1. Study of the Different Algorithms for Poincaré Map of Parametrically Forced Pendulum
Eui-Sun Lee
Department of Physics
Kangwon National University
• The Four Different Algorithms to Solve the SDEs
1. Euler Algorithm.
2. 2nd–order Range Kutta Algorithm.
3. Splitting 2nd – order Range Kutta Algorithm.
4. Splitting 4th – order Range Kutta Algorithm.

2. 1. Euler algorithm
Critical Scaling Behavior near the Critical Point in the Stochastic Parametrically Forced Pendulum
• The Stochastic Parametrically Forced Pendulum(PFP)
Random numbers :

3. Case of 1/100 time step
In the Case of the 1/100 Time Step, the Magnified Pictures are Not Self-Similar.
Accumulation point :A= ,Parameter Scaling factor:=4.6,Orbital scalng factor: =-2.5.
a. Sequence of the close-ups of Bifurcation diagram(No. of division:350, No. of transient:500, No. of map plots: 250.)
b. Sequence of the close-ups of Lyapunov Exponent(No. of division:350, No. of transient: 500, No. of average:10,000.)

4. Case of 1/400 time step
In the Case of the 1/400 Time Step, the Magnified Pictures are Self-Similar.
Accumulation point :A= ,Parameter Scaling factor:=4.66,Orbital scalng factor: =-2.5.
1. Sequence of the close-ups of Bifurcation diagram(No. of division:350, No. of transient:500, No. of map plots: 250.)
2. Sequence of the close-ups of Lyapunov Exponent(No. of division:350, No. of transient: 500, No. of average:10,000.)

5. Summary
We Study about the Euler Algorithm for Poincaré map of Stochastic PFP.
2. In Case of 1/400 Time Step of Euler Algorithm, the Stochastic PFP Exhibits
the Critical Scaling Behavior near the Accumulation Point A.