Computing Lyapunov constants for random recurrences with smooth coefficients. (English) Zbl 0984.65135
Summary: 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 nonsmooth dynamics (nondifferentiable invariant measures), making computations hard. Here, however, we consider recurrences with smooth random coefficients and smooth invariant measures. By computing discretized invariant measures and applying Richardson extrapolation, we can compute Lyapunov constants to 10 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 \((1.0574735537\dots)^n\). 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 \((1.1149200917\dots)^n\) and \((0.9949018837\dots)^n\), respectively.
MSC:
65P40 | Numerical nonlinear stabilities in dynamical systems |
65C40 | Numerical analysis or methods applied to Markov chains |
37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
37M25 | Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) |
Keywords:
Lyapunov constants; Markov chain; nondifferentiable invariant measures; random recurrences; Fibonacci sequence; nonsmooth dynamics; Richardson extrapolationOnline Encyclopedia of Integer Sequences:
Decimal expansion of the growth constant of sequences x_{n+1} = x_n + c_{n+1} *x_{n-1} with normally distributed random coefficients c_n.Decimal expansion of a constant related to the asymptotics of A048634.
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