The Dynamics of the Pendulum By Tori Akin and Hank Schwartz

An Introduction What is the behavior of idealized pendulums? What types of pendulums will we discuss? – Simple – Damped vs. Undamped – Uniform Torque – Non-uniform Torque

Parameters To Consider m-mass (or lack thereof) L-length g-gravity α-damping term I -applied torque Result: v’=-g*sin(θ)/L θ‘=v

Methods Nondimensionalization Linearization XPP/Phase Plane analysis Bifurcation Analysis Theoretical Analysis

Nondimensionalization Let ω=sqrt(g/L) and d τ/dt= ω θ‘=v→v v’=-g*sin(θ)/L →-sin(θ)

Systems and Equations Simple Pendulum – θ‘=v – v‘=-sin(θ) Simple Pendulum with Damping – θ‘= v – v‘=-sin(θ)- αv Simple Pendulum with constant Torque – θ‘= v – v‘=-sin(θ)+I

Hopf Bifurcation Simple Pendulum with Damping – θ‘= v – v‘=-sin(θ)- αv Jacobian: Trace=- α Determinant=cos( θ ) Vary α from positive to zero to negative

The Simple Pendulum with Constant Torque and No Damping The theta null cline: v = 0 The v null cline: θ=arcsin(I) Saddle Node Bifurcation I=1 Jacobian: θ‘= v v‘=-sin(θ)+I

Driven Pendulum with Damping θ’ = v v’ = -sin(θ) –αv + I Limit Cycle The theta null cline: v = 0 The v null cline: v = [ I – sin(θ)] / α I = sin(θ) and as cos 2 (θ) = 1 – sin 2 (θ) we are left with cos(θ) = ±√(1-I 2 ) Characteristic polynomial- λ 2 + α λ + √(1-I 2 ) = 0 which impliesλ = { ‒α±√ [α 2 - 4√(1-I 2 ) ] } / 2 Jacobian:

Homoclinic Bifurcation

Infinite Period Bifurcation

Bifurcation Diagram

Non-uniform Torque and Damped Pendulum τ’ = 1 θ’ = v v’ = -sin(θ) –αv + Icos(τ)

Double Pendulum

Results Basic Workings Various Oscillating Systems Hopf Bifurcation-Simple Pendulum Homoclinic Global Bifurcation-Uniform Torque Chaotic Behavior Saddle Node Bifurcation Infinite Period Bifurcation Applications to the real world Thank You!