{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
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"outputs": [
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"data": {
"text/html": [
""
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%html\n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 1-mõõtmeline dünaamiline süsteem"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Faasiportree"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Ühemõõtmelise dünaamilise süsteemi faasiruum $Q$ võib olla $S^1$, $\\mathbb{R}$ või alamhulk $\\mathbb{R}$'ist. Siin me eeldame, et $Q = \\mathbb{R}$. Seega on faasiportree ka ühemõõtmeline, ja võib olla näiteks selline:\n",
"\n",
"\n",
"\n",
"Kuna sellest graafikust on raske näha, mis toimub, joonistatakse ühemõõtmeliste süsteemide faasiportreesid hoopis teisel kujul. Selleks aitab meelde tuletada, et sellise süsteemi dünaamikat kirjeldab diferentsiaalvõrrand $\\dot{x} = f(x)$, ja et funktsioon $f$ määrab dünaamikat. Seega on kõige lihtsam ja arusaadavam viis joonistada funktsiooni $f$, mis võib olla sellisel kujul:\n",
"\n",
"\n",
"\n",
"See graafik näitab palju paremini süsteemi käitumist. Täpsemalt näeme:\n",
"\n",
"* Kui $f(x) > 0$, siis liigub süsteem paremale.\n",
"* Kui $f(x) < 0$, siis liigub süsteem vasakule.\n",
"* Kui $f(x) = 0$, siis süsteem ei liigu - see on püsipunkt."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Püsipunktid ja nende stabiilsus"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Faasiportree on kasulik mitte ainult püsipunktide leidmiseks, vaid ka nende liigitamiseks. Seda näitab kõige lihtsamini ühemõõtmeline näide:\n",
"\n",
"\n",
"\n",
"Antud näites on kolm püsipunktide tüüpi:\n",
"\n",
"
\n",
"
Atraktor (stabiilne sõlm) ●
\n",
"
Vool siseneb punkti mõlemalt poolt.
\n",
"
Repeller (ebastabiilne sõlm) ○
\n",
"
Vool väljub punktist mõlemalt poolt.
\n",
"
Sadul ◐
\n",
"
Vool siseneb ühelt poolt ja väljub teiselt poolt.
\n",
"
\n",
"\n",
"Kui dünaamilist süsteemi kirjeldab diferentsiaalvõrrand $\\dot{x} = f(x)$, siis näitavad funktsiooni $f$ väärtused püsipunkti $x_*$ ümbruses tema omadusi:\n",
"\n",
"
\n",
"
Atraktor (stabiilne sõlm) ●
\n",
"
$f(x) > 0$ kui $x < x_*$ ja $f(x) < 0$ kui $x > x_*$
\n",
"
Repeller (ebastabiilne sõlm) ○
\n",
"
$f(x) < 0$ kui $x < x_*$ ja $f(x) > 0$ kui $x > x_*$
\n",
"
Sadul ◐
\n",
"
Funktsioonil $f(x)$ märk on sama mõlemal pool püsipunktist $x_*$
\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Lahendite kvalitatiivne käitumine"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Faasiportree on ka kasulik süsteemi lahendite kvalitatiivse käitumise leidmiseks. Selleks on kõige lihtsam joonistada trajektoore funktsioonidena x(t):\n",
"\n",
"\n",
"\n",
"Diagramm näitab, et trajektoorid **koonduvad** atraktori ümbruses ja **lahknevad** repelleri ümbruses. Sadula ümbruses nad ühelt poolt koonduvad ja teiselt poolt lahknevad.\n",
"\n",
"Graafiliselt saab ka näidata üldise ühemõõtmelise dünaamilise süsteemi lahendite omadusi:\n",
"\n",
"* Trajektoorid kunagi ei lähe läbi püsipunkti.\n",
"* Tuletise $\\dot{x}$ märk trajektooril kunagi ei muutu.\n",
"\n",
"Viimasest omadusest järeldub ka, et ühemõõtmeline dünaamiline süsteem ei saa võnkuda (ehk perioodiliselt või ebaperioodiliselt edasi ja tagasi minna). Ainus võimalus, kuidas selline süsteem saab olla perioodiline, on faasiruum $Q = S^1$ ja trajektoor käib ringi."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Interaktiivne näide: liikumine ringjoone peal"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Selles näites kasutame NumPy numbrilisteks arvutusteks ja PyPlot joonistamiseks."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"\n",
"import matplotlib\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Kõigepealt defineerime süsteemi dünaamikat. Siin eeldame et $x \\in S^1$, ehk faasiruum on $Q = S^1$. Koordinaatideks võib valida vahemikku $[0, 2\\pi)$. Antud juhul vaatame dünaamikat $\\dot{x} = f_k(x) = \\sin x + k$, kus $k \\in \\mathbb{R}$ on konstantne parameeter."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def dxdt(x, k):\n",
" return np.sin(x) + k"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Süsteemi voolu joonistamiseks valime 20 punkti ringjoone peal."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"nx = 20\n",
"X = np.linspace(2 * np.pi / nx, 2 * np.pi, nx)\n",
"PX = 0.9 * np.cos(X)\n",
"PY = 0.9 * np.sin(X)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lisaks joonistame ka ringjoone, milleks on rohkem punkte vaja."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"XX = np.linspace(0, 2 * np.pi, 201)\n",
"PPX = np.cos(XX)\n",
"PPY = np.sin(XX)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lõpuks defineerime joonistamisfunktsiooni, mis arvutab $f_k(x)$ valitud punktides ja joonistab faasiportreet."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def fp(k):\n",
" # arvutame \\dot{x}\n",
" R = np.zeros((nx))\n",
" for i in range(nx):\n",
" R[i] = dxdt(X[i], k)\n",
" U = -R * PY\n",
" V = R * PX\n",
"\n",
" # joonistus\n",
" fig = plt.figure()\n",
" ax = fig.add_subplot(111)\n",
" ax.set_aspect('equal', 'box')\n",
" ax.axis('off')\n",
"\n",
" # ringjoon\n",
" ax.plot(PPX, PPY, 'k-')\n",
"\n",
" # nooled\n",
" ax.quiver(PX, PY, U, V, facecolor='b', width=0.01)\n",
"\n",
" # püsipunktid\n",
" if np.fabs(k) < 1.0:\n",
" pp = -np.arcsin(k)\n",
" ax.plot([0.9 * np.cos(pp)], [0.9 * np.sin(pp)], marker='o', markeredgecolor='k', markerfacecolor='w', markersize=8)\n",
" ax.plot([-0.9 * np.cos(pp)], [0.9 * np.sin(pp)], marker='o', markeredgecolor='k', markerfacecolor='k', markersize=8)\n",
" elif np.fabs(k) == 1.0:\n",
" ax.plot([0.0], [-0.9 * k], marker='o', markeredgecolor='k', markerfacecolor='gray', markersize=8)\n",
"\n",
" # näita joonistust\n",
" plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Kui me valime $k = 0$, on süsteemil 2 püsipunkti. Üks neist on repeller $x = 0$, teine on atraktor $x = \\pi$."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
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3Ixs7FgOB24kFJGcBE9zRZigiGTCzTwM3AZOAndI4f2Ep8DNmZqOI6ZfnAXt5g79gMz4F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\nAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fp(0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Kui $k$ on nullist erinev, aga $|k| < 1$, muutuvad püsipunktide asukohad."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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udRxGgd+HN9PMtohI3roHfD3j+K9nUUh3RQn8d/IuQEQk6R7wG5uxVI3HaEqm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\ncxktFTpe8wAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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XIyINMWNZ4kJr18CQNdxrW9bYzOYHXgZOdPejsq2wZ4Vt4bv7k8A/gJ3qe73C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\nFxEpCQW+iEhJKPBFREpCgS8iUhL/H40WNuC3U4OLAAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fp(0.5)\n",
"fp(-0.5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Juhul kui $|k| = 1$, jääb ainult üks püsipunkt alles, mis on sadul."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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ZVpTAXwIFvoi0ATP6MD/wBwNjazzFq9CcrVWLEvigwBeR9rAJsGLF32sa1nH3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\nfBGRklDgi4iUxP8DrB7QSatci7EAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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TsipqYQob+MlFwC5mVu/HLRGR3lRe2zu3zhPEycCzwF3ZlLRwZQj8ZeluTCQi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\nYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fp(1.0)\n",
"fp(-1.0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Kui me valime $|k| > 1$, püsipunkte enam ei ole."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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1a0qrmKKM8Gv551gMeNKMs4HBFc/vBVxMLKs6LcPaRERq1ZRsLt0IH7gq/fz+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\nAI8yY4tmvpZIb9JUzV3AfN2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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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Dx6ta8X6lDHx3fxP4O/HJWY8Lav5ZgS8izbId4MD/teLNShn4yVXAFmZWz/aF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\nRha+RLQXvxfYMK9V6VWjIZ0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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fp(2.0)\n",
"fp(-2.0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Dünaamilise süsteemi kvalitatiivse käitumise sõltuvust konstantsest parameetriest käsitleme lähemalt hiljem loengus, bifurkatsiooni teemal."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Küsimused"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Stabiilsete püsipunktide asukohad\n",
"\n",
"Kus asuvad süsteemi $\\dot{x} = \\cos(x)$ stabiilsed püsipunktid?\n",
"\n",
"* \n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Atraktor ja repellor\n",
"\n",
"Kui ● on atraktor ja ○ on repeller, kumb on õigesti tähistatud faasiportree?\n",
"\n",
"| Pilt | Faasiportree |\n",
"|---|---|\n",
"| |  |\n",
"| |  |"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Repellerite arv\n",
"\n",
"Mitu repellerit on süsteemil $\\dot{x} = x³ - x$?\n",
"\n",
"* \n",
"* \n",
"* \n",
"* "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Sadula asukoht\n",
"\n",
"Kus asub süsteemi $\\dot{x} = x³ - x² - x + 1$ sadul?\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
}