment of MathematicsUCSB California NanoSystems Institute

Courtesy Google Images

Chaos is “unpredictability”Think “Butterfly Effect”

Animation by Brian Weinstein

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

Dynamical Systems𝑓ሺ𝑥ሻ= 2𝑥 𝑓2ሺ𝑥ሻ= 2ሺ2𝑥ሻ= 4𝑥 𝑓3ሺ𝑥ሻ= 8𝑥 𝑓𝑛ሺ𝑥ሻ= 2𝑛𝑥

Note that any point x greater than zero eventually “blows up” to infinity

Computational ApproachUsing Matlab, I designed bifurcation diagrams for each subclass of time-evolution maps

Bifurcation diagram for logistic family

Lyapunov Exponents

Measures how quickly orbits move apartNegative → asymptotically stableZero → Lyapunov stablePositive → chaotic

Globally Stable SubclassesOrbits converge on fixed pointMost favorable physical case

Attractor SubclassesOrbits remain near an attractorAlso physically favorable

Period-Doubling SubclassesSideways “family tree”Leads to chaotic regime

Feigenbaum CascadesBecause the map has a negative Schwarzian

derivative locally, location of bifurcations given by Feigenbaum constant

Chaotic DynamicsFeigenbaum cascade “route” to chaosLyapunov exponents are positive in

chaotic regimes

Physical Model RecommendationsGlobally stable

subclasses are optimal

Attractor subclasses work well too

Avoid period-doubling cascades

So, what next?Higher-order polynomialsMay include exponentials and

trigonometric equationsMore than one parameter

Special Thanks to…

Jens-Uwe KuhnMariateresa NapoliArica LubinBjörn BirnirOfelia Aguirre