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A graph of periodic function f(x) that has period L exhibits the same pattern every L units along the xaxis, so that f(xL) = f(x) for every value of x If we know what the function looks like over one complete period, we can thus sketch a graph of the function over a wider interval of x (that may contain many periods) f(x) x PERIOD = L Toc JJ. The LVC coordinate system is shown in the image above, using the conventions defined in Adamo 1The basis vectors {Ô 1, Ô 2, Ô 3} are shown centered on the primary Spacecraft and are aligned with the LVLH frame The angle θ is the same as defined in the RIC Cylindrical frame and is measured in the orbital plane of the primary Spacecraft The is measured form the orbital plane. Theorem If s\ipp€Mvxp~x$ < 1 and Eu e C then E" e C°° The same holds for the weak unstable distribution of Anosov flows Katok obtained a similar result via nonstationary normal forms (oral communication) Since our technique is standard HK, LMM, we only sketch the proof We first list corollaries Considering f~x yields Corollary 1.
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` 5 ¹ p ¹ d ( ± 3\ % )" Ê Á ¡ Ý. Let’s return to the equation Ax = x Let’s look at another example Example A= 2 4 0 3 This is a 2 by 2 matrix, so we know that 1 2 = tr(A) = 5 1 2 = det(A) = 6 6 The eigenvalues are 1 = 2 and 2 = 3In fact, because this matrix was upper triangular, the eigenvalues are on the diagonal!. \0ñ ô1 ¦8o @ ô I S u b0 ó j g _ \0ñ ô1 b) Ý 1 ô w M 4 z X Ç)r 4 z b1 ô b X ¶ 4 z)r S ¥ r b 1 ô / Q ¦ S/ Q 4 z)r \ ¥ _ 1 ô / Q 1 ô / Q 0£ 4 z X Ç Ç Ç 1 ô / Q X ¶ d ¶ ¶ d 1 L b1 ô / Q X ¦ S/ Q b U q a b v b '¨ 1 L 0°3U K Z > ~ r M 1 L ¦ S/ Q 1 L b 1 ô / Q 0£ _. 1 Ø b%, ¿ Ý î b v) _ X 8 Z ¹ B>1> º>3 v>0>6 ¥ %, ¿ Ý î í%,2( ;.
