{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%%html\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Üldine $N$-mõõtmeline lineaarne süsteem" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Käesolev tööleht näitab kuidas määrata üldise $N$-mõõtmelise lineaarse süsteemi püsipunkti stabiilsust, ja kuidas lahendada liikumisvõrrandeid, et leida süsteemi trajektoore." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Omaväärtused ja püsipunkti stabiilsus" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Üldisel lineaarsel süsteemil $\\underline{\\dot{x}} = \\underline{\\underline{M}} \\cdot \\underline{x}$ on alati püsipunkt $\\underline{x} = (0, \\ldots, 0)$. Püsipunkti stabiilsus sõltub maatriksi $\\underline{\\underline{M}}$ **omaväärtustest**, täpsemalt nende reaalosadest. Lineaarse süsteemi püsipunkti stabiilsuse määramisel kehtivad sellised reeglid:\n", "\n", "* Kui kõikide omaväärtuste reaalosad on negatiivsed, siis püsipunkt on **atraktor**.\n", "* Kui kõikide omaväärtuste reaalosad on positiivsed, siis püsipunkt on **repeller**.\n", "* Kui on nii omaväärtused, mille reaalosad on positiivsed, kui omaväärtused, mille reaalosad on negatiivsed, siis püsipunkt on **sadul**.\n", "\n", "Kui kõikide omaväärtuste reaalosad on nullist erinevad, siis nimetatakse püsipunkti **hüperboolseks**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Üldine lahend" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Lineaarse dünaamilise süsteemi lahendeid saab leida **maatriksi eksponendi** abil. Kui algtingimus on $\\underline{x}(0) = \\underline{x}_0$, siis on algtingimusele vastav lahend\n", "\n", "$$\\underline{x}(t) = e^{\\underline{\\underline{M}}t} \\cdot \\underline{x}_0$$\n", "\n", "Maatriksi eksponent on defineeritud Taylori reana:\n", "\n", "$$e^{\\underline{\\underline{M}}t} = \\sum_{k = 0}^{\\infty}\\frac{(\\underline{\\underline{M}}t)^k}{k!}$$\n", "\n", "Eksponendi saab kõige paremini arvutada baasil, milles maatriks $\\underline{\\underline{M}}$ on **Jordani normaalkujul**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Jordani plokid ja lineaarse süsteemi normaalkuju" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Igat maatriksit $\\underline{\\underline{M}}$ võib viia **Jordani normaalkujju** lineaarse teisenduse abil, see tähendab et on olemas (mitteunikaalne) pööratav maatriks $\\underline{\\underline{P}}$ nii, et kehtib $\\underline{\\underline{M}} = \\underline{\\underline{P}} \\cdot \\underline{\\underline{J}} \\cdot \\underline{\\underline{P}}^{-1}$, ja $\\underline{\\underline{J}}$ on kujul\n", "\n", "$$\\underline{\\underline{J}} = \\begin{pmatrix}\n", "\\lambda_1 & 1 & & & & & & & & & & & \\\\\n", "& \\ddots & \\ddots & & & & & & & & & & \\\\\n", "& & \\ddots & 1 & & & & & & & & & \\\\\n", "& & & \\lambda_1 & & & & & & & & & \\\\\n", "& & & & \\lambda_2 & 1 & & & & & & & \\\\\n", "& & & & & \\ddots & \\ddots & & & & & & \\\\\n", "& & & & & & \\ddots & 1 & & & & & \\\\\n", "& & & & & & & \\lambda_2 & & & & & \\\\\n", "& & & & & & & & \\ddots & & & & & \\\\\n", "& & & & & & & & & \\lambda_k & 1 & & \\\\\n", "& & & & & & & & & & \\ddots & \\ddots & \\\\\n", "& & & & & & & & & & & \\ddots & 1 \\\\\n", "& & & & & & & & & & & & \\lambda_k \\\\\n", "\\end{pmatrix}.$$\n", "\n", "Siin $\\lambda_1, \\ldots, \\lambda_k$ on maatriksi $\\underline{\\underline{M}}$ **omaväärtused**. Jordani normaalkuju koosneb plokidest\n", "\n", "$$\\underline{\\underline{J}}_i = \\begin{pmatrix}\n", "\\lambda_i & 1 & & \\\\\n", "& \\ddots & \\ddots & \\\\\n", "& & \\ddots & 1 \\\\\n", "& & & \\lambda_i\n", "\\end{pmatrix},$$\n", "\n", "mille suurus vastab omaväärtuse $\\lambda_i$ kordsusele. Jordani maatriksi eksponent on samuti plokikujuline, ja iga plokk on vastava Jordani ploki eksponent. Selle eksponendi leidmiseks kirjutame Jordani ploki kujul $\\underline{\\underline{J}}_i = \\underline{\\underline{D}}_i + \\underline{\\underline{N}}_i$, kus\n", "\n", "$$\\underline{\\underline{D}}_i = \\begin{pmatrix}\n", "\\lambda_i & & \\\\\n", "& \\ddots & \\\\\n", "& & \\lambda_i\n", "\\end{pmatrix}, \\quad \\underline{\\underline{N}}_i = \\begin{pmatrix}\n", "0 & 1 & & \\\\\n", "& \\ddots & \\ddots & \\\\\n", "& & \\ddots & 1 \\\\\n", "& & & 0\n", "\\end{pmatrix}.$$\n", "\n", "Kuna need maatriksid kommuteeruvad, $\\underline{\\underline{D}}_i\\underline{\\underline{N}}_i = \\underline{\\underline{N}}_i\\underline{\\underline{D}}_i$, kehtib valem\n", "\n", "$$e^{(\\underline{\\underline{D}}_i + \\underline{\\underline{N}}_i)t} = e^{\\underline{\\underline{D}}_it}e^{\\underline{\\underline{N}}_it}.$$\n", "\n", "Nende maatriksi eksponendid on antud valemitega\n", "\n", "$$e^{\\underline{\\underline{D}}_it} = \\begin{pmatrix}\n", "e^{\\lambda_it} & & \\\\\n", "& \\ddots & \\\\\n", "& & e^{\\lambda_it}\n", "\\end{pmatrix}, \\quad e^{\\underline{\\underline{N}}_it} = \\begin{pmatrix}\n", "1 & t & \\cdots & \\frac{t^{p_i - 1}}{(p_i - 1)!}\\\\\n", "& \\ddots & \\ddots & \\vdots \\\\\n", "& & \\ddots & t \\\\\n", "& & & 1\n", "\\end{pmatrix},$$\n", "\n", "kus $p_i$ on Jordani ploki $\\underline{\\underline{J}}_i$ suurus. Sellest järeldub, et Jordani maatriksi eksponendi kohta kehtib valem\n", "\n", "$$e^{\\underline{\\underline{J}}t} = \\begin{pmatrix}\n", "e^{\\lambda_1t} & e^{\\lambda_1t}t & \\cdots & \\frac{e^{\\lambda_1t}t^{p_1 - 1}}{(p_1 - 1)!} & & & & & & & & & \\\\\n", "& \\ddots & \\ddots & \\vdots & & & & & & & & & \\\\\n", "& & \\ddots & e^{\\lambda_1t}t & & & & & & & & & \\\\\n", "& & & e^{\\lambda_1t} & & & & & & & & & \\\\\n", "& & & & e^{\\lambda_2} & e^{\\lambda_2}t & \\cdots & \\frac{e^{\\lambda_2t}t^{p_2 - 1}}{(p_2 - 1)!} & & & & & \\\\\n", "& & & & & \\ddots & \\ddots & \\vdots & & & & & \\\\\n", "& & & & & & \\ddots & e^{\\lambda_2}t & & & & & \\\\\n", "& & & & & & & e^{\\lambda_2} & & & & & \\\\\n", "& & & & & & & & \\ddots & & & & & \\\\\n", "& & & & & & & & & e^{\\lambda_k} & e^{\\lambda_k}t & \\cdots & \\frac{e^{\\lambda_kt}t^{p_k - 1}}{(p_k - 1)!} \\\\\n", "& & & & & & & & & & \\ddots & \\ddots & \\vdots \\\\\n", "& & & & & & & & & & & \\ddots & e^{\\lambda_k}t \\\\\n", "& & & & & & & & & & & & e^{\\lambda_k} \\\\\n", "\\end{pmatrix}.$$\n", "\n", "Lõpuks leiame, et maatriksi $\\underline{\\underline{M}}$ on\n", "\n", "$$\\begin{split}\n", "e^{\\underline{\\underline{M}}t} &= \\sum_{j = 0}^{\\infty}\\frac{(\\underline{\\underline{M}}t)^j}{j!}\\\\\n", "&= \\sum_{j = 0}^{\\infty}\\frac{(\\underline{\\underline{P}} \\cdot \\underline{\\underline{J}} \\cdot \\underline{\\underline{P}}^{-1}t)^j}{j!}\\\\\n", "&= \\sum_{j = 0}^{\\infty}\\underline{\\underline{P}} \\cdot \\frac{(\\underline{\\underline{J}}t)^j}{j!} \\cdot \\underline{\\underline{P}}^{-1}\\\\\n", "&= \\underline{\\underline{P}} \\cdot \\left(\\sum_{j = 0}^{\\infty}\\frac{(\\underline{\\underline{J}}t)^j}{j!}\\right) \\cdot \\underline{\\underline{P}}^{-1}\\\\\n", "&=\\underline{\\underline{P}} \\cdot e^{\\underline{\\underline{J}}t} \\cdot \\underline{\\underline{P}}^{-1}.\n", "\\end{split}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Interaktiivne näide: maatriksi eksponent" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Järgmisena rakendame oma teadmisi praktikale. Alljärgnev kood näitab, kuidas saab maatriksi eksponendi leida SymPy abil, ja sellega ka lineaarse süsteemi trajektoore leida. Kõigepealt on meil selleks SymPy paketti vaja." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import sympy as sp\n", "from sympy.interactive.printing import init_printing\n", "init_printing(use_unicode=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Siis defineerime sümboleid: aega tähistame $t$-ga, ja algtingimus on $(x_{10}, x_{20}, x_{30})$." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "t = sp.symbols('t', real=True)\n", "x0 = sp.Matrix(3, 1, sp.symbols('x_10 x_20 x_30', real=True))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Näide #1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Esimene näide on süsteem, mida kirjeldab dünaamika\n", "\n", "$$\\begin{pmatrix}\n", "\\dot{x}_1\\\\\n", "\\dot{x}_2\\\\\n", "\\dot{x}_3\n", "\\end{pmatrix} = \\begin{pmatrix}\n", "1 & -\\frac{2}{3} & 0\\\\\n", "2 & 3 & 1\\\\\n", "0 & -\\frac{2}{3} & 1\\\\\n", "\\end{pmatrix} \\cdot \\begin{pmatrix}\n", "x_1\\\\\n", "x_2\\\\\n", "x_3\n", "\\end{pmatrix}.$$\n", "\n", "Maatriks $\\underline{\\underline{M}}$ on seega selline:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAABNCAMAAAB0SHFsAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n", "text/latex": [ "$$\\left[\\begin{matrix}1 & - \\frac{2}{3} & 0\\\\2 & 3 & 1\\\\0 & - \\frac{2}{3} & 1\\end{matrix}\\right]$$" ], "text/plain": [ "⎡1 -2/3 0⎤\n", "⎢ ⎥\n", "⎢2 3 1⎥\n", "⎢ ⎥\n", "⎣0 -2/3 1⎦" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = sp.Matrix([[1, -sp.Rational(2, 3), 0], [2, 3, 1], [0, -sp.Rational(2, 3), 1]])\n", "M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vaatame, millised on maatriksi omaväärtused." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAAAVBAMAAAC08kb3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nT6B+nxvWJvc/8WQL+As39pQippZRLwAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left \\{ 1 : 1, \\quad 2 - i : 1, \\quad 2 + i : 1\\right \\}$$" ], "text/plain": [ "{1: 1, 2 - ⅈ: 1, 2 + ⅈ: 1}" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ev = M.eigenvals()\n", "ev" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Maatriksil on üks reaalne omaväärtus $\\lambda_1 = 1$ ja kaaskompleksne paar $\\lambda_{2,3} = 2 \\pm i$. Reaalosad on positiivsed, seega on püsipunkt $(0, 0, 0)$ **repeller**. Iga omaväärtuse kordsus on 1, seega kaasneb iga omaväärtusega üks omavektor." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVcAAABNCAMAAAAl1ddjAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ \\left[\\begin{matrix}- \\frac{1}{2}\\\\0\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\- \\frac{3}{2} + \\frac{3 i}{2}\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\- \\frac{3}{2} - \\frac{3 i}{2}\\\\1\\end{matrix}\\right]\\right ]$$" ], "text/plain": [ "⎡⎡-1/2⎤, ⎡ 1 ⎤, ⎡ 1 ⎤⎤\n", "⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥\n", "⎢⎢ 0 ⎥ ⎢ 3 3⋅ⅈ⎥ ⎢ 3 3⋅ⅈ⎥⎥\n", "⎢⎢ ⎥ ⎢- ─ + ───⎥ ⎢- ─ - ───⎥⎥\n", "⎢⎣ 1 ⎦ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥⎥\n", "⎢ ⎢ ⎥ ⎢ ⎥⎥\n", "⎣ ⎣ 1 ⎦ ⎣ 1 ⎦⎦" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "[(M - y * sp.eye(3)).nullspace()[0].applyfunc(sp.nsimplify) for y in ev]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Omavektorid moodustavad teisendusmaatriksi $\\underline{\\underline{P}}$." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOoAAABNCAMAAACfUwgqAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}- \\frac{1}{2} & 1 & 1\\\\0 & - \\frac{3}{2} + \\frac{3 i}{2} & - \\frac{3}{2} - \\frac{3 i}{2}\\\\1 & 1 & 1\\end{matrix}\\right]$$" ], "text/plain": [ "⎡-1/2 1 1 ⎤\n", "⎢ ⎥\n", "⎢ 3 3⋅ⅈ 3 3⋅ⅈ⎥\n", "⎢ 0 - ─ + ─── - ─ - ───⎥\n", "⎢ 2 2 2 2 ⎥\n", "⎢ ⎥\n", "⎣ 1 1 1 ⎦" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P, jc = M.jordan_cells()\n", "P = P.applyfunc(sp.nsimplify)\n", "P" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Selle abil leiame ka Jordani normaalkuju $\\underline{\\underline{J}}$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJoAAABLCAMAAAC/Q26vAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\noNv31M9iSaPwmiDzl+G+h1+msC5nJT61jgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 2 - i & 0\\\\0 & 0 & 2 + i\\end{matrix}\\right]$$" ], "text/plain": [ "⎡1 0 0 ⎤\n", "⎢ ⎥\n", "⎢0 2 - ⅈ 0 ⎥\n", "⎢ ⎥\n", "⎣0 0 2 + ⅈ⎦" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "J = (P.inv() * M * P).applyfunc(sp.nsimplify)\n", "J" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nüüd on ka lihtne leida selle maatriksi eksponenti $e^{\\underline{\\underline{J}}t}$." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAALwAAABMCAMAAADgBS2AAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n7jX5HzA9WgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left[\\begin{matrix}e^{t} & 0 & 0\\\\0 & e^{t \\left(2 - i\\right)} & 0\\\\0 & 0 & e^{t \\left(2 + i\\right)}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ t ⎤\n", "⎢ℯ 0 0 ⎥\n", "⎢ ⎥\n", "⎢ t⋅(2 - ⅈ) ⎥\n", "⎢0 ℯ 0 ⎥\n", "⎢ ⎥\n", "⎢ t⋅(2 + ⅈ)⎥\n", "⎣0 0 ℯ ⎦" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eJ = (t * J).exp()\n", "eJ" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Lõpuks kasutame lineaarset teisendust, et leida maatriksi eksponenti $e^{\\underline{\\underline{M}}t}$." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\left[\\begin{matrix}- \\frac{2}{3} e^{2 t} \\sin{\\left (t \\right )} + \\frac{2}{3} e^{2 t} \\cos{\\left (t \\right )} + \\frac{e^{t}}{3} & - \\frac{2}{3} e^{2 t} \\sin{\\left (t \\right )} & - \\frac{e^{2 t}}{3} \\sin{\\left (t \\right )} + \\frac{e^{2 t}}{3} \\cos{\\left (t \\right )} - \\frac{e^{t}}{3}\\\\2 e^{2 t} \\sin{\\left (t \\right )} & e^{2 t} \\sin{\\left (t \\right )} + e^{2 t} \\cos{\\left (t \\right )} & e^{2 t} \\sin{\\left (t \\right )}\\\\- \\frac{2}{3} e^{2 t} \\sin{\\left (t \\right )} + \\frac{2}{3} e^{2 t} \\cos{\\left (t \\right )} - \\frac{2 e^{t}}{3} & - \\frac{2}{3} e^{2 t} \\sin{\\left (t \\right )} & - \\frac{e^{2 t}}{3} \\sin{\\left (t \\right )} + \\frac{e^{2 t}}{3} \\cos{\\left (t \\right )} + \\frac{2 e^{t}}{3}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ 2⋅t 2⋅t t 2⋅t 2⋅t \n", "⎢ 2⋅ℯ ⋅sin(t) 2⋅ℯ ⋅cos(t) ℯ -2⋅ℯ ⋅sin(t) ℯ ⋅si\n", "⎢ - ───────────── + ───────────── + ── ─────────────── - ───────\n", "⎢ 3 3 3 3 3 \n", "⎢ \n", "⎢ 2⋅t 2⋅t 2⋅t \n", "⎢ 2⋅ℯ ⋅sin(t) ℯ ⋅sin(t) + ℯ ⋅cos(t) \n", "⎢ \n", "⎢ 2⋅t 2⋅t t 2⋅t 2⋅t \n", "⎢ 2⋅ℯ ⋅sin(t) 2⋅ℯ ⋅cos(t) 2⋅ℯ -2⋅ℯ ⋅sin(t) ℯ ⋅sin\n", "⎢- ───────────── + ───────────── - ──── ─────────────── - ────────\n", "⎣ 3 3 3 3 3 \n", "\n", " 2⋅t t ⎤\n", "n(t) ℯ ⋅cos(t) ℯ ⎥\n", "──── + ─────────── - ── ⎥\n", " 3 3 ⎥\n", " ⎥\n", " 2⋅t ⎥\n", " ℯ ⋅sin(t) ⎥\n", " ⎥\n", " 2⋅t t⎥\n", "(t) ℯ ⋅cos(t) 2⋅ℯ ⎥\n", "─── + ─────────── + ────⎥\n", " 3 3 ⎦" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eM = (P * eJ * P.inv()).applyfunc(lambda y: sp.expand(sp.expand_trig(sp.simplify(sp.simplify(y).as_real_imag()[0]))))\n", "eM" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Algtingimusele $(x_{10}, x_{20}, x_{30})$ vastav lahend on siis selline." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Y6cZrX8ATlQwNsg023US7Il6bRUB7A0YCgBuTvDLmga2XWHSyPLwyTZtiKhxeSdTFhLxp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\nGcb/HpaUsP9/iv8HcpsXNSrjHmYAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left[\\begin{matrix}\\frac{e^{t}}{3} \\left(2 \\sqrt{2} x_{10} e^{t} \\cos{\\left (t + \\frac{\\pi}{4} \\right )} + x_{10} - 2 x_{20} e^{t} \\sin{\\left (t \\right )} + \\sqrt{2} x_{30} e^{t} \\cos{\\left (t + \\frac{\\pi}{4} \\right )} - x_{30}\\right)\\\\\\left(2 x_{10} \\sin{\\left (t \\right )} + \\sqrt{2} x_{20} \\sin{\\left (t + \\frac{\\pi}{4} \\right )} + x_{30} \\sin{\\left (t \\right )}\\right) e^{2 t}\\\\\\frac{e^{t}}{3} \\left(- 2 x_{10} \\left(- \\sqrt{2} e^{t} \\cos{\\left (t + \\frac{\\pi}{4} \\right )} + 1\\right) - 2 x_{20} e^{t} \\sin{\\left (t \\right )} + x_{30} \\left(\\sqrt{2} e^{t} \\cos{\\left (t + \\frac{\\pi}{4} \\right )} + 2\\right)\\right)\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ ⎛ ___ t ⎛ π⎞ t ___ t ⎛ π\n", "⎢ ⎜2⋅╲╱ 2 ⋅x₁₀⋅ℯ ⋅cos⎜t + ─⎟ + x₁₀ - 2⋅x₂₀⋅ℯ ⋅sin(t) + ╲╱ 2 ⋅x₃₀⋅ℯ ⋅cos⎜t + ─\n", "⎢ ⎝ ⎝ 4⎠ ⎝ 4\n", "⎢ ───────────────────────────────────────────────────────────────────────────\n", "⎢ 3 \n", "⎢ \n", "⎢ ⎛ ___ ⎛ π⎞ ⎞ 2⋅t \n", "⎢ ⎜2⋅x₁₀⋅sin(t) + ╲╱ 2 ⋅x₂₀⋅sin⎜t + ─⎟ + x₃₀⋅sin(t)⎟⋅ℯ \n", "⎢ ⎝ ⎝ 4⎠ ⎠ \n", "⎢ \n", "⎢⎛ ⎛ ___ t ⎛ π⎞ ⎞ t ⎛ ___ t ⎛ \n", "⎢⎜- 2⋅x₁₀⋅⎜- ╲╱ 2 ⋅ℯ ⋅cos⎜t + ─⎟ + 1⎟ - 2⋅x₂₀⋅ℯ ⋅sin(t) + x₃₀⋅⎜╲╱ 2 ⋅ℯ ⋅cos⎜t \n", "⎢⎝ ⎝ ⎝ 4⎠ ⎠ ⎝ ⎝ \n", "⎢─────────────────────────────────────────────────────────────────────────────\n", "⎣ 3 \n", "\n", "⎞ ⎞ t ⎤\n", "⎟ - x₃₀⎟⋅ℯ ⎥\n", "⎠ ⎠ ⎥\n", "─────────── ⎥\n", " ⎥\n", " ⎥\n", " ⎥\n", " ⎥\n", " ⎥\n", " ⎥\n", " π⎞ ⎞⎞ t⎥\n", "+ ─⎟ + 2⎟⎟⋅ℯ ⎥\n", " 4⎠ ⎠⎠ ⎥\n", "─────────────⎥\n", " ⎦" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = (eM * x0).applyfunc(sp.simplify)\n", "x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Igaks juhuks kontrollime, kas dünaamiline võrrand an rahuldatud." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIlmzd0i77ts7uXj/QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAMtJREFUOBHtVcsS\nwiAMDOVhBVqK+f9/lRClZETPztBcSndZCLAQUFhjgRY7IwAKtSlhGwOR/hckSp3o2YqCcjrE8O4n\nqa0M6vbESkHFg8CwDahck7ToKidUWCmPnGtPOdTU3aP5UCUMhCn+QK9KrBpRPwYETsMO0oCcaa44\nSp6XrEdLhp026nhtYp9hgUM5i/H20lQtpKrB1Jiemsait/XOJy8t+li7u3xZVDxff3JTprFoqynS\nouKNuixaLfq1ZDsq0Mb407u1ZBsDT+aoGdQmGjgBAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$$" ], "text/plain": [ "⎡0⎤\n", "⎢ ⎥\n", "⎢0⎥\n", "⎢ ⎥\n", "⎣0⎦" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(x.applyfunc(lambda y: sp.diff(y, t)) - M * x).applyfunc(sp.simplify)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Näide #2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Teine näide on süsteem, mida kirjeldab dünaamika\n", "\n", "$$\\begin{pmatrix}\n", "\\dot{x}_1\\\\\n", "\\dot{x}_2\\\\\n", "\\dot{x}_3\n", "\\end{pmatrix} = \\begin{pmatrix}\n", "1 & 2 & 0\\\\\n", "1 & 1 & 2\\\\\n", "0 & -1 & 1\\\\\n", "\\end{pmatrix} \\cdot \\begin{pmatrix}\n", "x_1\\\\\n", "x_2\\\\\n", "x_3\n", "\\end{pmatrix}.$$\n", "\n", "Maatriks $\\underline{\\underline{M}}$ on seega selline:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAABLCAMAAABEOFk1AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nkntzLCX7BXmHJVkYbEbnAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}1 & 2 & 0\\\\1 & 1 & 2\\\\0 & -1 & 1\\end{matrix}\\right]$$" ], "text/plain": [ "⎡1 2 0⎤\n", "⎢ ⎥\n", "⎢1 1 2⎥\n", "⎢ ⎥\n", "⎣0 -1 1⎦" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = sp.Matrix([[1, 2, 0], [1, 1, 2], [0, -1, 1]])\n", "M" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vaatame, millised on maatriksi omaväärtused." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAVBAMAAADhmjDFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQ3Ynvq81UMrtE\ndiLw+n06AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVQoFWNgYGBUZMACFl0GCTK7YpECCh1b\nACQYP6BL8oVGT2BgYGoAirMmgCSnGYBICHjEwPkRKO4A5IEkmYvqkSTvT2D4AtUE0XkfSfLYAeYf\nOCUZIMYCTYTaCdXpDhQAApMAqDh7A4gLkWTOALEZnjoBLQH7YtoCEB/ZTqCuHKBQEwPD6iyQHJok\nQxpQK7svNCRQJN8xMOyfwMDQDtSDbCfYEIb/BiBJsJ3gkEDRmc3A4P8A4gvkQIC4VpCB8y/UK2BJ\n9fx2AZCZYH/yKGkfQJaEWIZCgjVBjEURB3PA4pjxCVHI1ACiWyEcdPLYAZAIJ/Y0FMzAAAADVjx4\nITqjfAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left \\{ 1 : 3\\right \\}$$" ], "text/plain": [ "{1: 3}" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ev = M.eigenvals()\n", "ev" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Maatriksil on üks reaalne omaväärtus $\\lambda_1 = 1$, mille kordsus on 3. Reaalosa on positiivne, seega on püsipunkt $(0, 0, 0)$ **repeller**. Sellega kaasneb üks omavektor." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAABLCAMAAAD6bgFoAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left [ \\left[\\begin{matrix}-2\\\\0\\\\1\\end{matrix}\\right]\\right ]$$" ], "text/plain": [ "⎡⎡-2⎤⎤\n", "⎢⎢ ⎥⎥\n", "⎢⎢0 ⎥⎥\n", "⎢⎢ ⎥⎥\n", "⎣⎣1 ⎦⎦" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "[(M - y * sp.eye(3)).nullspace()[0].applyfunc(sp.nsimplify) for y in ev]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Omavektor ja üldistatud omavektorid moodustavad teisendusmaatriksi $\\underline{\\underline{P}}$." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAABNCAMAAACmGHs0AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nXyAMQ/cfUlcu4Bt4SJUAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left[\\begin{matrix}5 & 0 & \\frac{1}{2}\\\\0 & \\frac{5}{2} & 0\\\\- \\frac{5}{2} & 0 & 1\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ 5 0 1/2⎤\n", "⎢ ⎥\n", "⎢ 0 5/2 0 ⎥\n", "⎢ ⎥\n", "⎣-5/2 0 1 ⎦" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P, jc = M.jordan_cells()\n", "P = P.applyfunc(sp.nsimplify)\n", "P" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Selle abil leiame ka Jordani normaalkuju $\\underline{\\underline{J}}$." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n9F/BH/rAMvgPIsZcNUrrGcsHFmPoF+h8JUzMd2cWAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}1 & 1 & 0\\\\0 & 1 & 1\\\\0 & 0 & 1\\end{matrix}\\right]$$" ], "text/plain": [ "⎡1 1 0⎤\n", "⎢ ⎥\n", "⎢0 1 1⎥\n", "⎢ ⎥\n", "⎣0 0 1⎦" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "J = (P.inv() * M * P).applyfunc(sp.nsimplify)\n", "J" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nüüd on ka lihtne leida selle maatriksi eksponenti $e^{\\underline{\\underline{J}}t}$." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAABOCAMAAAA5KyV5AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n", "text/latex": [ "$$\\left[\\begin{matrix}e^{t} & t e^{t} & \\frac{t^{2} e^{t}}{2}\\\\0 & e^{t} & t e^{t}\\\\0 & 0 & e^{t}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ 2 t⎤\n", "⎢ t t t ⋅ℯ ⎥\n", "⎢ℯ t⋅ℯ ─────⎥\n", "⎢ 2 ⎥\n", "⎢ ⎥\n", "⎢ t t ⎥\n", "⎢0 ℯ t⋅ℯ ⎥\n", "⎢ ⎥\n", "⎢ t ⎥\n", "⎣0 0 ℯ ⎦" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eJ = (t * J).exp()\n", "eJ" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Lõpuks kasutame lineaarset teisendust, et leida maatriksi eksponenti $e^{\\underline{\\underline{M}}t}$." ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAASoAAABQCAMAAACZKsgOAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left[\\begin{matrix}\\left(t^{2} + 1\\right) e^{t} & 2 t e^{t} & 2 t^{2} e^{t}\\\\t e^{t} & e^{t} & 2 t e^{t}\\\\- \\frac{t^{2} e^{t}}{2} & - t e^{t} & \\left(- t^{2} + 1\\right) e^{t}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡⎛ 2 ⎞ t t 2 t ⎤\n", "⎢⎝t + 1⎠⋅ℯ 2⋅t⋅ℯ 2⋅t ⋅ℯ ⎥\n", "⎢ ⎥\n", "⎢ t t t ⎥\n", "⎢ t⋅ℯ ℯ 2⋅t⋅ℯ ⎥\n", "⎢ ⎥\n", "⎢ 2 t ⎥\n", "⎢ -t ⋅ℯ t ⎛ 2 ⎞ t⎥\n", "⎢ ─────── -t⋅ℯ ⎝- t + 1⎠⋅ℯ ⎥\n", "⎣ 2 ⎦" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eM = (P * eJ * P.inv()).applyfunc(sp.simplify)\n", "eM" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Algtingimusele $(x_{10}, x_{20}, x_{30})$ vastav lahend on siis selline." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\left[\\begin{matrix}2 t^{2} x_{30} e^{t} + 2 t x_{20} e^{t} + x_{10} \\left(t^{2} + 1\\right) e^{t}\\\\t x_{10} e^{t} + 2 t x_{30} e^{t} + x_{20} e^{t}\\\\- \\frac{t^{2} x_{10}}{2} e^{t} - t x_{20} e^{t} + x_{30} \\left(- t^{2} + 1\\right) e^{t}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ 2 t t ⎛ 2 ⎞ t⎤\n", "⎢2⋅t ⋅x₃₀⋅ℯ + 2⋅t⋅x₂₀⋅ℯ + x₁₀⋅⎝t + 1⎠⋅ℯ ⎥\n", "⎢ ⎥\n", "⎢ t t t ⎥\n", "⎢ t⋅x₁₀⋅ℯ + 2⋅t⋅x₃₀⋅ℯ + x₂₀⋅ℯ ⎥\n", "⎢ ⎥\n", "⎢ 2 t ⎥\n", "⎢ t ⋅x₁₀⋅ℯ t ⎛ 2 ⎞ t⎥\n", "⎢- ───────── - t⋅x₂₀⋅ℯ + x₃₀⋅⎝- t + 1⎠⋅ℯ ⎥\n", "⎣ 2 ⎦" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = eM * x0\n", "x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Igaks juhuks kontrollime, kas dünaamiline võrrand an rahuldatud." ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIlmzd0i77ts7uXj/QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAMtJREFUOBHtVcsS\nwiAMDOVhBVqK+f9/lRClZETPztBcSndZCLAQUFhjgRY7IwAKtSlhGwOR/hckSp3o2YqCcjrE8O4n\nqa0M6vbESkHFg8CwDahck7ToKidUWCmPnGtPOdTU3aP5UCUMhCn+QK9KrBpRPwYETsMO0oCcaa44\nSp6XrEdLhp026nhtYp9hgUM5i/H20lQtpKrB1Jiemsait/XOJy8t+li7u3xZVDxff3JTprFoqynS\nouKNuixaLfq1ZDsq0Mb407u1ZBsDT+aoGdQmGjgBAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$$" ], "text/plain": [ "⎡0⎤\n", "⎢ ⎥\n", "⎢0⎥\n", "⎢ ⎥\n", "⎣0⎦" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(x.applyfunc(lambda y: sp.diff(y, t)) - M * x).applyfunc(sp.simplify)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Küsimused" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Hüperboolne püsipunkt\n", "\n", "Millised on Jacobi maatriksi omaväärtuste reaalosad hüperboolse püsipunkti juures?\n", "\n", "* \n", "* \n", "* \n", "* \n", "* " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Polünoomid maatriksi eksponendis\n", "\n", "Millisel juhul sisaldab maatriksi eksponent $e^{\\underline{\\underline{M}}t}$ polünoome ($t, t^2, \\ldots$)?\n", "\n", "* \n", "* \n", "* " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Püsipunkti tüüp\n", "\n", "Milline on järgneva süsteemi püsipunkt $\\underline{x}_* = (0, 0, 0)$?\n", "\n", "$$\\begin{pmatrix}\n", "\\dot{x}_1\\\\\n", "\\dot{x}_2\\\\\n", "\\dot{x}_3\n", "\\end{pmatrix} = \\begin{pmatrix}\n", "-5 & -4 & 9 \\\\\n", "3 & 2 & -3 \\\\\n", "-4 & -4 & 8\n", "\\end{pmatrix} \\cdot \\begin{pmatrix}\n", "x_1\\\\\n", "x_2\\\\\n", "x_3\n", "\\end{pmatrix}$$\n", "\n", "* \n", "* \n", "* " ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.2" } }, "nbformat": 4, "nbformat_minor": 1 }