Sistemas aut\303\264nomos e o retrato de fase Plotar o Campo de Dire\303\247\303\265es para um sistema de primeira ordem aut\303\264noma x' = f(x,y), y' = g(x,y) em um conjunto de N pontos (x1 , y1), ...,(xN,yN) dados, significa desenhar, para cada ponto (xJ,yJ), um vetor com origem neste ponto e com a dire\303\247\303\243o do vetor V = (x',y') = ( f(x0,y0), g(x0,y0) ). O vetor V representa a "velocidade " com que a solu\303\247\303\243o (x(t),y(t)) passa por este ponto. O Maple V tra\303\247a o Campo de Dire\303\247\303\265es de um sistema de equa\303\247\303\265es diferenciais atrav\303\251s do comando dfieldplot da biblioteca DEtools

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

Na op\303\247\303\243o "dirgrid" escolhemos uma grade de "N = m x n" pontos (o defaul \303\251 [20,20] ) e em "arrows=OP\303\207\303\203O" tamb\303\251m podemos escolher THIN (seta fina), SLIM (seta delgada), THICK (seta espessa), LINE (segmento de reta) , NONE (nada), para o desenho das linhas tangentes. Plotar o Retrato de Fase (ou Espa\303\247o de Fase) de um sistemea aut\303\264nomo x' = f(x,y) , y' = g(x,y) em um conjunto em conjunto N pontos (x1 , y1), ...,(xN,yN) dados, significa plotar simultaneamente as N solu\303\247\303\265es ((x(t),y(t)) que passam por cada ponto (xJ,yJ) no instante [Maple Math] , isto \303\251, desenhar as solu\303\247\303\265es que satisfazem a condi\303\247\303\243o inicial x(0)=xJ e y(0)=yJ Com o comando phaseportrait da biblioteca DEtools o Maple V plota o Retrato de Fase da equa\303\247\303\243o nos pontos e dom\303\255nios especificados. Este comando plota automaticamente tamb\303\251m o Campo de Dire\303\247\303\265es. A sintaxe do comando \303\251 dada por EspFase := phaseportrait( [ D(x)(t)=f(x,y), D(y)(t)=g(x,y)], [x(t), y(t)], t=a..b, [ [x(0)=a1, y(0)=b1] , ..., [x(0)=aM, y(0)=bM] ], x=c..d, y=e..f , dirgrid[m,m], arrows=OP\303\207\303\203O, title= `titulo`, color= cor, op\303\247\303\265es de dsolve/numeric, e outras op\303\247\303\265es de plot) Vejamos alguns exemplos simples

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKERFdG9vbHNGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=

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

Exemplo com algumas op\303\247\303\265es adicionais

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

Exemplo da equa\303\247\303\243o de Lotka-Volterra

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

Oscila\303\247\303\265es for\303\247adas e amortecidas Um corpo de massa M est\303\241 preso a uma mola de constante el\303\241stica K e oscila em um meio viscoso tal que a for\303\247a de atrito \303\251 proporcional \303\240 velocidade com constante L >0. Supomos que y(t) denota a coordenada do corpo no instante t. Vamos supor ainda que o corpo est\303\241 sujeito a uma for\303\247a externa peri\303\263dica F(t) = A cos ( w t). A equa\303\247\303\243o do movimento \303\251 dada por M y''+ L y'+ K y =F(t) Sabemos que a solu\303\247\303\243o desta equa\303\247\303\243o \303\251 dada por y(t) = T(t) + R cos (w t - d) na qual T(t), \303\251 solu\303\247\303\243o da equa\303\247\303\243o homog\303\252nea e [Maple Math] . O termo T(t) \303\251 chamado de transiente . A solu\303\247\303\243o y(t) tende para R cos(w t - d) que \303\251 chamada de solu\303\247\303\243o permanente . A amplitude \303\251 chamada de R e depende das constantes M, L, K, A e w . Para efeito de simplifica\303\247\303\243o vamos supor que M = K = 1, e, neste caso temos

\303\211 f\303\241cil ver que R(0) =1 Problema : a) Estudar graficamente a varia\303\247\303\243o de R(w) para os valores de L= 1.5, 1.2 , 1.0 e 0.25 e, em cada caso, determinar o valor aproximado da frequ\303\252ncia w para a qual a amplitude R \303\251 m\303\241xima. b) Obter a solu\303\247\303\243o exata do problema e, em cada caso, comparar os gr\303\241ficos da solu\303\247\303\243o exata com a da solu\303\247\303\243o aproximada Solu\303\247\303\243o (a) Escrevendo o valor de R em fun\303\247\303\243o de w e L

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

Vamos escrever os valores de L dados na forma de uma lista :

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEkTGlzRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRistRiw2LVEqJmNvbG9uZXE7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRlNGKy1JI21uR0YkNiRRJTAuMjVGJ0YvLUYsNi1RIixGJ0YvRjIvRjZGTEY3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJ0YrLUZWNiRRJDEuMEYnRi9GWUYrLUZWNiRRJDEuMkYnRi9GWUYrLUZWNiRRJDEuNUYnRi8tRiw2LVEiO0YnRi9GMkZmbkY3RjlGO0Y9Rj9GQUZULyUrZXhlY3V0YWJsZUdGNEYv

Vamos escrever uma seq\303\274Rw := seq( subs(L=Lis[i], RLw), i=1..4);encia de fun\303\247\303\265es R(w) para cada valor de L dados na lista acima

O gr\303\241fico de R(w) para os valores de L solicitados est\303\243o plotados abaixo. Clicando-se o mouse no ponto de m\303\241ximo de cada gr\303\241fico obtemos os pontos e os valores de maximos aproximados L = 0,25 L = 1,0 L = 1,2 L = 1,5 Se L = 0,25 ent\303\243o para w = 0,97 R(w) tem um valor m\303\241ximo 4,0 Se L = 1,0 ent\303\243o para w = 0,72 R(w) tem um valor m\303\241ximo de 1,2. Se L =1,2 ent\303\243o para w = 0,52 R(w) tem um valor m\303\241ximo de 1,05. Se L = 1,5 ent\303\243o R(w ) \303\251 decrescente se [Maple Math] , logo tem um m\303\241ximo em w=0 As figuras acima podem ser obtidas com

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

(b) Vamos calcular a solu\303\247\303\243o exata da equa\303\247\303\243o amortecida for\303\247ada para L = 0,25. Vamos trocar o coeficiente L=0.25, dado na forma de um float, para a forma fracion\303\241ria L= 1/4 :

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

Vamos simplificar a resposta fornecida com combine(express\303\243o,trig)

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

A primeira parcela de SolOsc2 (que apresenta a exp(-1/8 x) ) tende a zero quando x tende para infinito. A solu\303\247\303\243o particular 4 sen(x) \303\251 a solu\303\247\303\243o permanente. Vamos comparar o gr\303\241fico da solu\303\247\303\243o transiente Solosc2, com o o gr\303\241fico da solu\303\247\303\243o permanente 4 sin(x) e depois plotar a diferen\303\247a Sol2 - 4 sin(x)

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

Sol Exata = SolOsc2 Sol. Peman.= 4*sin(x) Diferen\303\247a = SolOsc2-4*sin(x)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

JSFH