使用保测多项式基计算一维映射的不变测度和Lyapunov指数
Computing the Invariant Measure and the Lyapunov Exponent for One-Dimensional Maps using a Measure-Preserving Polynomial Basis
AI总结：
本文推广了Ulam方法，通过保测多项式基逼近一维混沌映射的不变密度，并证明该方法在光滑密度逼近中的误差结果，同时利用多项式密度计算Lyapunov指数，收敛阶比密度本身高一阶，结合三次多项式实现高效高精度估计。
摘要：
我们考虑Ulam方法的一种推广，用于逼近一维混沌映射的不变密度。我们不再使用分段常数多项式来逼近密度，而是使用由要求它们在n+1个相邻子区间上保持测度所定义的n次多项式。在整个区间上，这导致了对密度的不连续分段多项式逼近。我们证明了该方法用于逼近光滑密度时的误差结果。我们还考虑了使用多项式密度计算Lyapunov指数，并表明其收敛阶比密度本身高一阶。结合在密度逼近中使用三次多项式，这产生了一种非常高效的方法，用于计算Lyapunov指数的高精度估计。我们通过一些例子说明了理论发现。
英文摘要：
We consider a generalisation of Ulam's method for approximating invariant densities of one-dimensional chaotic maps. Rather than use piecewise constant polynomials to approximate the density, we use polynomials of degree n which are defined by the requirement that they preserve the measure on n+1 neighbouring subintervals. Over the whole interval, this results in a discontinuous piecewise polynomial approximation to the density. We prove error results where this approach is used to approximate smooth densities. We also consider the computation of the Lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself. Together with using cubic polynomials in the density approximation, this yields a very efficient method for computing highly accurate estimates of the Lyapunov exponent. We illustrate the theoretical findings with some examples.
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