{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Introdução à Teoria Qualitativa de Equações Diferenciais\n", "Neste documento vamos explorar algums aspectos básicos da análise de sistemas dinâmicos representados por equações diferenciais ordinárias. Muito do que será explorado aqui tem relação com o que se conhece como \"teoria qualitativa da equações diferenciais\". Esta área da matemática, criada por [Henri Poincaré](https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9) e [Aleksandr Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov) envolve a análise das propriedades das soluções de EDOs sem necessariamente ter que explicitá-las.\n", "\n", "Vamos aprender como definir um sistema de EDOs no SAGE e resolvê-lo analiticamente, quando possível e numericamente usando o método de Runge-Kutta Prince-Dormand." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%display typeset" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def ODEsys(t,y,params):\n", " k1,k2 = params\n", " A,B = y\n", " return[-k1*A+k2*B,\n", " k1*A-k2*B]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "Agora fazemos a Integração numérica, Runge-Kutta 8/9 Prince-Dormand. \n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "T=ode_solver()\n", "T.algorithm=\"rk8pd\"\n", "T.function=ODEsys\n", "T.ode_solve(y_0=[500,0],t_span=[0,50],params=[.3,.25],num_points=200)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vejamos agora o formato de saída da solução." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(0, \\left[500, 0\\right]\\right), \\left(0.25, \\left[464.9639136355884, 35.03608636441148\\right]\\right), \\left(0.5, \\left[434.42876087953675, 65.57123912046313\\right]\\right), \\left(0.75, \\left[407.81632637022733, 92.18367362977253\\right]\\right), \\left(1.0, \\left[384.6226755583144, 115.37732444168543\\right]\\right), \\left(1.25, \\left[364.4086121738929, 135.591387826107\\right]\\right), \\left(1.5, \\left[346.7913615813497, 153.2086384186502\\right]\\right), \\left(1.75, \\left[331.43732253744054, 168.56267746255938\\right]\\right), \\left(2.0, \\left[318.0557500994762, 181.94424990052374\\right]\\right), \\left(2.25, \\left[306.39325006281507, 193.6067499371849\\right]\\right)\\right]$$" ], "text/plain": [ "[(0, [500, 0]),\n", " (0.25, [464.9639136355884, 35.03608636441148]),\n", " (0.5, [434.42876087953675, 65.57123912046313]),\n", " (0.75, [407.81632637022733, 92.18367362977253]),\n", " (1.0, [384.6226755583144, 115.37732444168543]),\n", " (1.25, [364.4086121738929, 135.591387826107]),\n", " (1.5, [346.7913615813497, 153.2086384186502]),\n", " (1.75, [331.43732253744054, 168.56267746255938]),\n", " (2.0, [318.0557500994762, 181.94424990052374]),\n", " (2.25, [306.39325006281507, 193.6067499371849])]" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T.solution[:10]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A saída é uma lista de tuplas com o valor da variável indepentente ($t$) na primeira posição, e um vetor com o estado do sistema naquele instante." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "a=list_plot([(i[0],i[1][0]) for i in T.solution],color='red', pointsize=20, legend_label='A', alpha=.8)\n", "b=list_plot([(i[0],i[1][1]) for i in T.solution],color='blue', pointsize=20, legend_label='B', alpha=.8)\n", "a.legend()\n", "b.legend()\n", "show(a+b)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "### Solução Analítica\n", "\n", "Como o nosso sistema trata-se de um sistema de EDOs lineares, Também é possível uma solução analítica:\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[A\\left(t\\right) = \\frac{500 \\, k_{1} e^{\\left(-{\\left(k_{1} + k_{2}\\right)} t\\right)}}{k_{1} + k_{2}} + \\frac{500 \\, k_{2}}{k_{1} + k_{2}}, B\\left(t\\right) = -\\frac{500 \\, k_{1} e^{\\left(-{\\left(k_{1} + k_{2}\\right)} t\\right)}}{k_{1} + k_{2}} + \\frac{500 \\, k_{1}}{k_{1} + k_{2}}\\right]$$" ], "text/plain": [ "[A(t) == 500*k_1*e^(-(k_1 + k_2)*t)/(k_1 + k_2) + 500*k_2/(k_1 + k_2),\n", " B(t) == -500*k_1*e^(-(k_1 + k_2)*t)/(k_1 + k_2) + 500*k_1/(k_1 + k_2)]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "var('t k_1 k_2')\n", "\n", "A = function('A')(t)\n", "B = function('B')(t)\n", "de1 = diff(A,t) == -k_1*A+k_2*B\n", "de2 = diff(B,t) == k_1*A-k_2*B\n", "sol = desolve_system([de1,de2],[A,B],ics=[0,500,0], ivar=t)\n", "show(sol)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Atribuindo valores para as taxas de conversão:\n", "# k1=1/3\n", "# k2=1/4\n", "solA, solB = sol[0].rhs(), sol[1].rhs()\n", "plot((solA(k_1=1/3, k_2=1/4),solB(k_1=1/3, k_2=1/4)),(t,0,30))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Valor de A no Equilíbrio:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}214.285714285714$$" ], "text/plain": [ "214.285714285714" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k1=1/3\n", "k2=1/4\n", "n(k2*500/(k1+k2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "### Equilíbrios\n", "\n", "O equilíbrio de um sistema dinâmico se dá quando todas as suas derivadas são igual a zero, ou seja, é um estado do qual sistema não sairá a menos que perturbado.\n", "\n", "No caso acima, $dA/dt=0$ e $dB/dt=0$ ou $k_1A=k_2B$\n", "\n", "Para um modelo simples como esse, escrever o equilíbrio é trivial, mas também podemos usar o Sage para encontrar os equilíbrios:\n" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "