The Mathematical Institute, University of Oxford, Eprints Archive

Computing Lyapunov constants for random recurrences with smooth coefficients

Wright, Thomas G. and Trefethen, Lloyd N. (2000) Computing Lyapunov constants for random recurrences with smooth coefficients. Technical Report. Unspecified. (Submitted)

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Abstract

In recent years, there has been much interest in the growth and decay rates (Lyapunov constants) of solutions to random recurrences such as the random Fibonacci sequence $x_{n+1} = \pm x_{n} \pm x_{n-1}$. Many of these problems involve non-smooth dynamics (nondifferentiable invariant measures), making computations hard. Here, however, we consider recurrences with smooth random coefficients and smooth invariant measures. By computing discretised invariant measures and applying Richardson extrapolation, we can compute Lyapunov constants to ten digits of accuracy. In particular, solutions to the recurrence $x_{n+1} = x_{n} + c_{n+1} x_{n-1}$, where the $\{c_{n}\}$ are independent standard normal variables, increase exponentially (almost surely) at the asymptotic rate $(1.0574735537...)^{n}$. Solutions to the related recurrences $x_{n+1} = c_{n+1}x_{n} + x_{n-1}$, and $x_{n+1} = c_{n+1}x_{n} + d_{n+1}x_{n-1}$ (where the $\{d_{n}\}$ are also independent standard normal variables) increase (decrease) at the rates $(1.1149200917...)^{n}$ and $(0.9949018837...)^(n)$ respectively.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1280
Deposited By:Lotti Ekert
Deposited On:01 Jun 2011 09:20
Last Modified:01 Jun 2011 09:20

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