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""
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""
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"source": [
"%%html\n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Mõisted pidevate dünaamiliste süsteemide kohta"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Faasiruum"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Dünaamiliste süsteemide kõige algsem mõiste on **faasiruum**. Faasiruum on hulk, mis koosneb kõikidest süsteemil võimalikutest olekutest. Iga faasiruumi punkt vastab identselt süsteemi olekule, ja on kirjeldatav sõltumatute muutujate kaudu. Sõltumatute muutujate arv on faasiruumi **mõõde**.\n",
"\n",
"Näide: Ühemõõtmeline lineaarne pendel. Kui pendel liigub ainult ühes suunas, siis kirjeldab tema positsiooni üks reaalarv $x$. Aga sellest ei piisa, et pendli olekut kirjeldada, sest pendlil on lisaks positsioonile ka kiirus $v$. Positsioon $x$ ja kiirus $v$ on sõltumatud muutujad - pendlil võib olla igasugune kiirus, sõltumatu tema asukohast. Kiirendus aga ei ole neist sõltumatu, vaid järeldub pendli liikumisvõrrandist. Ühemõõtmelise pendli faasiruum on $\\mathbb{R}^2$, faasiruumi mõõde on 2.\n",
"\n",
"Antud näide näitab, et faasiruumi mõõde tavaliselt ei ole liikumissuundade arvuga võrdne. Tihti on faasiruum lihtsalt eukleidiline ruum $\\mathbb{R}^n$, või alamhulk eukleidilisest ruumist, aga üldjuhul võib ka keerulisem olla. Näiteks, kui dünaamiline süsteem on jäik füüsikaline pendel, mis pöörleb ühes suunas ümber punkti, siis on tema positsioonide ruum hoopis ring $S^1$. Kuna kiirus on siiski reaalarv, on tema faasiruum $S^1 \\times \\mathbb{R}$. (Diferentsiaalgeomeetria abil võib ka õigemini öelda, et faasiruum on puutujate kihtkond $TS^1$ - mis on antud juhul võrdne ruum.)\n",
"\n",
"Siit edaspidi tähistame faasiruumi tähega $Q$, ja kirjutame faasiruumi elemente kas vektori kujul või indeksitega, $\\underline{x} = (x_1, \\ldots, x_n)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Trajektoorid"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Kui dünaamiline süsteem areneb ajas, siis vastab igale ajahetkele $t$ olek $\\underline{x}(t) \\in Q$. Funktsiooni $\\underline{x}: T \\to Q$, kus $T$ on lubatud ajade hulk, nimetatakse **trajektooriks**. Kui dünaamiline süsteem on **pidev**, siis on $T$ pidev ruum, ehk kas $\\mathbb{R}$ või lahtine vahemik $(a, b)$.\n",
"\n",
"Näide: Kui ühemõõtmeline lineaarne pendel võngub, siis on tal igal ajahetkel $t \\in T = \\mathbb{R}$ positsioon $x(t)$ ja kiirus $v(t)$. Võnkumise trajektoor faasiruumis on funktsioon $\\underline{x}: T \\to Q, t \\mapsto \\underline{x}(t) = (x(t), v(t))$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Dünaamika"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lisaks faasiruumile koosneb dünaamiline süsteem ka **dünaamikast**. Dünaamika määrab, kuidas dünaamiline süsteem käitub, ja millised trajektoorid on lubatud. Pideva dünaamilise süsteemi dünaamikat kirjeldab diferentsiaalvõrrand. Dünaamilisi süsteeme jagatakse kahesse tüübisse:\n",
"\n",
"* **Autonoomse** dünaamilise süsteemi dünaamikat kirjeldab diferentsiaalvõrrand $\\underline{\\dot{x}} = f(\\underline{x})$, kus funktsioon $f$ sõltub ainult süsteemi olekust $\\underline{x}$.\n",
"* **Mitteautonoomse** dünaamilise süsteemi dünaamikat kirjeldab diferentsiaalvõrrand $\\underline{\\dot{x}} = f(\\underline{x}, t)$, kus funktsioon $f$ sõltub lisaks süsteemi olekule $\\underline{x}$ ka ajast $t$.\n",
"\n",
"Alati on võimalik ümberkirjutada mitteautonoomset süsteemi, mille faasiruum on $Q$, hoopis autonoomse süsteemina, mille faasiruum on $Q' = Q \\times T$ ja mille dünaamika on $(\\underline{\\dot{x}}, \\dot{t}) = (f(\\underline{x}, t), 1)$.\n",
"\n",
"Antud definitsioon väidab, et dünaamilist süsteemi kirjeldab esimest järku diferentsiaalvõrrand, kuigi näiteks pendli liikumist kirjeldab teist järku diferentsiaalvõrrand $\\ddot{x} = a(x, \\dot{x})$. Siin aga on juba näha et selle võrrandi muutuja on vaid positsioon $x$, kuigi faasiruum koosneb lisaks sellele ka kiirusest $v$. Dünaamilise süsteemi kujul on pendli liikumisvõrrand hoopis $(\\dot{x}, \\dot{v}) = (v, a(x, v))$, mis on esimest järku diferentsiaalvõrrand. Selline ümberkirjutamine on alati võimalik, kui on antud kõrgemat järku diferentsiaalvõrrand."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Algtingimused"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lisaks dünaamikale sõltub dünaamilise süsteemi käitumine ka tema olekust $\\underline{x}_0 = \\underline{x}(t_0)$ valitud ajahetkel $t_0$. Sellist olekut nimetatakse **algtingimuseks**. Dünaamika ja algtingimus koos määravad unikaalset trajektoori, mis on süsteemi liikumisvõrrandi lahendus."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Püsipunktid"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Punkti $\\underline{x}_* \\in Q$ faasiruumis nimetatakse **püsipunktiks**, kui antud punktis kehtib $\\underline{\\dot{x}} = f(\\underline{x}) = 0$ (autonoomse süsteemi kohta) või $\\underline{\\dot{x}} = f(\\underline{x}, t) = 0$ (mitteautonoomse süsteemi kohta).\n",
"\n",
"Näide: Ühemõõtmelist lineaarset pendli kirjeldab diferentsiaalvõrrandite süsteem:\n",
"\n",
"* $\\dot{x} = v$\n",
"* $\\dot{v} = -(hv + kx)/m$\n",
"\n",
"Siin $m$ on pendli mass, $k$ on vedru parameeter ja $h$ on hõõrdetegur. Kui me eeldame et need parameetrid on positiivsed, on süsteemil ainult üks püsipunkt $x_* = 0, v_* = 0$.\n",
"\n",
"Kui dünaamilisel süsteemil on püsipunkt $\\underline{x}_*$, siis $\\underline{x}(t) = \\underline{x}_*$ on unikaalne trajektoor mis vastab algtingimusele $\\underline{x}_0 = \\underline{x}_*$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Vool ja faasiportree"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Siit edaspidi eeldame, et dünaamiline süsteem on autonoomne, kuna igat süsteemi võib kirjutada autonoomsel kujul $\\underline{\\dot{x}} = f(\\underline{x})$. Antud liikusmisvõrrand määrab iga faasiruumi punkti $\\underline{x}$ kohta vektorit $\\underline{\\dot{x}}$, mis on trajektoori puutujavektor selles punktis. Kogu dünaamika on sellepärast kirjeldatav vektoriväljana faasiruumis. See vektoriväli on dünaamilise süsteemi **vool**. Joonistus, mis näitab dünaamilise süsteemi voolu, on **faasiportree**.\n",
"\n",
"Näide: Ühemõõtmelise lineaarse pendli\n",
"\n",
"* $\\dot{x} = v$\n",
"* $\\dot{v} = -(hv + kx)/m$\n",
"\n",
"faasiportree on selline:\n",
" \n",
"\n",
"\n",
"Faasiportree võimaldab leida süsteemi trajektoore graafilisel meetodil: Selleks on vaja vektoreid ühendada niimoodi, et nad on trajektooride puutujavektorid. Pendli kohta on tulemus selline:\n",
"\n",
"\n",
"\n",
"Faasiportree aitab ka leida süsteemi püsipunkte. Pendli faasiportree näitab, et ainus püsipunkt on $(x, v) = (0, 0)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Interaktiivne näide: ühemõõtmeline pendel"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Pendli simulatsioonis kasutame SciPy integreerimiseks, NumPy numbrilisteks arvutusteks ja PyPlot joonistamiseks."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"\n",
"from scipy.integrate import odeint\n",
"import matplotlib\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Järgmisena defineerime pendli dünaamikat. See on funktsioon, mis sõltub faasiruumipunktist $y = (x, v)$, ajast $t$, hõõrdetegurist $h$, vedru parameetrist $k$ ja massist $m$:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def pendel(y, t, h, k, m):\n",
" x, v = y\n",
" dydt = [v, -(h * v + k * np.sin(x)) / m]\n",
" return dydt"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Konstantsed pendli parameetrid on järgnevalt."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"h = 0.5\n",
"k = 0.5\n",
"m = 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Alustame faasiportreest. Selleks on meil võrku vaja, mille peale nooli joonistada. Võrku defineeritakse NumPy funktioonidega linspace ja meshgrid."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"xmin = -10\n",
"xmax = 10\n",
"nx = 21\n",
"X = np.linspace(xmin, xmax, nx)\n",
"\n",
"vmin = -10\n",
"vmax = 10\n",
"nv = 21\n",
"V = np.linspace(vmin, vmax, nv)\n",
"\n",
"XX, VV = np.meshgrid(X, V)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Järgmisena arvutame $(\\dot{x}, \\dot{v})$ igas võrgupunktis."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"dxdt = np.zeros((nx, nv))\n",
"dvdt = np.zeros((nx, nv))\n",
"\n",
"for i in range(nx):\n",
" for j in range(nv):\n",
" dxdt[i, j], dvdt[i, j] = pendel([XX[i, j], VV[i, j]], 0, h, k, m)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lõpuks joonistame faasiportreet quiver funktsiooniga."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
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RokVMS0tTrDUYDDx9+jR79OjBpk2bcsuWLYpSSwtJTEzktGnTWL9+fU6fPp0PHz5UrM3N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\nAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = plt.figure()\n",
"ax = fig.add_subplot(111)\n",
"ax.set_aspect('equal', 'box')\n",
"ax.set_xlabel('x')\n",
"ax.set_ylabel('v')\n",
"ax.quiver(XX, VV, dxdt, dvdt)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lisaks arvutame ka trajektoori. Selleks on meil algtingimust $y_0 = (x_0, v_0)$ vaja:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"y0 = [10, 0]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Järgmisena defineerime ajavahemiku $T = (t_0, t_1)$, mille vahel me süsteemi lahendame."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"t = np.linspace(0, 100, 1000)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lahenduse annab meile SciPy funktsioon odeint. Funktsiooni argumendid on dünaamika, algtingimus, ajavahemik ja konstantsed parameetrid."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"sol = odeint(pendel, y0, t, args = (h, k, m))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lõpuks joonistame trajektoori."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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2s2a+xlxGBpxxRu3qE2MGmZlaUiBZYpXAmFk9IBt4vXBfCCEAE4EepZzWI/F4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\nYERERCR2/j9DvsRzO9ko+gAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = plt.figure()\n",
"ax = fig.add_subplot(111)\n",
"ax.set_aspect('equal', 'box')\n",
"ax.set_xlabel('x')\n",
"ax.set_ylabel('v')\n",
"ax.plot(sol[:, 0], sol[:, 1])\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Küsimused"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Faasiruum ja selle dimensioon"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Sfääriline pendel\n",
"\n",
"Jäik füüsikaline pendel on fikseeritud ühes punktis ja võib vabalt liikuda vaid tingimusel, et vaba pendli otsa kaugus fikseeritud otsast on konstantne (ja võrdub pendli pikkusega). Mis on süsteemi faasiruumi mõõde?\n",
"\n",
"* \n",
"* \n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Kaheosaline jäik füüsikaline pendel\n",
"\n",
"Loengus vaatasime videosid, kus ühe füüsikalise pendli otsas oli fikseeritud teine füüsikaline pendel, ja iga pendel sai vabalt pöörelda ühes suunas ümber paigaldamispunkti:\n",
"\n",
"[Video](http://www.youtube.com/watch?v=dhZxdV2naw8)\n",
"\n",
"Milline on selle süsteemi faasiruum?\n",
"\n",
"* \n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Trajektoorid"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Palli viskamine\n",
"\n",
"Keegi viskab palli ja vaatab selle liikumist. Vabalangemise tõttu jälgib pall parabooli. Kas see parabool on dünaamilise süsteemi trajektoor?\n",
"\n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Trajektoori mõiste\n",
"\n",
"Trajektoor on kirjeldatav funktsioonina…\n",
"\n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Dünaamika"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Dünaamika kirjeldamine võrrandi abil\n",
"\n",
"Mis kirjeldab pideva dünaamilise süsteemi dünaamikat?\n",
"\n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Autonoomne ja mitteautonoomne süsteem\n",
"\n",
"Mis on erinevus autonoomse ja mitteautonoomse süsteemi vahel?\n",
"\n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Kolmandat järku diferentsiaalvõrrand\n",
"\n",
"Teile on antud diferentsiaalvõrrand $\\dddot{x} = f(x, \\dot{x}, \\ddot{x})$, mida te soovite dünaamilise süsteemi kujul kirjutada. Mis sümbol peab olema küsimärgi asemel, et vastus oleks õige?\n",
"\n",
"* $\\dot{x} = v$\n",
"* $\\dot{v} = a$\n",
"* $\\dot{a} = f(x, v, ?)$\n",
"\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Algtingimused"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Algtingimuse mõiste\n",
"\n",
"Mis on algtingimus?\n",
"\n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Seos algtingimuste ja muude mõistete vahel\n",
"\n",
"Milline väide on õige?\n",
"\n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Püsipunktid"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Püsipunktide olemasolek\n",
"\n",
"Kas igal süsteemil on vähemalt üks püsipunkt?\n",
"\n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Tuumareaktsiooni püsipunkt\n",
"\n",
"Teil on tuumareaktor, milles tekib radioaktiivne aine. Olgu $x$ selle aine kogus reaktoris. Reaktoris toimuvad järgmised protsessid, mis mõjutavad aine kogust:\n",
"\n",
"* Konstantne teke $a = 0,1 \\mathrm{g}/\\mathrm{s}$.\n",
"* Lagunemine $-kx$, kus $k$ on konstantne lagunemistegur $k = 0,01 \\mathrm{s}^{-1}$.\n",
"\n",
"Leidke süsteemi püsipunkti $x$.\n",
"\n",
"* \n",
"* \n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Vool ja faasiportree"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Dünaamika → faasiportree\n",
"\n",
"Teile on antud dünaamiline süsteem:\n",
"\n",
"* $\\dot{x} = \\cos y$\n",
"* $\\dot{y} = \\cos x$\n",
"\n",
"Milline on süsteemi faasiportree?\n",
"\n",
"| Pilt | Faasiportree |\n",
"|---|---|\n",
"| |  |\n",
"| |  |\n",
"| |  |"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Faasiportree → dünaamika\n",
"Teile on antud faasiportree:\n",
"\n",
"\n",
"\n",
"Milline on selle süsteemi dünaamika?\n",
"\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
}