二維度的函數及其在類神經網路的應用

Abstract

考慮二維的函數T( x, y ) = ( y, F(y) – bx). 此處的F是一個三片的線性函數所結合而成的函數。第一個部分我們證明出T的Semiconjugate 條件可推得Smale Horseshoe的存在，接下來將此定理應用在一維的CNNs上。此時的混沌現象是發生在所謂的 [3,3] 區。第二個部分的結果是藉由L. S. Young 的論文證明出T在某些參數範圍內會有Bowen-Reulle-Sinai 測度，這是另一種觀點的混沌現象，將其應用在CNNs中，這些參數產生在 [3,3], [3,2], [2,3] 中的某些部分。

Of concern is a two-dimensional map T of the form. T( x , y ) = ( y , F(y) – bx ). Here F is a three-piece linear map. This thesis contains two parts. In part one, we first prove a theorem which states that a semiconjugate condition for T implies the existence of Smale horseshoe. Second, the theorem is applied to show the spatial chaos of one-dimensional Cellular Neural Networks. We improve a result of Hsu [2000]. Such horseshoe type of chaos corresponds to the spatial chaos in the so called [3,3] region which has spatial entropy ln 2 . In part two, a Theorem of L. S. Young is used to show that in certain parameters' range T has a Bowen-Reulle-Sinai measure. In the case of CNNs, this amounts to the fact that whenever the parameters are in the parts of [3,3], [3,2], and [2,3], then the spatial chaos of one-dimensional Cellular Neural Networks exists in the Bowen-Reulle measure's sense.