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Comparative Analysis of Poincaré Map Algorithms for Parametrically Forced Pendulum

This study explores four algorithms (Euler, 2nd-order Range Kutta, Splitting 2nd-order Range Kutta, Splitting 4th-order Range Kutta) for solving SDEs in the Stochastic Parametrically Forced Pendulum system. Critical scaling behavior near the accumulation point A∞ is investigated through bifurcation diagrams and Lyapunov Exponents. The influence of different time steps (1/100 and 1/400) on system behavior is analyzed.

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Comparative Analysis of Poincaré Map Algorithms for Parametrically Forced Pendulum

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  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)

  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=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.)

  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=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.)

  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.

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