Scaling Behavior in the Stochastic 1D Map

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Presentation transcript:

Scaling Behavior in the Stochastic 1D Map Eui-Sun Lee Department of Physics Kangwon National University Stochastic 1D map x : state variable A : parameter controlling the degree of nonlinearity  : parameter controlling the amplitude of the additive noise Random numbers :  : random variable, n is statistically independent bounded random number, for any n, for nm.

Effect of Noise for period doubling Transition to Chaos Bifurcation diagram & Lyapunov exponent n is uniformly distributed over [-0.5,0.5], so Bifurcation diagram Lyapunov exponent - <  < 0 : Gray,  > 0(Chaos) : Black.

Scaling factor for the stochastic 1D map Parameter scaling factor: = 4.6692... . Orbital scaling factor: = - 2.5029… . Amplitude of noise scaling factor: = 6.62903… . Through the renormalization group analysis , the amplitude of noise scaling factor is obtained.

Summary = 6.62903… . Amplitude of noise scaling factor: 1. Through the renormalization group analysis , The Amplitude of noise scaling factor(µ) is obtained. Amplitude of noise scaling factor: = 6.62903… .