From 06c00fed8933a85c6e0846968afe463655c77114 Mon Sep 17 00:00:00 2001 From: Michael Uleysky Date: Fri, 18 Jan 2019 14:09:35 +1000 Subject: [PATCH] Text about the evolution matrix --- Evolution_matrix/SVD.eps | 13402 ++++++++++++++++++++++++++++++++++++ Evolution_matrix/lyap.tex | 476 ++ 2 files changed, 13878 insertions(+) create mode 100644 Evolution_matrix/SVD.eps create mode 100644 Evolution_matrix/lyap.tex diff --git a/Evolution_matrix/SVD.eps b/Evolution_matrix/SVD.eps new file mode 100644 index 0000000..0d9f8d6 --- /dev/null +++ b/Evolution_matrix/SVD.eps @@ -0,0 +1,13402 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: SVD.eps +%%DocumentFonts: (atend) +%%BoundingBox: 65 200 455 585 +%%Orientation: Portrait +%%EndComments + +%%BeginFont: SFTI1200 +%!FontType1-1.0: SFTI1200 0.3 +%%CreationDate: Wed Sep 12 2001 +% Copyright (c) 2001 Vladimir Volovich . +% See the file COPYING (GNU General Public License) for license conditions. +% Converted from METAFONT EC/TC and LH fonts: +% ecti1200, 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Distributed under the GNU General Public License (http://www.gnu.org/copyleft/gpl.txt). As a special exception, permission is granted to include this font program in a PostScript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def +/FullName (Pazo Math Italic) readonly def +/FamilyName (PazoMath) readonly def +/ItalicAngle -9.50 def +/isFixedPitch false def +/UnderlinePosition -100 def +/UnderlineThickness 50 def +/Weight (Regular) readonly def +end readonly def +/FontName /PazoMath-Italic def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put } for +dup 32 /space put +dup 35 /epsilon put +dup 36 /rho1 put +dup 68 /Delta put +dup 70 /Phi put +dup 71 /Gamma put +dup 74 /theta1 put +dup 76 /Lambda put +dup 80 /Pi put +dup 81 /Theta put +dup 83 /Sigma put +dup 85 /Upsilon put +dup 86 /sigma1 put +dup 87 /Omega put +dup 88 /Xi put +dup 89 /Psi put +dup 97 /alpha put +dup 98 /beta put +dup 99 /chi put +dup 100 /delta put +dup 101 /epsilon1 put +dup 102 /phi put +dup 103 /gamma put +dup 104 /eta put +dup 105 /iota put +dup 106 /phi1 put +dup 107 /kappa put +dup 108 /lambda put +dup 109 /mu put +dup 110 /nu put +dup 112 /pi put +dup 113 /theta put +dup 114 /rho put +dup 115 /sigma put +dup 116 /tau put +dup 117 /upsilon put +dup 118 /pi1 put +dup 119 /omega put +dup 120 /xi put +dup 121 /psi put +dup 122 /zeta put +dup 160 /Euro put +dup 182 /partialdiff put +dup 224 /tie put +dup 226 /dotlessj put +readonly def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.00100 0 0 0.00100 0 0] readonly def +/UniqueID 5116634 def +/FontBBox {-70 -277 902 733} readonly def +currentdict end +currentfile eexec +e8dc61728071abd4d5bcd230c4e186dce095a0ca84ffc476adfaa8212426 +8e60d2cc570c5f52c10ed4a85a71ce8e5bcf26aa13b27d7fce1cc0c72c3b +848dd5ffb2d56d06a1920fade4ffa3b35336627e9ed1622c1f9652956d4a +424488b4c4ffda9936442bbe0ca749700ae98a2bf964fad088f0cc64b7c1 +d76742f25e90e759aca624a321402b1d20863faf5e04b86c1ae66f450b15 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def +%/HArrow {/HeadHeight 10 /HeadWidth 5 /LineWidth 10} def +/HArrw 0.8 def +/HArrhd 0.2 def +/circlinewidth 1.5 def +/phi1 30 def +/phi2 40 def +/sigma1 1.3 def +/sigma2 1.0 sigma1 div def +/fs 16 def +/subfs fs 0.6 mul def +/subsh -0.1 fs mul def +/phir 0.5 u def +/phirlinewidth 0.5 def + +/XYArrows { + <> Arrow + <> Arrow + /SFRM1200 findfont fs scalefont setfont + XYArrowlen u 0 moveto + [ [/SFRM1200 fs 0 fs -0.4 mul (x) true] ] <> Show + 0 XYArrowlen u moveto + [ [/SFRM1200 fs 0.3 fs mul 5 (y) true] ] <> Show +} def + +/strparams {20 dict begin /inp exch def gsave newpath 0 0 moveto + /stringtype {inp false charpath} def + /arraytype + { + inp {dup dup 0 get findfont exch 1 get scalefont setfont + dup 2 get /x exch def dup 3 get /y exch def + dup 5 get {x y rmoveto 4 get false charpath}{pop} ifelse + } forall + } def + inp type exec pathbbox + /ury exch def /urx exch def /lly exch def /llx exch def + % Return values + urx llx sub ury lly sub llx neg lly neg +grestore end} def +/GetParam { 3 dict begin + /default exch def /key exch def /dict exch def + dict key known {dict key get}{default} ifelse + end +} def +/Show {20 dict begin /Params exch def /String exch def + /HAlign Params /HAlign -1 GetParam def + /VAlign Params /VAlign -1 GetParam def + /UseHShift Params /UseHShift false GetParam def + /UseVShift Params /UseVShift false GetParam def + String strparams /VShift exch def /HShift exch def /Height exch def /Width exch def + % Correct HAlign and VAlign + HAlign 0 lt {/HAlign -1 def}{} ifelse HAlign 0 gt {/HAlign 1 def}{} ifelse + VAlign 0 lt {/VAlign -1 def}{} ifelse VAlign 0 gt {/VAlign 1 def}{} ifelse + % Move initial point + UseHShift {HShift 0 rmoveto}{} ifelse UseVShift {0 VShift rmoveto}{} ifelse + HAlign 0 eq {Width -0.5 mul 0 rmoveto}{} ifelse HAlign 1 eq {Width neg 0 rmoveto}{} ifelse + VAlign 0 eq {0 Height -0.5 mul rmoveto}{} ifelse VAlign 1 eq {0 Height neg rmoveto}{} ifelse + /stringtype {show} def + /arraytype {{ + dup dup 0 get findfont exch 1 get scalefont setfont currentpoint 3 -1 roll dup dup 2 get exch 3 get rmoveto dup 4 get show + dup length 5 eq {pop pop pop}{ + dup length 6 eq {5 get {pop pop}{moveto} ifelse}{ + dup 5 get exch 6 get currentpoint 4 -2 roll % x0 y0 x y ux uy + {1}{3}ifelse index exch % x0 y0 x y ye ux + {2}{4}ifelse index exch %x0 y0 x y xe ye + moveto pop pop pop pop + }ifelse}ifelse + }forall} def + String dup type exec +end} def +/Arrow +{ 20 dict begin /Params exch def + /X0 Params /X0 0 GetParam def + /Y0 Params /Y0 0 GetParam def + Params /X1 known + { + /X1 Params /X1 100 GetParam def + /Y1 Params /Y1 100 GetParam def + /Length X1 X0 sub dup mul Y1 Y0 sub dup mul add sqrt def + /Angle Y1 Y0 sub X1 X0 sub atan def + } + { + /Length Params /Length 100 GetParam def + /Angle Params /Angle 45 GetParam def + } ifelse + /HeadHeight Params /HeadHeight 7 GetParam def + /HeadWidth Params /HeadWidth 3 GetParam def + /Heads Params /Heads 1 GetParam def + /SolidHead Params /SolidHead false GetParam def + + gsave + % Setting line parameters + 1 setlinejoin 1 setlinecap + Params /LineWidth known {Params /LineWidth get setlinewidth} if + % Transform coordinates + X0 Y0 translate + Angle rotate + % Draw main line + newpath + 0 0 moveto + Length 0 lineto + stroke + % Draw front head + Heads 0 eq not { + newpath + Length HeadHeight sub HeadWidth moveto + Length 0 lineto + Length HeadHeight sub HeadWidth neg lineto + SolidHead {closepath true upath fill ustroke}{stroke} ifelse + } if + % Draw back head + Heads 2 eq { + newpath + HeadHeight HeadWidth moveto + 0 0 lineto + HeadHeight HeadWidth neg lineto + SolidHead {closepath true upath fill ustroke}{stroke} ifelse + } if +grestore end} def +/lineplot +{20 dict begin /Params exch def + Params /Yx known + { + /Xt {} def + /Yt Params /Yx {} GetParam def + /Tb Params /Xb 0 GetParam def + /Te Params /Xe 1 GetParam def + } + { + /Xt Params /Xt {} GetParam def + /Yt Params /Yt {} GetParam def + /Tb Params /Tb 0 GetParam def + /Te Params /Te 1 GetParam def + } ifelse + /Samples Params /Samples 1000 GetParam def + /Xscale Params /Xscale 1 GetParam def + /Yscale Params /Yscale 1 GetParam def + /FromCurX Params /FromCurX true GetParam def + /FromCurY Params /FromCurY true GetParam def + + % Move initial point. + FromCurX {0}{Tb Xt Xscale mul} ifelse + FromCurY {0}{Tb Yt Yscale mul} ifelse + rmoveto + % Current point. + currentpoint /y0 exch def /x0 exch def + /X0 Tb Xt def + /Y0 Tb Yt def + % Step calculation. + /Step Te Tb sub Samples div def + % Main cycle + 1 1 Samples { /i exch def + /T i Step mul Tb add def + T Xt X0 sub Xscale mul x0 add + T Yt Y0 sub Yscale mul y0 add + lineto + } for +end} def +/HatchArea +{ 20 dict begin /Params exch def + /LX Params /LX 0 GetParam def + /BY Params /BY 0 GetParam def + /RX Params /RX 100 GetParam def + /TY Params /TY 100 GetParam def + /LineWidth Params /LineWidth 0.5 GetParam def + /Angle Params /Angle 45 GetParam def + /Dist Params /Dist 2 GetParam def + + gsave + LineWidth setlinewidth + newpath + /AngleAbs Angle abs def + /AngleSgn Angle 0 eq {1}{Angle AngleAbs div} ifelse def + AngleAbs 45 gt + { + /yc TY BY add 0.5 mul def + /yd TY BY sub 0.5 mul def + /Dx 2 yd mul AngleAbs cos mul AngleAbs sin div def + /RDist Dist AngleAbs sin div def + % Init value + Dx RDist div ceiling neg + % Increment + 1 + % End value + RX LX sub RDist div ceiling + {RDist mul LX add yc yd AngleSgn mul sub moveto + Dx 2 yd mul AngleSgn mul rlineto + } for + } + { + /xc RX LX add 0.5 mul def + /xd RX LX sub 0.5 mul def + /Dy 2 xd mul AngleAbs sin mul AngleAbs cos div def + /RDist Dist AngleAbs cos div def + % Init value + Dy RDist div ceiling neg + % Increment + 1 + % End value + TY BY sub RDist div ceiling + {RDist mul BY add xc xd AngleSgn mul sub exch moveto + 2 xd mul AngleSgn mul Dy rlineto + } for + } + ifelse + stroke + grestore end} def + /HatchArrow +{ 20 dict begin /Params exch def + /X0 Params /X0 0 GetParam def + /Y0 Params /Y0 0 GetParam def + Params /X1 known + { + /X1 Params /X1 100 GetParam def + /Y1 Params /Y1 100 GetParam def + /Length X1 X0 sub dup mul Y1 Y0 sub dup mul add sqrt def + /Angle Y1 Y0 sub X1 X0 sub atan def + } + { + /Length Params /Length 100 GetParam def + /Angle Params /Angle 45 GetParam def + } ifelse + /HeadHeight Params /HeadHeight 7 GetParam def + /HeadWidth Params /HeadWidth 5 GetParam def + /ArrowWidth Params /ArrowWidth 5 GetParam def + /Heads Params /Heads 1 GetParam def + /HatchWidth Params /HatchWidth 0.5 GetParam def + /HatchAngle Params /HatchAngle 45 GetParam def + /HatchDist Params /HatchDist 2 GetParam def + + gsave + % Setting line parameters + 1 setlinejoin 1 setlinecap + Params /LineWidth known {Params /LineWidth get setlinewidth} if + % Transform coordinates + X0 Y0 translate + Angle rotate + newpath + % Draw front head + /xb 0 def + /xe Length def + Heads 0 eq not { + /xe Length HeadHeight sub def + Length HeadHeight sub HeadWidth moveto + Length 0 lineto + Length HeadHeight sub HeadWidth neg lineto + Length HeadHeight sub HeadWidth lineto + } if + % Draw back head + Heads 2 eq { + /xb HeadHeight def + HeadHeight HeadWidth moveto + 0 0 lineto + HeadHeight HeadWidth neg lineto + HeadHeight HeadWidth lineto + } if + stroke + % Draw main line + newpath + xb ArrowWidth moveto + xe ArrowWidth lineto + xe ArrowWidth neg lineto + xb ArrowWidth neg lineto + xb ArrowWidth lineto + stroke + % Hatching + HatchWidth setlinewidth + gsave + newpath + xb ArrowWidth moveto + xe ArrowWidth lineto + xe ArrowWidth neg lineto + xb ArrowWidth neg lineto + xb ArrowWidth lineto + clip + <> HatchArea + grestore + Heads 0 eq not { + newpath + Length HeadHeight sub HeadWidth moveto + Length 0 lineto + Length HeadHeight sub HeadWidth neg lineto + Length HeadHeight sub HeadWidth lineto + Heads 2 eq { + HeadHeight HeadWidth moveto + 0 0 lineto + HeadHeight HeadWidth neg lineto + HeadHeight HeadWidth lineto + } if + clip + <> HatchArea + <> HatchArea + } if +grestore end} def + + + +% First +XYArrows +<> Arrow +1 u phi1 sin mul 1 u phi1 cos mul moveto +[ + [/SFBI1200 fs 0 0 (v) true] + [/SFBI1200 subfs -0.01 fs mul subsh (2) false] +] <> Show +<> Arrow +1 u phi1 cos mul 1 u phi1 sin neg mul moveto +[ + [/SFBI1200 fs 3 3 (v) true] + [/SFBI1200 subfs -0.01 fs mul subsh (1) false] +] <> Show +newpath +0 0 moveto +circlinewidth setlinewidth +<> lineplot +closepath stroke +newpath +0 0 moveto +phirlinewidth setlinewidth +<> lineplot +0 0 moveto +<> lineplot +stroke +phir 0.3 u add dup phi1 0.5 mul sin mul exch phi1 0.5 mul cos mul moveto +[ + [/PazoMath-Italic fs -3 0 (j) true] + [/SFBI1200 subfs -0.1 fs mul subsh (1) false] +] <> Show + +% Translation to second +<> HatchArrow +XYArrowlen HArrhd add HArrw 0.5 mul add u 0 moveto +[ + [/SFTI1200 fs 1.5 mul -5 fs (V) true] +] <> Show + +% Translation to fourth +<> HatchArrow +0 XYArrowlen HArrhd add HArrw 0.5 mul add neg u moveto +[ + [/SFTI1200 fs 1.5 mul 3 5 (U) true] + [/PazoMath-Italic fs 1.5 mul 1 0 (S) true] + [/SFTI1200 fs 1.5 mul -4 0 (V) true] +] <> Show + +XYArrowlen HArrhd 2 mul add HArrw add XYArrowlen add u 0 translate + +% Second +XYArrows +newpath +0 0 moveto +circlinewidth setlinewidth +<> lineplot +closepath stroke +<> Arrow +1 u 0 moveto +[ + [/SFBI1200 fs 3 3 (i) true] +] <> Show +<> Arrow +0 1 u moveto +[ + [/SFBI1200 fs 3 6 (j) true] +] <> Show + +% Translation to third +<> HatchArrow +0 XYArrowlen HArrhd add HArrw 0.5 mul add neg u moveto +[ + [/PazoMath-Italic fs 1.5 mul 7 5 (S) true] +] <> Show + +0 XYArrowlen HArrhd 2 mul add HArrw add XYArrowlen add neg u translate + +% Third +XYArrows +newpath +0 0 moveto +circlinewidth setlinewidth +<> lineplot +closepath stroke +<> Arrow +sigma1 u 0 moveto +[ + [/PazoMath-Italic fs 1 8 (s) true] + [/SFBI1200 subfs -0.2 fs mul subsh (1) true] + [/SFBI1200 fs 1 0 (i) true] +] <> Show +<> Arrow +0 sigma2 u moveto +[ + [/PazoMath-Italic fs 3 6 (s) true] + [/SFBI1200 subfs -0.2 fs mul subsh (2) true] + [/SFBI1200 fs 1 0 (j) true] +] <> Show + +% Translation to forth +<> HatchArrow +XYArrowlen HArrhd add HArrw 0.5 mul add neg u 0 moveto +[ + [/SFTI1200 fs 1.5 mul 3 fs (U) true] +] <> Show + +XYArrowlen HArrhd 2 mul add HArrw add XYArrowlen add neg u 0 translate + +% Forth +XYArrows +<> Arrow +sigma1 u phi2 cos mul sigma1 u phi2 sin mul moveto +[ + [/SFBI1200 fs 0 0 (u) true] + [/SFBI1200 subfs -0.01 fs mul subsh (1) false] +] <> Show +<> Arrow +sigma2 u phi1 sin neg mul sigma2 u phi2 cos mul moveto +[ + [/SFBI1200 fs -5 5 (u) true] + [/SFBI1200 subfs -0.01 fs mul subsh (2) true] +] <> Show +newpath +0 0 moveto +circlinewidth setlinewidth +<> lineplot +closepath stroke +newpath +0 0 moveto +phirlinewidth setlinewidth +<> lineplot +0 0 moveto +<> lineplot +stroke +phir 0.3 u add dup phi2 0.5 mul cos mul exch phi2 0.5 mul sin mul moveto +[ + [/PazoMath-Italic fs -5 2 (j) true] + [/SFBI1200 subfs -0.1 fs mul subsh (2) false] +] <> Show + +grestore +end +showpage diff --git a/Evolution_matrix/lyap.tex b/Evolution_matrix/lyap.tex new file mode 100644 index 0000000..0602511 --- /dev/null +++ b/Evolution_matrix/lyap.tex @@ -0,0 +1,476 @@ +\documentclass[12pt]{article} +\usepackage{graphicx} +\usepackage{color} +\usepackage[utf8]{inputenc} +\usepackage[T2A]{fontenc} +\usepackage[russian]{babel} +\usepackage{amsmath} +\usepackage{amssymb} + +\topmargin=-1.8cm +\oddsidemargin=-15mm +\evensidemargin=-15mm +\textheight=24.5cm +\textwidth=19cm +\tolerance=1000 + +\parskip=5pt plus 4pt minus 2pt +\tolerance=9000 +% +\begin{document} +\section{Сингулярное разложение матрицы} +Сингулярное разложение (singular value decomposition)~--- представление произвольной матрицы $m\times n$ в виде +% +\begin{equation} +M=U\Sigma V, +\label{SVD} +\end{equation} +% +где $U$ и $V$~--- унитарные матрицы $m\times m$ и $n\times n$, соответственно, $\Sigma$~--- диагональная матрица $m\times n$. +Числа, стоящие на диагонали $\Sigma$, называются сингулярными числами матрицы $M$. Единичные вектора $u$ и $v$ такие, что $Mv=\sigma u$ и $M^*u=\sigma v$, +$\sigma$~--- сингулярное число $M$, называются, соответственно, левыми и правыми сингулярными векторами матрицы $M$. +Если матрица $M$ является вещественной, то её сингулярные числа также вещественны, а $U$ и $V$~--- ортогональные матрицы. Так как матрица $\Sigma$ и, соответственно, сингулярное разложение определены с точностью до перестановки сингулярных чисел, то можно потребовать упорядочения сингулярных чисел на диагонали $\Sigma$ по невозрастанию. Такое разложение является единственным. + +Если матрица $M$ является квадратной, то сингулярное разложение имеет очень простой геометрический смысл. Действие любой матрицы на вектор можно представить в виде трёх последовательных операций: поворот/отражение матрицей $V$, растяжение/сжатие по координатным осям матрицей $\Sigma$, второй поворот/отражение матрицей $U$. Таким образом, матрица $M$ переводит сферу единичного радиуса в эллипсоид с полуосями, равными сингулярным числам и направленными вдоль левых сингулярных векторов. Правые сингулярные вектора~--- это, соответственно, прообразы полуосей эллипсоида. + +Рассмотрим сингулярное разложение вещёственной матрицы $2\times 2$: +% +\begin{equation} +M=U\Sigma V \Rightarrow +\begin{pmatrix} +a&b\\c&d +\end{pmatrix} += +\begin{pmatrix} +\cos\phi_2&-\sin\phi_2\\ +\sin\phi_2&\cos\phi_2 +\end{pmatrix} +\begin{pmatrix} +\sigma_1&0\\ +0&\sigma_2 +\end{pmatrix} +\begin{pmatrix} +\cos\phi_1&-\sin\phi_1\\ +\sin\phi_1&\cos\phi_1 +\end{pmatrix}. +\label{SVD2x2} +\end{equation} +% +Последовательное действие этих матриц на единичную окружность и сингулярные векторы показано на рис.~\ref{SVD_geom}. +В данном разложении не используются матрицы отражения, что приводит к возможности сингулярным числам быть отрицательными. Однако, из общих соображений понятно, что матрица эволюции непрерывного потока не может содержать отражений. Для матрицы \eqref{sol_linsys} это можно показать непосредственно. +% +\begin{figure}[!htb] +\centerline{\includegraphics[width=0.8\textwidth]{SVD.eps}} +\caption{Геометрический смысл сингулярного разложения матрицы.} +\label{SVD_geom} +\end{figure} +% + +Перемножая матрицы, получаем систему из четырёх уравнений с четырьмя неизвестными +% +\begin{equation} +\begin{pmatrix} +a&b\\c&d +\end{pmatrix} += +\begin{pmatrix} +\sigma_1\cos\phi_1\cos\phi_2-\sigma_2\sin\phi_1\sin\phi_2& +-\sigma_1\sin\phi_1\cos\phi_2-\sigma_2\cos\phi_1\sin\phi_2\\ +\sigma_1\cos\phi_1\sin\phi_2+\sigma_2\sin\phi_1\cos\phi_2& +-\sigma_1\sin\phi_1\sin\phi_2+\sigma_2\cos\phi_1\cos\phi_2 +\end{pmatrix}. +\label{SVD2x2eq} +\end{equation} +% +Введём обозначения +% +\begin{equation} +\begin{aligned} +\alpha&=a+d, &\beta&=a-d, &\gamma&=c+b, &\delta&=c-b,\\ +\xi&=\sigma_1+\sigma_2, &\eta&=\sigma_1-\sigma_2, &\Phi&=\phi_1+\phi_2, &\Psi&=\phi_2-\phi_1. +\end{aligned} +\label{SVD2x2def} +\end{equation} +% +Вычитая и складывая уравнения \eqref{SVD2x2eq} и используя введённые обозначения \eqref{SVD2x2def}, получаем +% +\begin{equation} +\alpha=\xi\cos\Phi,\quad \beta=\eta\cos\Psi,\quad \gamma=\eta\sin\Psi,\quad \delta=\xi\sin\Phi. +\label{SVD2x2fineq} +\end{equation} +% +Решение системы \eqref{SVD2x2fineq} имеет вид +% +\begin{equation} +\xi=\sqrt{\alpha^2+\delta^2},\quad \eta=\sqrt{\beta^2+\gamma^2},\quad +\Phi=\operatorname{arctg2}{(\delta,\alpha)}, \quad \Psi=\operatorname{arctg2}{(\gamma,\beta)}. +\label{SVD2x2sol} +\end{equation} +% +Здесь $\operatorname{arctg2}{(y,x)}$~--- это угол между вектором $(x,y)$ и осью $x$. Он может быть определён как +% +\begin{equation} +\operatorname{arctg2}{(y,x)}=\left\{ +\begin{aligned} +&\arctg{(y/x)}, &x\ge 0,\\ +&\arctg{(y/x)}+\pi, &x<0. +\end{aligned}\right. +\label{arctg2} +\end{equation} +% +Окончательный вид решения +% +\begin{equation} +\begin{gathered} +\sigma_1=\frac{\sqrt{(a+d)^2+(c-b)^2}+\sqrt{(a-d)^2+(b+c)^2}}{2},\quad +\sigma_2=\frac{\sqrt{(a+d)^2+(c-b)^2}-\sqrt{(a-d)^2+(b+c)^2}}{2},\\ +\phi_1=\frac{\operatorname{arctg2}{(c-b,\,a+d)}-\operatorname{arctg2}{(c+b,\,a-d)}}{2},\quad +\phi_2=\frac{\operatorname{arctg2}{(c-b,\,a+d)}+\operatorname{arctg2}{(c+b,\,a-d})}{2}. +\end{gathered} +\label{SVD2x2finsol} +\end{equation} +% +Из решения очевидно, что сингулярные числа упорядочены по невозрастанию, то есть $\sigma_1\ge\sigma_2$. +Произведение $\sigma_1\sigma_2$ определяет отношение конечной и начальной площадей и равно, естественно, $\operatorname{Det}M$. +Из определения сингулярного разложения следует, что +% +\begin{equation} +\sigma_1\geqslant\frac{\Vert M\mathbf{x}\Vert}{\Vert\mathbf{x}\Vert}\geqslant\sigma_2, +\label{SVD_mainprop} +\end{equation} +% +где $\Vert\cdot\Vert$ обозначает евклидову норму. То есть, {\em длина произвольного вектора $\mathbf{x}$ изменяется матрицей $M$ минимум в $\sigma_2$ раз и максимум в $\sigma_1$ раз}. + +\section{Связь собственных и сингулярных чисел} +Устойчивость или неустойчивость движения определяется не сингулярными числами, а собственными. Для двухмерной матрицы \eqref{SVD2x2} собственные числа можно выразить через сингулярные. Запишем характеристическое уравнение +% +\begin{equation} +\operatorname{Det}{(M-\lambda)}=0 \Longrightarrow \lambda^2-(a+d)\lambda+ad-bc=0. +\label{ch_eq} +\end{equation} +% +Подставив значения $a$, $b$, $c$ и $d$ из \eqref{SVD2x2eq} +% +\begin{displaymath} +\begin{gathered} +a+d=\sigma_1(\cos\phi_1\cos\phi_2-\sin\phi_1\sin\phi_2)+\sigma_2(\cos\phi_1\cos\phi_2-\sin\phi_1\sin\phi_2)=(\sigma_1+\sigma_2)\cos{(\phi_1+\phi_2)},\\ +ad-bc=\operatorname{Det}M=\sigma_1\sigma_2, +\end{gathered} +\end{displaymath} +% +получаем +% +\begin{equation} +\lambda^2-(\sigma_1+\sigma_2)\cos{(\phi_1+\phi_2)}\lambda+\sigma_1\sigma_2=0. +\label{ch_eq_sigma} +\end{equation} +% +Детерминант характеристического уравнения: +% +\begin{equation} +D=(\sigma_1+\sigma_2)^2\cos^2{(\phi_1+\phi_2)}-4\sigma_1\sigma_2. +\label{det_ch_eq} +\end{equation} +% +Знак детерминанта определяет, являются ли собственные числа комплексными (комплексно-сопряжённая пара) или вещественными. Для любых сингулярных чисел, если сумма углов $\phi_1$ и $\phi_2$ равна $\pi/2+n\pi$, $n$~--- произвольное целое, детерминант будет отрицательным. И наоборот, если сумма углов кратна $\pi$, детерминант неотрицателен для любых сингулярных чисел. +Сами собственные числа равны +% +\begin{equation} +\lambda_{1,2}=\frac{(\sigma_1+\sigma_2)\cos{(\phi_1+\phi_2)}\pm\sqrt{D}}{2}. +\label{lambda_ch_eq} +\end{equation} +% +Произведение собственных чисел равно произведению сингулярных и равно детерминанту матрицы. + +\section{Инвариантный эллипс} +В случае, если действие матрицы $M$ сохраняет площадь ($\operatorname{Det} M=1$), существуют инвариантные кривые второго порядка, которые под действием матрицы переходят сами в себя. В этой главе мы будем искать инвариантный эллипс. + +Общее уравнение эллипса с центром в начале координат имеет вид: +% +\begin{equation} +\begin{gathered} +Ax^2+By^2+Cxy=1,\\ +A=\frac{\cos^2\theta}{a^2}+\frac{\sin^2\theta}{b^2},\qquad +B=\frac{\sin^2\theta}{a^2}+\frac{\cos^2\theta}{b^2},\\ +C=\frac{b^2-a^2}{a^2b^2}\sin2\theta, +\end{gathered} +\label{ellbase} +\end{equation} +% +где $a$ и $b$~--- полуоси эллипса, а $\theta$~--- угол между осью абсцисс и полуосью $a$. Так как матрица $M$~--- линейный оператор, достаточно найти один инвариантный эллипс, остальные можно получить путём масштабирования. Будем искать эллипс с площадью, равной площади единичного круга и большой полуосью $k\ge 1$. В этом случае параметры в уравнении \eqref{ellbase} имеют вид +% +\begin{equation} +\begin{gathered} +A=\frac{\cos^2\theta}{k^2}+k^2\sin^2\theta,\qquad +B=\frac{\sin^2\theta}{k^2}+k^2\cos^2\theta,\\ +C=-\frac{k^4-1}{k^2}\sin2\theta. +\end{gathered} +\label{ellpars0} +\end{equation} +% +Угол $\theta$ определяет направление большой полуоси. Сумма коэффициентов $A+B=(k^4+1)/k^2$ не зависит от ориентации эллипса и определяется только его формой. + +Матрица $M=U\Sigma V$ последовательно поворачивает ($V$), растягивает-сжимает ($\Sigma$) и снова поворачивает ($U$) эллипс. При этом большая полуось может изменится только под действием растяжения-сжатия. Рассмотрим действие матрицы $\Sigma$. Так как, $\operatorname{Det}\Sigma=1$, +% +\begin{displaymath} +\Sigma= +\begin{pmatrix} +\sigma & 0\\ +0& 1/\sigma +\end{pmatrix},\quad \sigma\ge 1. +\end{displaymath} +% +Действие $\Sigma$ преобразует координаты $x$ и $y$ +% +\begin{gather} +x'=\sigma x,\qquad y'=y/\sigma,\\ +x=x'/\sigma,\qquad y=\sigma y'. +\end{gather} +% +Подставив $x$ и $y$ в уравнение \eqref{ellbase} с учётом \eqref{ellpars0} получаем уравнение эллипса в новых координатах $x'$ и $y'$ с новыми параметрами +% +\begin{equation} +\begin{gathered} +A'=\frac{\cos^2\theta'}{k'^2}+k'^2\sin^2\theta'=\frac{\cos^2\theta}{\sigma^2k^2}+\frac{k^2}{\sigma^2}\sin^2\theta,\qquad +B'=\frac{\sin^2\theta'}{k'^2}+k'^2\cos^2\theta'=\frac{\sigma^2}{k^2}\sin^2\theta+\sigma^2k^2\cos^2\theta,\\ +C'=-\frac{k'^4-1}{k'^2}\sin2\theta'=-\frac{k^4-1}{k^2}\sin2\theta. +\end{gathered} +\label{ellpars1} +\end{equation} +% +По условию инвариантности $k'=k$. Тогда, из выражения для $C'$ следует, что $\sin2\theta'=\sin2\theta$. Это уравнение имеет два решения, $\theta'=\theta$ и $\theta'=\pi/2-\theta$. Первое решение не имеет смысла, так как эллипс переходит сам в себя, что возможно только при $s=1$. Таким образом, остаётся только второе решение. Изменение угла ориентации не зависит от $k$, поэтому мы теперь можем найти наклон инвариантного эллипса $\phi_0$ +% +\begin{equation} +\phi_0\xrightarrow{V}\theta=\phi_0+\phi_1\xrightarrow{\Sigma}\theta'=\frac{\pi}{2}-(\phi_0+\phi_1)\xrightarrow{U}\frac{\pi}{2}-(\phi_0+\phi_1)+\phi_2=\left\{ +\begin{aligned} +&\phi_0\\ +&\phi_0+\pi +\end{aligned} +\right., +\label{phi0transform} +\end{equation} +% +откуда получаем +% +\begin{equation} +\phi_0=\frac{\phi_2-\phi_1}{2}\pm\frac{\pi}{4}. +\label{phi0} +\end{equation} +% +К выбору знака мы вернёмся ниже. + +Вернёмся к поиску $k$. Так как $A+B$ является инвариантом относительно поворота, $k$ найдём из уравнения $A+B=A'+B'$ +% +\begin{equation} +\frac{k^4+1}{k^2}=\frac{\sigma^4k^4+1}{\sigma^2k^2}\cos^2{\theta}+\frac{k^4+\sigma^4}{\sigma^2k^2}\sin^2\theta. +\label{inveq} +\end{equation} +% +После тривиальных преобразований получаем +% +\begin{equation} +\begin{gathered} +\sigma^2k^4+\sigma^2=(\sigma^4k^4+1)\cos^2\theta+(k^4+\sigma^4)(1-\cos^2\theta),\\ +k^4(\sigma^2-1-(\sigma^4-1)\cos^2\theta)=\sigma^2(\sigma^2-1)-(\sigma^4-1)\cos^2\theta,\\ +k^4(1-(\sigma^2+1)\cos^2\theta)=\sigma^2-(\sigma^2+1)\cos^2\theta. +\end{gathered} +\label{inveq1} +\end{equation} +% +% +\begin{equation} +k^4=\frac{\sigma^2-(\sigma^2+1)\cos^2\theta}{1-(\sigma^2+1)\cos^2\theta}=1+\frac{\sigma^2-1}{1-(\sigma^2+1)\cos^2\theta}=1+\frac{2(\sigma^2-1)}{1-\sigma^2-(\sigma^2+1)\cos2\theta}. +\label{inveq2} +\end{equation} +% +Так как $k\ge 1$, $1-\sigma^2-(\sigma^2+1)\cos2\theta> 0$, или +% +\begin{equation} +-\cos2\theta>\frac{\sigma^2-1}{\sigma^2+1}, +\label{thetacrit} +\end{equation} +% +из чего следует, что $\cos2\theta<0$. Это условие позволяет определить знак в формуле \eqref{phi0} +% +\begin{equation} +\cos2\theta=\cos2(\phi_0+\phi_1)=\cos2\left(\frac{\phi_2-\phi_1}{2}\pm\frac{\pi}{4}+\phi_1\right)=\cos\left(\frac{\pi}{2}\pm(\phi_1+\phi_2)\right)=\mp\sin(\phi_1+\phi_2), +\end{equation} +% +Можно показать, что при возведении в квадрат условие \eqref{thetacrit} эквивалентно условию $D<0$, где $D$ определяется формулой \eqref{det_ch_eq}. Таким образом, параметры инвариантного эллипса опредяляются через сингулярное разложение как +% +\begin{equation} +\phi_0=\frac{\phi_2-\phi_1}{2}-\xi\frac{\pi}{4},\quad k=\sqrt[4]{1+\frac{2(\sigma^2-1)}{1-\sigma^2+\xi(\sigma^2+1)\sin(\phi_1+\phi_2)}},\quad \xi=\operatorname{sgn}\sin(\phi_1+\phi_2). +\end{equation} +% + +\section{Показатель Ляпунова} +Пусть дана система нелинейных уравнений +% +\begin{equation} +\mathbf{\dot x}=\mathbf{f}(\mathbf{x},t),\quad +\mathbf{x}=(x_1,\dotsc,x_n),\quad +\mathbf{f}(\mathbf{x},t)=(f_1(x_1,\dotsc,x_n,t),\dots,f_n(x_1,\dotsc,x_n,t)). +\label{nonlinsys} +\end{equation} +% +Показатель Ляпунова в точке $\mathbf{x_0}$ определяется как +% +\begin{equation} +\Lambda(\mathbf{x_0})=\lim_{t\to\infty}\lim_{\Vert\delta \mathbf{x}(0)\Vert\to 0}\frac{\ln(\Vert\delta \mathbf{x}(t)\Vert/\Vert\delta \mathbf{x}(0)\Vert)}{t}, +\label{lyap_def} +\end{equation} +% +где $\delta \mathbf{x}(t)=\mathbf{x_1}(t)-\mathbf{x_0}(t)$, $\mathbf{x_0}(t)$ и $\mathbf{x_1}(t)$~--- решения системы \eqref{nonlinsys}, $\mathbf{x_0}(0)=\mathbf{x_0}$. Данный предел существует и одинаков почти для любого выбора $\delta \mathbf{x}(0)$. Геометрический смысл показателя Ляпунова состоит в том, что две близлежащих траектории расходятся в пространстве в среднем экспоненциально по времени с показателем экспоненты, равным показателю Ляпунова. + +В силу малости $\Vert\delta \mathbf{x}(0)\Vert$ и, соответственно, $\Vert\delta \mathbf{x}(t)\Vert$ можно линеаризовать +систему \eqref{nonlinsys} в окрестности траектории $\mathbf{x_0}(t)$, получив систему нестационарных линейных уравнений +% +\begin{equation} +\begin{pmatrix} +\delta\dot x_1\\ +\hdotsfor{1}\\ +\delta\dot x_n +\end{pmatrix}=J(t) +\begin{pmatrix} +\delta x_1\\ +\hdotsfor{1}\\ +\delta x_n +\end{pmatrix}, +\label{linearization} +\end{equation} +% +где $J(t)$~--- якобиан системы \eqref{nonlinsys} вдоль траектории $\mathbf{x_0}(t)$ +% +\begin{equation} +J(t)= +\begin{pmatrix} +\dfrac{\partial f_1(\mathbf{x_0}(t),t)}{\partial x_1}&\dots&\dfrac{\partial f_1(\mathbf{x_0}(t),t)}{\partial x_n}\\ +\hdotsfor{3}\\ +\dfrac{\partial f_n(\mathbf{x_0}(t),t)}{\partial x_1}&\dots&\dfrac{\partial f_n(\mathbf{x_0}(t),t)}{\partial x_n} +\end{pmatrix}. +\label{jacobian} +\end{equation} +% +Решение системы \eqref{linearization} можно представить в виде +% +\begin{equation} +\begin{pmatrix} +\delta x_1(t)\\ +\hdotsfor{1}\\ +\delta x_n(t) +\end{pmatrix}=G(t) +\begin{pmatrix} +\delta x_1(0)\\ +\hdotsfor{1}\\ +\delta x_n(0) +\end{pmatrix}, +\label{evol_mat} +\end{equation} +% +где $G(t)$~--- матрица эволюции. Матрица эволюции подчиняется дифференциальному уравнению, которое получается подстановкой \eqref{evol_mat} в \eqref{linearization} +% +\begin{equation} +\dot G=JG, +\label{evol_mat_diffur} +\end{equation} +% +с начальным условием $G(0)=I$, где $I$~--- единичная матрица. + +Максимальное значение $\lim\limits_{\Vert\delta \mathbf{x}(0)\Vert\to 0}\dfrac{\Vert\delta \mathbf{x}(t)\Vert}{\Vert\delta \mathbf{x}(0)\Vert}$ для системы \eqref{linearization} будет равно $\sigma_1(G(t))$~--- максимальному сингулярному числу матрицы $G(t)$. Если $\lim\limits_{t\to\infty}\dfrac{\sigma_2(G(t))}{\sigma_1(G(t))}=0$, где $\sigma_2(G(t))$~--- следующее по величине сингулярное число матрицы $G(t)$, то +определение \eqref{lyap_def} может быть переписано как +% +\begin{equation} +\Lambda=\lim_{t\to\infty}\frac{\ln\sigma_1(G(t))}{t}=\lim_{t\to\infty}\Lambda(t). +\label{lyap_def_sigma} +\end{equation} +% +Величину $\Lambda(t)$ будем называть {\em показателем Ляпунова за время $t$}. Это есть {\em отношение логарифма максимально возможного растяжения вектора ко времени}. Также определим {\em мгновенный показатель Ляпунова} $\Lambda_0$ как показатель Ляпунова системы стационарных линейных уравнений. +% +\begin{equation} +\begin{pmatrix} +\delta\dot x_1\\ +\hdotsfor{1}\\ +\delta\dot x_n +\end{pmatrix}=J(0) +\begin{pmatrix} +\delta x_1\\ +\hdotsfor{1}\\ +\delta x_n +\end{pmatrix}. +\label{linearization_stat} +\end{equation} +% +Мгновенный показатель Ляпунова определяет скорость экспоненциального разбегания траекторий в данной точке пространства в данный момент времени. + +\section{Решение системы стационарных линейных уравнений} +Пусть задана система линейных уравнений с матрицей скоростей $J$, не зависящей от времени +% +\begin{equation} +\begin{pmatrix} +\dot x\\ +\dot y +\end{pmatrix}=J +\begin{pmatrix} +x\\y +\end{pmatrix},\quad +J= +\begin{pmatrix} +a&b\\ +c&d +\end{pmatrix}. +\label{linsys} +\end{equation} +% +Тогда, матрица эволюции $G$ системы \eqref{linsys} имеет вид +% +\begin{equation} +G(t)=\left\{ +\begin{aligned} +&A_1e^{\lambda_1t}+A_2e^{\lambda_2t}, &D>0,\\ +&e^{\lambda t}\left(I+\frac{K}{2}t\right), &D=0,\\ +&e^{\lambda t}\left(I\cos{(\omega t)}+\frac{K}{\sqrt{-D}}\sin{(\omega t)}\right), &D<0, +\end{aligned}\right. +\label{sol_linsys} +\end{equation} +% +где +% +\begin{equation} +\begin{gathered} +I=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad +K=\begin{pmatrix}a-d&2b\\2c&d-a\end{pmatrix},\quad +A_1=\frac{\sqrt{D}I+K}{2\sqrt{D}},\quad +A_2=\frac{\sqrt{D}I-K}{2\sqrt{D}},\\ +D=4bc+(a-d)^2,\quad \lambda=\frac{a+d}{2},\quad \omega=\frac{\sqrt{-D}}{2},\quad +\quad \lambda_1=\frac{a+d+\sqrt{D}}{2},\quad \lambda_2=\frac{a+d-\sqrt{D}}{2}. +\end{gathered} +\label{Amatrix} +\end{equation} +% +Собственные числа матрицы эволюции: $e^{\lambda_1t}$ и $e^{\lambda_2t}$. +Поток \eqref{linsys} в общем случае не сохраняет площадь. Отношение площади в момент времени $t$ к площади в начальный момент времени определяется детерминантом матрицы эволюции $\operatorname{Det}{G(t)}$ +% +\begin{equation} +\frac{S(t)}{S(0)}=\operatorname{Det}{G(t)}=e^{(a+d)t}=e^{\operatorname{Tr}Jt}. +\label{square} +\end{equation} +% +Таким образом, поток \eqref{linsys} сохраняет площадь, тогда и только тогда, когда след матрицы скоростей равен нулю. Это равносильно условию равенства нулю суммы собственных чисел матрицы скоростей. + +Точное выражение для максимального сингулярного числа $\sigma_1$ матрицы эволюции \eqref{sol_linsys} (и для показателя Ляпунова на конечное время, соответственно) весьма громоздко, поэтому приведём только его +асимптотику при $t\to\infty$: +% +\begin{equation} +\sigma_1(t)= +\left\{ +\begin{aligned} +&\sqrt{\frac{(b+c)^2+(a-d)^2}{4bc+(a-d)^2}}e^{\lambda_1t},&D>0,\\ +&|b-c|te^{\lambda t},&D=0,\\ +&F(t)e^{\lambda t},&D<0, +\end{aligned} +\right. +\label{sigma_of_evol} +\end{equation} +% +где $F(t)$~--- ограниченная периодическая функция с периодом $w/2$. Воспользовавшись определением \eqref{lyap_def_sigma}, нетрудно определить показатель Ляпунова системы \eqref{linsys}. +\end{document}