Eui-Sun Lee Department of Physics Kangwon National University

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Presentation transcript:

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.

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

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=0.351 003,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.)

Case of 1/400 time step In the Case of the 1/400 Time Step, the Magnified Pictures are Self-Similar. Accumulation point :A=3.559 562,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.)

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.