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Scaling Behavior in the Stochastic 1D Map

This study investigates the scaling behavior of the stochastic 1D map and its sensitivity to parameters and noise. The effect of noise on period doubling transition to chaos is examined through bifurcation diagrams and Lyapunov exponents. Renormalization group analysis is used to obtain the amplitude of noise scaling factor. The results show a scaling factor of 6.62903 for the stochastic 1D map.

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Scaling Behavior in the Stochastic 1D Map

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  1. 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 variable, nis statistically independent bounded random number, for any n, for nm.

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

  3. = 4.6692... . = - 2.5029… . = 6.62903… . Scaling factor for the stochastic 1D map • Parameter scaling factor: • Orbital scaling factor: • Amplitude of noise scaling factor: Through the renormalization group analysis , the amplitude of noise scaling factor is obtained.

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

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