1. Poincaré Map of Parametrically Forced Pendulum
Eui-Sun Lee
Department of Physics
Kangwon National University
Continuous time system
Parametrically forced pendulum(PFP) :
the angular position
the damping coefficient
the amplitude of the external driving force of period one
the undamped natural frequency of the unforced pendulum
These equation have the Inversion symmetry

2. Solving The Second-Order Differential Equation
A second-order differential equation is reduced to two first-order differential equation.
Forth-order Runge-Kutta method
Initial value:
Step width
a number of steps
for( i=0,1,2…,N)

3. Discrete time system Poincaré Map Parametrically forced pendulum
The Poincaré Map(Solid circle) is The Stroboscopic
sampling of Flow(Solid Line).
The Poincaré Map(T) is Convenient for Finding The Periodic Orbit,Analyzing the Stability of The
Orbit, Showing up Fractal Geometric Properties of Attractor.

4. Symmetry The Poincaré Map has an Inversion Symmetry such that
In The PFP, The Poincaré Map and flow has an Inversion Symmetry and Asymmetry.
Inversion Symmetry
Inversion Asymmetry

5. Summary
1.A dynamical system is described by differential equation mathematically.
2.The Poincaré Map(T) is Convenient for Finding The Periodic Orbit,Analyzing the Stability of The
Orbit, Showing up Fractal Geometric Properties of Attractor.