Computing Lyapunov constants for random recurrences with smooth coefficients
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), maki...
| Main Authors: | , |
|---|---|
| Format: | Report |
| Published: |
Unspecified
2000
|
| _version_ | 1797068482200731648 |
|---|---|
| author | Wright, T Trefethen, L |
| author_facet | Wright, T Trefethen, L |
| author_sort | Wright, T |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T22:11:21Z |
| format | Report |
| id | oxford-uuid:51ea119c-3c3c-4ccb-9a20-68de774b0e76 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:11:21Z |
| publishDate | 2000 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:51ea119c-3c3c-4ccb-9a20-68de774b0e762022-03-26T16:22:29ZComputing Lyapunov constants for random recurrences with smooth coefficientsReporthttp://purl.org/coar/resource_type/c_93fcuuid:51ea119c-3c3c-4ccb-9a20-68de774b0e76Mathematical Institute - ePrintsUnspecified2000Wright, TTrefethen, LIn 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. |
| spellingShingle | Wright, T Trefethen, L Computing Lyapunov constants for random recurrences with smooth coefficients |
| title | Computing Lyapunov constants for random recurrences with smooth coefficients |
| title_full | Computing Lyapunov constants for random recurrences with smooth coefficients |
| title_fullStr | Computing Lyapunov constants for random recurrences with smooth coefficients |
| title_full_unstemmed | Computing Lyapunov constants for random recurrences with smooth coefficients |
| title_short | Computing Lyapunov constants for random recurrences with smooth coefficients |
| title_sort | computing lyapunov constants for random recurrences with smooth coefficients |
| work_keys_str_mv | AT wrightt computinglyapunovconstantsforrandomrecurrenceswithsmoothcoefficients AT trefethenl computinglyapunovconstantsforrandomrecurrenceswithsmoothcoefficients |