Ott William, Rivas Mauricio A, West James
Department of Mathematics, University of Houston, URL : http://www.math.uh.edu/~ott/.
Department of Mathematics, Wake Forest University.
J Stat Phys. 2015 Dec;161(5):1098-1111. doi: 10.1007/s10955-015-1376-9. Epub 2015 Oct 1.
Can Lyapunov exponents of infinite-dimensional dynamical systems be observed by projecting the dynamics into ℝ using a 'typical' nonlinear projection map? We answer this question affirmatively by developing embedding theorems for compact invariant sets associated with maps on Hilbert spaces. Examples of such discrete-time dynamical systems include time- maps and Poincaré return maps generated by the solution semigroups of evolution partial differential equations. We make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite-dimensional dynamical system. In so doing, we adopt an empirical approach and formulate checkable conditions under which a Lyapunov exponent computed from experimental data will be a Lyapunov exponent of the infinite-dimensional dynamical system under study (provided the nonlinear projection map producing the data is typical in the sense of prevalence).
能否通过使用“典型”非线性投影映射将无限维动力系统的动力学投影到ℝ中，来观测其李雅普诺夫指数？我们通过为与希尔伯特空间上的映射相关的紧致不变集发展嵌入定理，对这个问题给出了肯定的回答。这类离散时间动力系统的例子包括由演化偏微分方程的解半群生成的时间映射和庞加莱返回映射。我们尽最大努力将假设置于投影动力学上，而非潜在的无限维动力系统上。这样做时，我们采用了一种经验方法，并制定了可检验的条件，在这些条件下，根据实验数据计算出的李雅普诺夫指数将是所研究的无限维动力系统的李雅普诺夫指数（前提是产生数据的非线性投影映射在普遍意义上是典型的）。
