1. Name: Ha Pham Chaos Theory: Double Pendulum

2. Methods Double pendulum: simple pendulum with another pendulum attached to its end Solve Lagrangian: L = Kinetic Energy – Potential Energy Resulting E.O.M : How to solve: ode45 ! (t, x’) = ode45(odefun,tspan,x) Now Convert Back to Spatial Coordinate: f = [theta1dot,theta1dbldot,theta2dot,theta2dbldot] Initial conditions: x = [theta1; theta1dot; theta2;theta2dot] Output state vector: x’=[theta1’; theta1dot’; theta2’;theta2dot’]

3. For initial conditions x = [0.5;0;0.5;0] Simple Pendulum This dynamical system is chaotic, which means small deviation grows really fast !

4. Poincare section of the previous initial condition

5. Using different initial conditions vector Extremely Sensitive to initial conditions!